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We compare closed and rigid monoidal categories. +Closedness is +defined by the tensor product having a right adjoint: the internal-hom functor. +Rigidity on the other hand generalises the concept of duals in the sense of finite- +dimensional vector spaces. A consequence of these axioms is that the internal-hom +functor is implemented by tensoring with the respective duals. This raises the +question: can one decide whether a closed monoidal category is rigid, simply by +verifying that the internal-hom is tensor-representable? At the Research School on +Bicategories, Categorification and Quantum Theory, Heunen suggested that this +is not the case. In this note, we will prove his claim by constructing an explicit +counterexample. +1. Introduction: Closed and Rigid Monoidal Categories +Monoidal categories are a ubiquitous tool in mathematics, physics, and computer +science [BS11]. +Often, they come equipped with additional structures, such as +braidings or twists, see the previously cited article. In the following, we will compare +two notions of duality for monoidal categories: closedness and rigidity. +We assume the reader’s familiarity with standard concepts of category theory; in +particular, adjunctions and monoidal categories as discussed for example in [ML98] +and [EGNO15]. As rigidity and closedness are preserved, as well as reflected, by +monoidal equivalences, see [Lin78], we restrict ourselves to the strict setting. As +such, let C be a strict monoidal category with − ⊗ −: C × C −→ C as its tensor +product and 1 ∈ C as its unit. +The category C is called (right) closed if it admits a functor [−, −]: Cop × C −→ C, +the (right) internal-hom, such that for all objects x ∈ C there exists an adjunction +(1.1) +− ⊗ x: C ⇄ C :[x, −]. +On the other hand, C is said to be (right) rigid if every object x ∈ C has a (right) +dual x∗ equipped with an evaluation and coevaluation morphism +evx : x∗ ⊗ x −→ 1 +and +coevx : 1 −→ x ⊗ x∗, +subject to the snake identities +(1.2) +idx = (idx ⊗ evx)(coevx ⊗idx) +and +idx∗ = (evx ⊗idx∗)(idx∗ ⊗ coevx). +Rigid monoidal categories are closed, see for example Section 2.10 of [EGNO15]. +Date: January 10, 2023. +2020 Mathematics Subject Classification. 18D15(primary), 18M10(secondary). +Key words and phrases. closed monoidal categories, rigid monoidal categories, autonomous +categories, Grothendieck–Verdier categories. +We would like to thank Robert Allen for fruitful discussions in the early stages of this project, +as well as Chris Heunen and Jean-Simon Lemay for their comments on a draft of this note. T.Z. is +supported by the DFG grant KR 5036/2-1. +1 +arXiv:2301.03545v1 [math.CT] 9 Jan 2023 + +DUALITY IN MONOIDAL CATEGORIES +2 +Lemma 1.1. If C is rigid, the internal-hom is implemented by the adjunction +(1.3) +− ⊗ x: C ⇄ C :− ⊗ x∗ +for all x ∈ C. +The main concern of this note is to show that the converse of the above result +does not hold. That is, we will prove that the internal-hom being given by tensoring +with the dual of an object does not imply rigidity. +In order to elucidate the underlying problem, let us assume that we are given +objects x, y ∈ C such that − ⊗ x: C ⇄ C :− ⊗ y. The unit and counit of the +adjunction provide us with natural candidates for the coevaluation and evaluation +morphisms: +coevx := η1 : 1 −→ x ⊗ y +and +evx := ε1 : y ⊗ x −→ 1. +The triangle identities of this adjunction evaluated at the monoidal unit state that +idx = εx ◦ (η1 ⊗ x) and idy = (ε1 ⊗ x) ◦ ηy. However, since we a priori do not know +whether εx ∼= idx ⊗ ε1 and ηy ∼= idy ⊗ η1, the snake identities do not necessarily +follow. +2. A counterexample +First, we define a strict monoidal category (D, ⊕, 0) in terms of generators and +relations. For details of this type of construction we refer the reader to [Kas98, +Chapter XII]. The objects of D are the natural numbers N0 with addition as the +tensor product and 0 ∈ N0 as monoidal unit.1 Its arrows are tensor products and +compositions of identities, and the generating morphisms +(2.1) +ηm,n : m −→ m ⊕ n ⊕ n, +εm,n : m ⊕ n ⊕ n −→ m, +n, m ∈ N0, n ≥ 1. +These are for all i, j, k, l, n ∈ N0 with n, k ≥ 1 subject to the relations +ηi+j+2k+l,n(idi ⊕ ηj,k ⊕ idl) = ((idi ⊕ ηj,k ⊕ idl) ⊕ id2n)ηi+j+l,n, +(2.2) +ηi+j+l,n(idi ⊕ εj,k ⊕ idl) = ((idi ⊕ εj,k ⊕ idl) ⊕ id2n)ηi+j+2k+l,n, +(2.3) +εi+j+2k+l,n((idi ⊕ ηj,k ⊕ idl) ⊕ id2n) = (idi ⊕ ηj,k ⊕ idl)εi+j+l,n, +(2.4) +εi+j+l,n((idi ⊕ εj,k ⊕ idl) ⊕ id2n) = (idi ⊕ εj,k ⊕ idl)εi+j+2k+l,n. +(2.5) +These relations are tailored to implement for any n ∈ N natural transformations +ηx,n : x −→ x ⊕ (n ⊕ n), +εx,n : x ⊕ (n ⊕ n) −→ x, +for all x ∈ D. +For example, let i, j, k, l, n be as above. Further, define x := i⊕j⊕l, y := i⊕j⊕2k⊕j, +and f := idi ⊕ ηj,k ⊕ idj : x −→ y. In this setting, Equation (2.2) translates to the +usual naturality condition, expressed by the commutativity of the following diagram: +x +y +x ⊕ (n ⊕ n) +y ⊕ (n ⊕ n) +f +ηy,n +ηx,n +f⊕(idn⊕idn) +By quotienting out the triangle identities, we obtain a category C in which tensoring +with any fixed object gives rise to a self-adjoint functor. Explained in more detail, +1A strict monoidal category whose monoid of objects is (isomorphic to) the natural numbers is +also called a PRO. + +DUALITY IN MONOIDAL CATEGORIES +3 +the monoidal category (C, ⊕, 0) has the same objects and generating morphisms as +D and the same identities hold. In addition, for any i, n ∈ N0 with n ≥ 1 we require +(2.6) +εi+n,n(ηi,n ⊕ idn) = idi+n, +and +(εi,n ⊕ idn)(ηi+n,n) = idi+n. +The next result succinctly summarises the observations made so far concerning the +internal-hom of C. +Lemma 2.1. The category C is closed monoidal; its internal-hom functor is given by +(2.7) +− ⊗ n: C ⇄ C :− ⊗ n, +for all n ∈ C. +In order to analyse the morphisms in C and show that it is not rigid monoidal, we +will rely on two tools. The first is the length of an arrow f ∈ C(n, m). It is defined as +the minimal number of generating morphisms needed to present f. The second tool +will be given by invariants for morphisms in C arising from functors into the category +vectk of finite-dimensional vector spaces over a field k. Note that for any such vector +space V there exists an isomorphism φ: V −→ V ∗ to its dual V ∗. The morphisms +coevV := (idV ⊗ φ−1) coevV : k −→ V ⊗ V, +evV := (φ ⊗ idV ) evV : V ⊗ V −→ k +satisfy the snake identities, turning V into its own dual. The next theorem is an +application of [Kas98, Proposition XII.1.4]. +Theorem 2.2. For any V ∈ vectk and isomorphism φ: V −→ V ∗ there exists a +strong monoidal functor F(V,φ) : C −→ vectk such that for all n, m ∈ N0 with n ≥ 1 +F(V,φ)(ηm,n) = idm ⊗ coevV ⊗n +and +F(V,φ)(εm,n) = idm ⊗ evV ⊗n. +To prove the statement, one has to show that relations in C are mapped to relations +in vectk. This amounts to verifying that V is its own right dual, in the rigid sense. +Corollary 2.3. The category C is skeletal. Furthermore, for any g ∈ C(m, n) the +following arrows cannot be isomorphisms: +(2.8) +(idj1 ⊗ ηl,m ⊗ idj2)g, +g(idi1 ⊗ εj,k, idi2). +Proof. Let V ∈ vectk of dimension at least 2 and fix an isomorphism φ: V −→ V ∗. +For any n, m ∈ C we have F(V,φ)(n) = V ⊗n = V ⊗m = F(V,φ)(m) if and only if n = m. +Thus, C must be skeletal. +Now suppose that g ∈ C(m, n) and consider the morphism f := g(idi1 ⊗ εj,k, idi2). +Applying F(V,φ) to f, we get F(V,φ)(f) = F(V,φ)(g)F(V,φ)(idi1 ⊗ εj,k, idi2). However, due +to the difference in the dimensions of its source and target, F(V,φ)(idi1 ⊗ εj,k, idi2) +must have a non-trivial kernel and thus f cannot be an isomorphism. +A similar argument involving the cokernel proves that (idj1 ⊗ ηl,m ⊗ idj2)g is not +invertible. +□ +We can now state and prove our main theorem. +Theorem 2.4. The category C is not rigid. +Proof. We assume that 1 ∈ C admits a right dual. Due to the uniqueness of adjoints, +there exist isomorphisms ϑ: 2n −→ 2n and θ: n −→ n such that the evaluation and +coevaluation morphisms are given by +coev1 := ϑη0,1 : 0 −→ 2, +ev1 := ε0,1(θ ⊗ idn): 2 −→ 0. +We now want to consider the following subset of homomorphisms of D: +S := +� +(id1 ⊗ ε0,1) φ (η0,1 ⊗ id1) ∈ D(1, 1) +��� φ ∈ D(3, 3) such that π(φ) is invertible +� +, + +DUALITY IN MONOIDAL CATEGORIES +4 +where π: D −→ C is the ‘projection’ functor. +By construction, the morphism +s = (id1 ⊗ ev1)(coev1 ⊗id1) corresponding to one of the two snake-identities is an +element of S. Furthermore, every element of S has length at least two.2 Thus, by +proving that S is closed under the relations arising from Equation (2.6), it follows +that π(s) ̸= id1, which concludes the proof. +To that end, let us consider an element x = (id1 ⊗ ε0,1) φ (η0,1 ⊗ id1) ∈ S. There +are two types of ‘moves’ we have to study. First, suppose we expand an identity into +one of the triangle-morphisms. This equates to either pre- or postcomposing φ with +an arrow ψ ∈ D(3, 3) which projects onto an isomorphism in C, leading to another +element in S. Second, a triangle-morphism might be contracted to an identity. A +priori, there are three ways in which this might occur +x = (id1 ⊗ ε0,1)ε1,1(η0,1 ⊗ id1), +where φ = φ′ ε1,1, or +(2.9) +x = (id1 ⊗ ε0,1)η1,1φ′′(η0,1 ⊗ id1), +with φ = η1,1 φ′′, or +(2.10) +x = (id1 ⊗ ε0,1)φ2tφ1(η0,1 ⊗ id1) +with φ = φ2tφ1 and π(t) = id. +(2.11) +Due to Corollary 2.3, neither π(φ′)π(ε1,1) nor π(η1,1)π(φ′′) are isomorphisms, contra- +dicting Cases (2.9) and (2.10). Now assume x = (id1 ⊗ ε0,1) φ2tφ1 (η0,1 ⊗ id1) and +φ = φ2tφ1. Using the functoriality of π: D −→ C, we get +π(φ) = π(φ2tφ1) = π(φ2)π(t)π(φ1) = π(φ2)π(φ1) = π(φ2φ1). +Since π(φ2φ1) is an isomorphism, (id1 ⊗ ε0,1)φ2φ1(η0,1 ⊗ id1) is an element of S. +□ +3. Tensor-Representability and Grothendieck–Verdier Categories +Although the internal-hom of a closed monoidal category C being tensor-represent- +able does not imply rigidity, C often admits additional structure. +Definition 3.1 ([BD13, Section 1.1]). A Grothendieck–Verdier category is a pair +(C, d) of a monoidal category C and an object d ∈ C, such that there exists an +antiequivalence D: C −→ Cop and for all x ∈ C the functor C(−⊗x, d) is representable +by D(x). +If d = 1 is the monoidal unit, one speaks of an r-category. +Symmetric Grothendieck–Verdier categories are also called ⋆-autonomous cate- +gories, see [Bar95]. Any rigid monoidal category is an instance of an r-category. The +converse does not hold, as shown by the counterexamples [BD13, Example 1.9] and +[BD13, Example 3.3]. +We conclude this note by showing that any monoidal category where tensoring +with an object has tensor-reprensentable left and right adjoints is an r-category. To +this end, we fix a monoidal category C such that for any x ∈ C there exist objects +L(x) and R(x) such that +− ⊗ L(x) ⊣ − ⊗ x ⊣ − ⊗ R(x). +Theorem 3.2. If C is as described above, it is an r-category. +Proof. By the parameter theorem, see for example [ML98, Theorem IV.7.3], the +object maps L, R: Ob(C) −→ Ob(C) can be promoted to functors +R: C −→ Cop +and +L: Cop −→ C. +2Note that the relations of D leave the number of generating morphisms in any presentation of a +given arrow invariant. + +DUALITY IN MONOIDAL CATEGORIES +5 +We verify that L and R are quasi-inverses of each other. By assumption, for all +y, z ∈ C we have +C(y ⊗ LR(x), z) ∼= C(y, z ⊗ R(x)) ∼= C(y ⊗ x, z). +Setting y = 1, the Yoneda embedding gives rise to a natural isomorphism LR −→ IdC. +A similar argument gives RL ∼= IdCop. +In order to show that C(− ⊗ x, 1) is representable by R(x), we have to prove that +for all y ∈ C there exists a natural isomorphism +C(y ⊗ x, 1) ∼= C(y, R(x)). +By assumption, we have C(y ⊗ x, z) ∼= C(y, z ⊗ Rx). The claim follows by setting +z = 1. +□ +References +[Bar95] Michael Barr. Nonsymmetric ∗-autonomous categories. Theor. Comput. Sci., 139(1- +2):115–130, 1995. +[BD13] Mitya Boyarchenko and Vladimir Drinfeld. +A duality formalism in the spirit of +Grothendieck and Verdier. Quantum Topol., 4(4):447–489, 2013. +[BS11] John Baez and Mike Stay. Physics, topology, logic and computation: a Rosetta Stone. +In New structures for physics, volume 813 of Lecture Notes in Phys., pages 95–172. +Springer, Heidelberg, 2011. +[EGNO15] Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, and Victor Ostrik. Tensor categories, +volume 205 of Mathematical Surveys and Monographs. American Mathematical Society, +Providence, RI, 2015. +[Kas98] Christian Kassel. Quantum groups. In Algebra and operator theory (Tashkent, 1997), +pages 213–236. Kluwer Acad. Publ., Dordrecht, 1998. +[Lin78] Harald Lindner. Adjunctions in monoidal categories. Manuscr. Math., 26:113–139, 1978. +[ML98] Saunders Mac Lane. Categories for the working mathematician, volume 5 of Graduate +Texts in Mathematics. Springer-Verlag, New York, second edition, 1998. +S.H., Philipps-Universit¨at Marburg, Arbeitsgruppe Algebraische Lie-Theorie, +Hans-Meerwein-Straße 6, 35043 Marburg +Email address: sebastian.halbig@uni-marburg.de +T.Z., Technische Universit¨at Dresden, Institut f¨ur Geometrie, Zellescher Weg +12–14, 01062 Dresden +Email address: tony.zorman@tu-dresden.de + diff --git a/-tE1T4oBgHgl3EQf8gUY/content/tmp_files/load_file.txt b/-tE1T4oBgHgl3EQf8gUY/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2ebbc9cb2044b3565fcf1f440115d66ad0b1e488 --- /dev/null +++ b/-tE1T4oBgHgl3EQf8gUY/content/tmp_files/load_file.txt @@ -0,0 +1,195 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf,len=194 +page_content='DUALITY IN MONOIDAL CATEGORIES SEBASTIAN HALBIG AND TONY ZORMAN Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' We compare closed and rigid monoidal categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' Closedness is defined by the tensor product having a right adjoint: the internal-hom functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' Rigidity on the other hand generalises the concept of duals in the sense of finite- dimensional vector spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' A consequence of these axioms is that the internal-hom functor is implemented by tensoring with the respective duals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' This raises the question: can one decide whether a closed monoidal category is rigid, simply by verifying that the internal-hom is tensor-representable?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' At the Research School on Bicategories, Categorification and Quantum Theory, Heunen suggested that this is not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' In this note, we will prove his claim by constructing an explicit counterexample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' Introduction: Closed and Rigid Monoidal Categories Monoidal categories are a ubiquitous tool in mathematics, physics, and computer science [BS11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' Often, they come equipped with additional structures, such as braidings or twists, see the previously cited article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' In the following, we will compare two notions of duality for monoidal categories: closedness and rigidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' We assume the reader’s familiarity with standard concepts of category theory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' in particular, adjunctions and monoidal categories as discussed for example in [ML98] and [EGNO15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' As rigidity and closedness are preserved, as well as reflected, by monoidal equivalences, see [Lin78], we restrict ourselves to the strict setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' As such, let C be a strict monoidal category with − ⊗ −: C × C −→ C as its tensor product and 1 ∈ C as its unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' The category C is called (right) closed if it admits a functor [−, −]: Cop × C −→ C, the (right) internal-hom, such that for all objects x ∈ C there exists an adjunction (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content='1) − ⊗ x: C ⇄ C :[x, −].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' On the other hand, C is said to be (right) rigid if every object x ∈ C has a (right) dual x∗ equipped with an evaluation and coevaluation morphism evx : x∗ ⊗ x −→ 1 and coevx : 1 −→ x ⊗ x∗, subject to the snake identities (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content='2) idx = (idx ⊗ evx)(coevx ⊗idx) and idx∗ = (evx ⊗idx∗)(idx∗ ⊗ coevx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' Rigid monoidal categories are closed, see for example Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content='10 of [EGNO15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' Date: January 10, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' 18D15(primary), 18M10(secondary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' closed monoidal categories, rigid monoidal categories, autonomous categories, Grothendieck–Verdier categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' We would like to thank Robert Allen for fruitful discussions in the early stages of this project, as well as Chris Heunen and Jean-Simon Lemay for their comments on a draft of this note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' is supported by the DFG grant KR 5036/2-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content='03545v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content='CT] 9 Jan 2023 DUALITY IN MONOIDAL CATEGORIES 2 Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' If C is rigid, the internal-hom is implemented by the adjunction (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content='3) − ⊗ x: C ⇄ C :− ⊗ x∗ for all x ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' The main concern of this note is to show that the converse of the above result does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' That is, we will prove that the internal-hom being given by tensoring with the dual of an object does not imply rigidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' In order to elucidate the underlying problem, let us assume that we are given objects x, y ∈ C such that − ⊗ x: C ⇄ C :− ⊗ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' The unit and counit of the adjunction provide us with natural candidates for the coevaluation and evaluation morphisms: coevx := η1 : 1 −→ x ⊗ y and evx := ε1 : y ⊗ x −→ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' The triangle identities of this adjunction evaluated at the monoidal unit state that idx = εx ◦ (η1 ⊗ x) and idy = (ε1 ⊗ x) ◦ ηy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' However, since we a priori do not know whether εx ∼= idx ⊗ ε1 and ηy ∼= idy ⊗ η1, the snake identities do not necessarily follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' A counterexample First, we define a strict monoidal category (D, ⊕, 0) in terms of generators and relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' For details of this type of construction we refer the reader to [Kas98, Chapter XII].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' The objects of D are the natural numbers N0 with addition as the tensor product and 0 ∈ N0 as monoidal unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content='1 Its arrows are tensor products and compositions of identities, and the generating morphisms (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content='1) ηm,n : m −→ m ⊕ n ⊕ n, εm,n : m ⊕ n ⊕ n −→ m, n, m ∈ N0, n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' These are for all i, j, k, l, n ∈ N0 with n, k ≥ 1 subject to the relations ηi+j+2k+l,n(idi ⊕ ηj,k ⊕ idl) = ((idi ⊕ ηj,k ⊕ idl) ⊕ id2n)ηi+j+l,n, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content='2) ηi+j+l,n(idi ⊕ εj,k ⊕ idl) = ((idi ⊕ εj,k ⊕ idl) ⊕ id2n)ηi+j+2k+l,n, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content='3) εi+j+2k+l,n((idi ⊕ ηj,k ⊕ idl) ⊕ id2n) = (idi ⊕ ηj,k ⊕ idl)εi+j+l,n, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content='4) εi+j+l,n((idi ⊕ εj,k ⊕ idl) ⊕ id2n) = (idi ⊕ εj,k ⊕ idl)εi+j+2k+l,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content='5) These relations are tailored to implement for any n ∈ N natural transformations ηx,n : x −→ x ⊕ (n ⊕ n), εx,n : x ⊕ (n ⊕ n) −→ x, for all x ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' For example, let i, j, k, l, n be as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' Further, define x := i⊕j⊕l, y := i⊕j⊕2k⊕j, and f := idi ⊕ ηj,k ⊕ idj : x −→ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' In this setting, Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content='2) translates to the usual naturality condition, expressed by the commutativity of the following diagram: x y x ⊕ (n ⊕ n) y ⊕ (n ⊕ n) f ηy,n ηx,n f⊕(idn⊕idn) By quotienting out the triangle identities, we obtain a category C in which tensoring with any fixed object gives rise to a self-adjoint functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' Explained in more detail, 1A strict monoidal category whose monoid of objects is (isomorphic to) the natural numbers is also called a PRO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' DUALITY IN MONOIDAL CATEGORIES 3 the monoidal category (C, ⊕, 0) has the same objects and generating morphisms as D and the same identities hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' In addition, for any i, n ∈ N0 with n ≥ 1 we require (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content='6) εi+n,n(ηi,n ⊕ idn) = idi+n, and (εi,n ⊕ idn)(ηi+n,n) = idi+n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' The next result succinctly summarises the observations made so far concerning the internal-hom of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' The category C is closed monoidal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' its internal-hom functor is given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content='7) − ⊗ n: C ⇄ C :− ⊗ n, for all n ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' In order to analyse the morphisms in C and show that it is not rigid monoidal, we will rely on two tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' The first is the length of an arrow f ∈ C(n, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' It is defined as the minimal number of generating morphisms needed to present f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' The second tool will be given by invariants for morphisms in C arising from functors into the category vectk of finite-dimensional vector spaces over a field k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' Note that for any such vector space V there exists an isomorphism φ: V −→ V ∗ to its dual V ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' The morphisms coevV := (idV ⊗ φ−1) coevV : k −→ V ⊗ V, evV := (φ ⊗ idV ) evV : V ⊗ V −→ k satisfy the snake identities, turning V into its own dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' The next theorem is an application of [Kas98, Proposition XII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' For any V ∈ vectk and isomorphism φ: V −→ V ∗ there exists a strong monoidal functor F(V,φ) : C −→ vectk such that for all n, m ∈ N0 with n ≥ 1 F(V,φ)(ηm,n) = idm ⊗ coevV ⊗n and F(V,φ)(εm,n) = idm ⊗ evV ⊗n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' To prove the statement, one has to show that relations in C are mapped to relations in vectk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' This amounts to verifying that V is its own right dual, in the rigid sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' The category C is skeletal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' Furthermore, for any g ∈ C(m, n) the following arrows cannot be isomorphisms: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content='8) (idj1 ⊗ ηl,m ⊗ idj2)g, g(idi1 ⊗ εj,k, idi2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' Let V ∈ vectk of dimension at least 2 and fix an isomorphism φ: V −→ V ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' For any n, m ∈ C we have F(V,φ)(n) = V ⊗n = V ⊗m = F(V,φ)(m) if and only if n = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' Thus, C must be skeletal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' Now suppose that g ∈ C(m, n) and consider the morphism f := g(idi1 ⊗ εj,k, idi2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' Applying F(V,φ) to f, we get F(V,φ)(f) = F(V,φ)(g)F(V,φ)(idi1 ⊗ εj,k, idi2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' However, due to the difference in the dimensions of its source and target, F(V,φ)(idi1 ⊗ εj,k, idi2) must have a non-trivial kernel and thus f cannot be an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' A similar argument involving the cokernel proves that (idj1 ⊗ ηl,m ⊗ idj2)g is not invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' □ We can now state and prove our main theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' The category C is not rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' We assume that 1 ∈ C admits a right dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' Due to the uniqueness of adjoints, there exist isomorphisms ϑ: 2n −→ 2n and θ: n −→ n such that the evaluation and coevaluation morphisms are given by coev1 := ϑη0,1 : 0 −→ 2, ev1 := ε0,1(θ ⊗ idn): 2 −→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' We now want to consider the following subset of homomorphisms of D: S := � (id1 ⊗ ε0,1) φ (η0,1 ⊗ id1) ∈ D(1, 1) ��� φ ∈ D(3, 3) such that π(φ) is invertible � , DUALITY IN MONOIDAL CATEGORIES 4 where π: D −→ C is the ‘projection’ functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' By construction, the morphism s = (id1 ⊗ ev1)(coev1 ⊗id1) corresponding to one of the two snake-identities is an element of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' Furthermore, every element of S has length at least two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content='2 Thus, by proving that S is closed under the relations arising from Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content='6), it follows that π(s) ̸= id1, which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' To that end, let us consider an element x = (id1 ⊗ ε0,1) φ (η0,1 ⊗ id1) ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' There are two types of ‘moves’ we have to study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' First, suppose we expand an identity into one of the triangle-morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' This equates to either pre- or postcomposing φ with an arrow ψ ∈ D(3, 3) which projects onto an isomorphism in C, leading to another element in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' Second, a triangle-morphism might be contracted to an identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' A priori, there are three ways in which this might occur x = (id1 ⊗ ε0,1)ε1,1(η0,1 ⊗ id1), where φ = φ′ ε1,1, or (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content='9) x = (id1 ⊗ ε0,1)η1,1φ′′(η0,1 ⊗ id1), with φ = η1,1 φ′′, or (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content='10) x = (id1 ⊗ ε0,1)φ2tφ1(η0,1 ⊗ id1) with φ = φ2tφ1 and π(t) = id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content='11) Due to Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content='3, neither π(φ′)π(ε1,1) nor π(η1,1)π(φ′′) are isomorphisms, contra- dicting Cases (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content='9) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' Now assume x = (id1 ⊗ ε0,1) φ2tφ1 (η0,1 ⊗ id1) and φ = φ2tφ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' Using the functoriality of π: D −→ C, we get π(φ) = π(φ2tφ1) = π(φ2)π(t)π(φ1) = π(φ2)π(φ1) = π(φ2φ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' Since π(φ2φ1) is an isomorphism, (id1 ⊗ ε0,1)φ2φ1(η0,1 ⊗ id1) is an element of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' Tensor-Representability and Grothendieck–Verdier Categories Although the internal-hom of a closed monoidal category C being tensor-represent- able does not imply rigidity, C often admits additional structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content='1 ([BD13, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' A Grothendieck–Verdier category is a pair (C, d) of a monoidal category C and an object d ∈ C, such that there exists an antiequivalence D: C −→ Cop and for all x ∈ C the functor C(−⊗x, d) is representable by D(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' If d = 1 is the monoidal unit, one speaks of an r-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' Symmetric Grothendieck–Verdier categories are also called ⋆-autonomous cate- gories, see [Bar95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' Any rigid monoidal category is an instance of an r-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' The converse does not hold, as shown by the counterexamples [BD13, Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content='9] and [BD13, Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' We conclude this note by showing that any monoidal category where tensoring with an object has tensor-reprensentable left and right adjoints is an r-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' To this end, we fix a monoidal category C such that for any x ∈ C there exist objects L(x) and R(x) such that − ⊗ L(x) ⊣ − ⊗ x ⊣ − ⊗ R(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' If C is as described above, it is an r-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' By the parameter theorem, see for example [ML98, Theorem IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content='3], the object maps L, R: Ob(C) −→ Ob(C) can be promoted to functors R: C −→ Cop and L: Cop −→ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' 2Note that the relations of D leave the number of generating morphisms in any presentation of a given arrow invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' DUALITY IN MONOIDAL CATEGORIES 5 We verify that L and R are quasi-inverses of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' By assumption, for all y, z ∈ C we have C(y ⊗ LR(x), z) ∼= C(y, z ⊗ R(x)) ∼= C(y ⊗ x, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' Setting y = 1, the Yoneda embedding gives rise to a natural isomorphism LR −→ IdC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' A similar argument gives RL ∼= IdCop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' In order to show that C(− ⊗ x, 1) is representable by R(x), we have to prove that for all y ∈ C there exists a natural isomorphism C(y ⊗ x, 1) ∼= C(y, R(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' By assumption, we have C(y ⊗ x, z) ∼= C(y, z ⊗ Rx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' The claim follows by setting z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' □ References [Bar95] Michael Barr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' Nonsymmetric ∗-autonomous categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=', 139(1- 2):115–130, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' [BD13] Mitya Boyarchenko and Vladimir Drinfeld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' A duality formalism in the spirit of Grothendieck and Verdier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' Quantum Topol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=', 4(4):447–489, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' [BS11] John Baez and Mike Stay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' Physics, topology, logic and computation: a Rosetta Stone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' In New structures for physics, volume 813 of Lecture Notes in Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=', pages 95–172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' Springer, Heidelberg, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' [EGNO15] Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, and Victor Ostrik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' Tensor categories, volume 205 of Mathematical Surveys and Monographs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' American Mathematical Society, Providence, RI, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' [Kas98] Christian Kassel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' Quantum groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' In Algebra and operator theory (Tashkent, 1997), pages 213–236.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' Kluwer Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=', Dordrecht, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' [Lin78] Harald Lindner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' Adjunctions in monoidal categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' Manuscr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=', 26:113–139, 1978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' [ML98] Saunders Mac Lane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' Categories for the working mathematician, volume 5 of Graduate Texts in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' Springer-Verlag, New York, second edition, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=', Philipps-Universit¨at Marburg, Arbeitsgruppe Algebraische Lie-Theorie, Hans-Meerwein-Straße 6, 35043 Marburg Email address: sebastian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content='halbig@uni-marburg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content='de T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content=', Technische Universit¨at Dresden, Institut f¨ur Geometrie, Zellescher Weg 12–14, 01062 Dresden Email address: tony.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content='zorman@tu-dresden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} +page_content='de' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'} diff --git a/.gitattributes b/.gitattributes index eeb7b4c34f8baf906cef4943ad90e16001fde272..6ceef1b2cd9cbab3f534bccad779b81384dc166a 100644 --- a/.gitattributes +++ b/.gitattributes @@ -6092,3 +6092,53 @@ rNA0T4oBgHgl3EQfKv_4/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -tex TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf filter=lfs diff=lfs merge=lfs 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xiang.li.implus, csjqian, junli, csjyang}@njust.edu.cn +Abstract +Image guidance is an effective strategy for depth super- +resolution. Generally, most existing methods employ hand- +crafted operators to decompose the high-frequency (HF) and +low-frequency (LF) ingredients from low-resolution depth +maps and guide the HF ingredients by directly concatenat- +ing them with image features. However, the hand-designed +operators usually cause inferior HF maps (e.g., distorted or +structurally missing) due to the diverse appearance of com- +plex depth maps. Moreover, the direct concatenation often re- +sults in weak guidance because not all image features have +a positive effect on the HF maps. In this paper, we de- +velop a recurrent structure attention guided (RSAG) frame- +work, consisting of two important parts. First, we introduce a +deep contrastive network with multi-scale filters for adaptive +frequency-domain separation, which adopts contrastive net- +works from large filters to small ones to calculate the pixel +contrasts for adaptive high-quality HF predictions. Second, +instead of the coarse concatenation guidance, we propose a +recurrent structure attention block, which iteratively utilizes +the latest depth estimation and the image features to jointly +select clear patterns and boundaries, aiming at providing re- +fined guidance for accurate depth recovery. In addition, we +fuse the features of HF maps to enhance the edge structures in +the decomposed LF maps. Extensive experiments show that +our approach obtains superior performance compared with +state-of-the-art depth super-resolution methods. +Introduction +Depth super-resolution (DSR) is a fundamental low-level vi- +sion topic in computer vision as it plays an important role in +a variety of applications, such as 3D reconstruction (Hou, +Dai, and Nießner 2019), autonomous driving (Caesar et al. +2020), and virtual reality (Meuleman et al. 2020). Gener- +ally, DSR is to recover a high-resolution depth map precisely +from a given low-resolution depth map. Recently, an image +guidance DSR framework becomes more and more popu- +lar since it has demonstrated remarkable progress by bor- +rowing the structures and boundaries in high-resolution im- +age to improve the depth map (Hui, Loy, and Tang 2016). +As shown in Fig. 1 (a), most efforts (Hui, Loy, and Tang +*These authors contributed equally. +†corresponding authors +Copyright © 2023, Association for the Advancement of Artificial +Intelligence (www.aaai.org). All rights reserved. +Hand-crafted HF +& LF Separation +RGB & HF +Feature Fusion +������������������������������������ +Adaptive HF & LF +Separation +Structure +Attention +(a) Image Guided Residual Framework +(b) Recurrent Structure Attention Guided Framework +HF & LF +Feature Fusion +������������������������������������ +������������������������������������ +������������������������������������ +������������������������������������ +������������������������������������ +Bicubic +Interpolation +������������������������������������ +������������������������������������ +������������������������������������ +������������������������������������ +Recurrent +Figure 1: Image guided DSR framework. (a) Popular image +guided residual DSR framework; (b) Our recurrent structure +attention guided DSR framework. +2016; Guo et al. 2019; Zuo et al. 2019b; Li et al. 2019) usu- +ally 1) use hand-crafted operators (e.g., hand-designed fil- +ters) to perform an early spectral decomposition of the low- +resolution depth map (i.e., high-frequency (HF) and low- +frequency (LF)), 2) coarsely implement the image guidance +by directly concatenating the image features into the HF +maps, and 3) run a simple up-sampling like bicubic interpo- +lation on the decomposed low-resolution LF map to a high- +resolution one. However, this framework still suffers from +three challenging problems as follows. +Firstly, the hand-designed operators often cause a weak +spectral decomposition as they are difficult to handle the di- +verse structures in the complex depth map, resulting in lost +object structures in the HF map of Fig. 2 (b). Secondly, the +direct feature concatenation results in weak image guidance +since the complex textures of the image usually produce in- +ferior features based on our observation. For example, Fig. 2 +(f-h) show clear features of the ceramic bottle (white box) +and poor features of the poster (yellow box), corresponding +to the complex and simple textures of the image in Fig. 2 (j), +respectively. Thirdly, the bicubic interpolation also results +in blurred edges in the LF map of Fig. 2 (d), because it is +unsuitable for up-sampling of all kinds of structures. +arXiv:2301.13419v1 [cs.CV] 31 Jan 2023 + +(c) HF (Ours) +(b) HF (hand-crafted) +(d) LF (hand-crafted) +(e) LF (Ours) +(f) Fea. map (DMSG) +(i) Fea. map (Ours) +(j) Color image +(a) LR depth +(h) Fea. map (Ours-direct) +(g) Fea. map (DJFR) +Figure 2: Visualizations of the decomposed HF&LF and guidance feature maps. (b) and (d) show a weak frequency-domain +separation using the hand-designed operators(Hui, Loy, and Tang 2016). (f-h) show image guidance features with redundant +textures and noise in DMSG (Hui, Loy, and Tang 2016), DJFR (Li et al. 2019) and our network using direct concatenation. +Compared with them, our method produces better HF structure in (c), sharper LF boundaries in (e), and clearer guidance +structure in (i). LR depth map and color image are plotted in (a) and (g). +To address these problems, we develop a novel recur- +rent structure attention guided (RSAG) framework for high- +quality DSR in Fig. 1 (b) through three aspects. First of all, +we introduce a deep contrastive network with multi-scale fil- +ters (DCN) to effectively decompose the HF and LF compo- +nents of the input depth, instead of the hand-designed opera- +tors. DCN is to subtly stack simple contrastive networks (Xu +et al. 2020) three times from large filters to small ones for a +coarse-to-fine HF prediction with contextual structures, and +to calculate the LF component by subtracting the HF predic- +tion from the input depth map. To better guide the depth fea- +tures, in addition, we propose a recurrent structure attention +(SA) block to select the useful image features, instead of the +direct concatenation guidance. The key step of SA is to add +absolute values of contrastive attention features of the image +and the latest depth prediction, and then calculate an atten- +tion map by employing channel and spatial attention oper- +ators. Finally, we present an HF&LF feature fusion (HLF) +block to improve the blurred edges in the LF component +by concatenating the HF feature produced by our SA block, +as its contextual structure can enhance the edges. Overall, +our RSAG framework has a significant improvement on the +hand-crafted spectral decomposition and image guidance. In +summary, our contributions are as follows: +• We introduce a deep contrastive network with multi-scale +filters (DCN) for the robust HF and LF reconstruction, +where the HF structure is implemented by stacking the +pixel-wise contrast from large to small kernels. +• We propose a novel recurrent structure attention (SA) +block by combining the latest depth prediction with the +image feature to select useful image guidance features. +• Extensive experiments on the benchmark datasets verify +the superior effectiveness of the proposed framework and +achieve state-of-the-art restoration performance. +Related work +In this section, we mainly review the previous spectral de- +composition and cross-modality fusion mechanisms used in +depth map super-resolution (DSR). +Spectral Decomposition in DSR +Since the HF component of the depth map can provide suf- +ficient structure information which coincides well with the +image boundaries, most methods adopt early spectral de- +composition for efficient DSR. A line of methods (Makarov, +Aliev, and Gerasimova 2017; Xiao et al. 2018; Li et al. 2019; +Zuo et al. 2019b; Guo et al. 2019) regard the interpolated +depth input as the LF component and add a jump connection +to transfer it to the end of the network. This global residual +learning forces the network to focus on recovering the HF +details. Another line of methods adopt the hand-designed +filters (Hui, Loy, and Tang 2016; Yang et al. 2017) or edge- +attention (Ye, Duan, and Li 2018; Chen and Jung 2018) +blocks to extract HF information. However, these methods +require additional completion operation, since the HF out- +puts always include broken edges and holes. Recently, oc- +tave convolution (Chen et al. 2019) is utilized for frequency +division operation in DSR network (He et al. 2021), which is +a plug-and-play convolutional unit. However, it separates the +frequency domain in embedding space, which does not guar- +antee that HF information is completely extracted. Instead, +we propose a simple, fast, and adaptive separation method +at the pixel level to provide reliable HF and LF maps. +Cross-modality Fusion Mechanism +Multi-path/scale Learning. Previous methods (Li et al. +2016; Lutio et al. 2019; Zhu et al. 2018; Chen and Jung +2018; Hao et al. 2019; Su et al. 2019) extract features in +color space and depth space through two independent paths + +Copy +Add +SA +Bicubic +������������������������������������ +������������������������������������������������ +������������������������ +������������������������ +DCN +������������������������������������ +������������������������������������ +SA +������������������������������������ +������������������������������������ +������������������������−������������ +������������������������ +������������������������������������ +Recurrent +������������������������������������ +Figure 3: The pipeline of our RSAG framework. It consists of a green DCN module for the adaptive frequency-domain separa- +tion, an orange recurrent SA module for the HF component recovery, and a blue module for the LF component recovery. +respectively, and transfer common structures through a joint +branch. However, the multi-path methods may cause details +missing since the cross-modality features are only fused in +one specific layer. To handle the abovementioned problem, +recent methods (Hui, Loy, and Tang 2016; Guo et al. 2019; +He et al. 2021; Zuo et al. 2019b,a; Yan et al. 2022) adopt a +multi-scale fusion strategy to merge the cross-modality fea- +tures at different levels. Although the multi-scale methods +have achieved considerable performance, the coarse aggre- +gation may cause texture copying and depth bleeding. +Recursive Learning. In order to generate higher-level +details without introducing excessive parameters, recursive +learning repeatedly applies similar modules for progres- +sive image reconstruction. Existing recursive DSR meth- +ods (Wen et al. 2019; Yang et al. 2019; Song et al. 2020) +construct the depth map in a coarse-to-fine manner by re- +garding the previous crude depth output as the input of the +DSR network. Even though the multi-supervision and resid- +ual learning avoid vanishing or exploding gradient problems +to a certain extent, there still exists the risk of falling into a +local optimum. However, we propose a recurrent guidance +for DSR, which considers the previous depth prediction as +the guidance information for the next recursion. As the re- +cursion progresses, the continuously refined guidance is a +strong constraint for better choosing the image features. +Attention Mechanism. In recent years, the attention +mechanism (Zhang et al. 2019; Guo et al. 2020; Wang et al. +2021) has achieved significant improvements in the low- +level vision field. In DSR task, Song et al. (Song et al. 2020) +utilize the channel attention to focus on HF depth. Mean- +while, Tang et al. (Tang et al. 2021) also design an HF +attention bridge to extract the useful HF information dur- +ing the depth estimation process and input it into the re- +construction network. Although these attention operations +selectively highlight the HF information, they do not es- +sentially solve the problem of texture copying and incon- +sistent boundaries in guidance images. The most related +to our method is (Zhong et al. 2021), which also aims to +find the consistent structure with an attention mechanism. +However, there are big differences between them. 1) The +proposed method uses contrastive networks to explore the +cross-modality correlation in the HF layer since the HF +modalities of the depth map and image are closer. 2) Com- +pared with single image guidance, we complement the guid- +ance with progressively refined depth prediction in a recur- +sive fashion to accurately mine the consistent structure. +Approach +In this section, we introduce our recurrent structure attention +guided (RSAG) framework for DSR. As shown in Fig. 3, +RSAG contains three modules, including a deep contrastive +network with multi-scale filters (DCN), a recurrent structure +attention module (SA), and an HF&LF feature fusion (HLF) +module. DCN adaptively learns the HF and LF decomposi- +tion by cascading contrastive networks from large filters to +small ones. Then, by introducing the last depth prediction +to complement the image guidance, recurrent SA jointly se- +lects the useful and clear structure features of the image for +accurate HF depth reconstruction. Furthermore, during the +reconstruction process, HF features guided by recurrent SA +are integrated with LF features to refine the LF edges. +Before presenting our method, we denote a high- +resolution (HR) image by Y hr ∈ RH×W , where H and +W are the sizes of the image, an HR depth map by +Dhr ∈ RH×W , a low-resolution (LR) depth map by Dlr ∈ +RpH×pW , where 0 < p ≤ 1 is the downscaling factor +(e.g., 1/4, 1/8, and 1/16). For Dhr, Dlf ∈ RH×W and +Dhf ∈ RH×W are denoted as its LF and HF components. +Deep Contrastive Network with Multi-scale Filters +As shown in Fig. 4 (a), we aim to explore a DCN network +for high-quality frequency components, instead of the hand- +designed operator for the frequency-domain decomposition. +Inspired by the contrast learning operator (Xu et al. 2020), +which is designed for RGB image decomposition, we stack + +Conv7 +������������������������������������������������ +������������������������������������ +������������������������������������ +(a) Deep Contrastive Network with Multi-scale Filters +Conv5 +Conv3 +Conv5 +Conv1 +Conv3 +LFE +Channel Attention +Space Attention +γ +c +������������������������������������ +������������������������−������������ +������������������������ +(b) Structure Attention +Conv1 +* +Conv1 +Conv3 +Conv1 +Conv3 +������������������������ +������������ +LFE +Figure 4: The architectures of (a) DCN and (b) SA. DCN +aims to decompose HF and LF maps of a depth map by +stacking three contrastive networks from large to small fil- +ters. SA tends to adaptively filter out unwanted textures and +highlight the useful HF regions of the image. +it three times to a DCN with multi-scale filters for extracting +high-quality HF components of the depth map. +Specifically, given an LR depth map Dlr ∈ RpH×pW as +input, we first upscale it to the desired resolution map Dbic ∈ +RH×W by bicubic interpolation. We denote the number of +layers of our DCN network as I, and the HF map Dhf is +defined as a recursive formulation: +Dhf = HI +I , +(1) +HI +i = Sigmoid +� +Convk(HI +i−1) − Convk−2(HI +i−1) +� +, +(2) +where HI +0 = Dbic; HI +i is the HF feature of the i-th layer +in the DCN network with I layers (1 ≤ i ≤ I); Convk(·) +represents a k × k convolutional operation followed by +PReLU (He et al. 2015) activation, k = 2(I −i)+3, and we +set I = 3 in this paper. Then, the LF map Dlf is calculated +by subtracting the HF map Dhf from Dbic: +Dlf = Dbic − Dhf. +(3) +To better understand the DCN network with different layers +(I = 1, 2, 3), Fig. 5 shows their HF features. Fig. 5 (a-c) plot +the HF features H1 +1, H2 +1 and H2 +2 of shallow DCN networks. +Compared to H1 +1 and H2 +2, the HF feature H3 +3 of deeper DCN +network are shown in Fig. 5 (f), which has the clearest and +most complete edges. According to Fig. 5 (d-f), it is worth +noticing that deeper DCN is prone to weaken depth informa- +tion and enhance structural information (e.g., edge of plaster +statue behind the teapot). +Recurrent Structure Attention +Removing textures while making full use of consistent +boundaries in the image is a key challenge for guided DSR. +Instead of trivial cross-modality feature concatenation, we +propose a novel recurrent structure attention (SA) mecha- +nism to bridge the modality gap between depth input and +image guidance. As shown in Fig. 4 (b), we put our efforts +into the following two aspects: (1) A cross-modality struc- +ture feature attention is designed, where the consistent struc- +tures are highlighted by contrast operators and the redundant +features (e.g., textures and inconsistent boundaries) are sup- +pressed in channel and space levels. (2) For better guiding +depth details restoration, useful image features are selected +with the progressively refined depth prediction recursively. +Structure Attention. Given the image Y hr ∈ RH×W +and the same size depth map Dhr ∈ RH×W as input, we +first use the learnable feature extractor to produce a set of +hierarchical image and depth features, which match with the +corresponding HF features in decoder path. Then, sharing +the same spirit as contrastive networks used in DCN, we ex- +ploit the contextual information under multiple-level recep- +tive fields and calculate the high contrastive features as HF +components. We further sum the depth and image contrast +maps and use absolute operations to enforce their consistent +structures and prevent HF smoothing caused by positive and +negative cancellations. This process can be formulated as: +J = |Fy1 − Fy2| + |Fd1 − Fd2|, +(4) +Fyi = Conv2i−1(LFE(Y hr)), +(5) +Fdi = Conv2i−1(LFE(Dhr)), i ∈ {1, 2} , +(6) +where LFE(·) denotes the learnable feature extractor for ini- +tial hierarchical features learning. Conv2i−1(·) are the con- +volutions with kernel size 2i − 1 followed by PReLU (He +et al. 2015) activation. Fyi and Fdi are extracted image fea- +tures and depth features under different receptive fields, re- +spectively. J denotes the joint HF features, which are further +fed into the channel and spatial attention blocks (Woo et al. +2018). Such a design encourages learning the interaction be- +tween different channels and focusing on the important spa- +tial locations. The features after the attention block denoted +as structure-aware features Sa, can be formulated as: +Sa = SpatA(CA(J)), +(7) +where SpatA(·) and CA(·) represent the spatial attention +and the channel attention blocks, respectively. At last, Sa +is added to the image features and combined with the depth +features. The SA process is formulated as follows: +G =SA(Dhr, Y hr) +=Cat(LFE(Dhr), Sa + γ ∗ LFE(Conv1(Y hr))), +(8) +where γ denotes a learnable parameter for controlling the +degree of highlighting and Cat(·) means concatenation of +features. Conv1(·) is a 1 × 1 convolutional kernel followed +by PReLU (He et al. 2015) activation. G represents the fused +guidance features for feeding into the decoder path. +Recurrent Mechanism with Refined Depth Guidance. +As mentioned above, compared to the single image guid- +ance, the HR depth guidance owns the same modality as the +LR depth input, which facilitates our attention module to lo- +cate and select consistent edge structures in image guidance. +More clear depth structures can achieve more accurate guid- +ance information for better details restoration, thence we re- +fine the depth guidance in a recursive manner. + +������������1 +1 +(a) +(b) +(c) +(d) +(e) +(f) +������������1 +2 +������������2 +2 +������������1 +3 +������������2 +3 +������������3 +3 +Figure 5: Visual HF features of our DCN network with dif- +ferent layers (I = 1, 2, 3). +Specifically, for the first recursion, the input depth guid- +ance is the up-sampled version of LR depth map Dbic ∈ +RH×W by bicubic interpolation, i.e., G0 = SA(Dbic, Y hr). +For the k-th recursion, the latest output of our DSR network +is taken as the input of the attention module next time. The +recurrent SA can be formulated as follows: +Gk =SA(HLF(Gk−1, Dlf, Dhf), Y hr), +(9) +where HLF(·) is the HF&LF feature fusion operation. As +shown in Fig. 6, the image feature map before being inputted +into the SA module contains complex patterns and unclear +boundaries. As the recursion progresses, complex textures +are removed (e.g., background pattern and cylinder label). +HF&LF feature fusion module +Different from previous methods directly up-sampling LF +component by bicubic interpolation, we propose an HF&LF +feature fusion (HLF) module to reconstruct the HF compo- +nent and improve the blurred LF edges. The HF reconstruc- +tion module is built upon the U-Net architecture, including +an encoder path, an attention-based guidance branch, and +a decoder branch (See orange blocks of Fig. 3). Rich hi- +erarchical features extracted from the guidance branch and +encoder-decoder structure are fused by using repeated resid- +ual convolutional block attention modules (Woo et al. 2018; +Guo et al. 2020). Then, the achieved contextual features in +the decoder branch are concatenated with the LF features at +multiple levels for the edges refining during the LF recon- +struction (See blue blocks of Fig. 3). +Loss Function +We train our model by minimizing the smooth-L1 loss be- +tween the network output Dhr of each recursion and the +ground-truth depth map Dgt. For the k-th recursion, the loss +function Lk(·) is defined as below: +Lk(Dhr +k , Dgt) = +N +� +i=1 +smoothL1(Dhr +k,i, Dgt +i ), +(10) +where smoothL1(x) = +� +0.5x2, +if |x| < 1 +|x| − 0.5, +otherwise. Dhr +k +de- +notes the network output of the k-th recursion. N and i in- +������������ = 0 +������������ = 1 +������������ = 2 +Figure 6: Visual image features calculated by the Eq. (8) +when the recursive step is varied from k = 0 to 2. +dicate the pixel number and the pixel index in the map, re- +spectively. We can obtain K depth outputs and the overall +loss is expressed as: +Ls = +K +� +k=1 +λkLk, +(11) +where λk is the weight coefficient of the k-th loss. +Experiment +Experimental Setting +To evaluate the performance of our framework, we conduct +sufficient experiments on five datasets: +• Middlebury (Hirschmuller and Scharstein 2007) & MPI +Sintel (Butler et al. 2012): Training dataset consists of 34 +RGB/D pairs from Middlebury dataset and 58 RGB/D +pairs from MPI Sintel dataset. Testing dataset includes +6 RGB/D pairs (Art, Books, Dolls, Laundry, Mobeius, +Reindeer) from Middlebury 2005. +• NYU-v2 (Silberman et al. 2012): Following the widely +used data splitting manner, we sample 1000 pairs for +training and the rest 449 pairs for testing. +• Lu (Lu, Ren, and Liu 2014): We test 6 RGB/D pairs from +this dataset with the training model on NYU-v2. +• RGB-D-D (He et al. 2021): Following FDSR (He et al. +2021), we use 405 RGB/D pairs for evaluation with the +training model on NYU-v2. +We compare our method with 3 traditional methods: TGV +(Ferstl et al. 2013), FBS (Barron and Poole 2016), SDF +(Ham, Cho, and Ponce 2017), 3 classical methods: DJF (Li +et al. 2016), DMSG (Hui, Loy, and Tang 2016), DGDIE +(Gu et al. 2017) and 12 state-of-the-art (SOTA) methods: +SVLRM (Pan et al. 2019), GSPRT (Lutio et al. 2019), DJFR +(Li et al. 2019), PacNet (Su et al. 2019), GbFT (AlBahar +and Huang 2019), CUNet (Deng and Dragotti 2020), PM- +BAN (Ye et al. 2020), DKN (Kim, Ponce, and Ham 2021), +FDKN (Kim, Ponce, and Ham 2021), FDSR (He et al. 2021), +AHMF (Zhong et al. 2021) and CTKT (Sun et al. 2021). +Mean Absolute Error (MAD) and Root Mean Squared Error +(RMSE) are used to evaluate the performance. +During training, we randomly extract patches with stride += {96, 96, 128} for the scale = {4, 8, 16} respectively as +ground truth and use bicubic interpolation to get LR in- +puts. The training and testing data are normalized to the +range [0, 1]. To balance the training time and network perfor- +mance, we set the recurrent steps of the SA blocks as k = 2 +in this paper. The loss weights are set as λk = 0.5. The + +Model +Art +Books +Dolls +Laundry +Mobeius +Reindeer +×4 +×8 +×16 +×4 +×8 +×16 +×4 +×8 +×16 +×4 +×8 +×16 +×4 +×8 +×16 +×4 +×8 +×16 +Bicbuic +1.15 +2.15 +4.04 +0.41 +0.72 +1.32 +0.44 +0.76 +1.31 +0.65 +1.17 +2.17 +0.41 +0.76 +1.37 +0.66 +1.16 +2.26 +DJF +0.40 +1.07 +2.78 +0.16 +0.45 +1.00 +0.20 +0.49 +0.99 +0.28 +0.71 +1.67 +0.18 +0.46 +1.02 +0.23 +0.60 +1.36 +DMSG +0.46 +0.76 +1.53 +0.15 +0.41 +0.76 +0.25 +0.51 +0.87 +0.30 +0.46 +1.12 +0.21 +0.43 +0.76 +0.31 +0.52 +0.99 +DGDIE +0.48 +1.20 +2.44 +0.30 +0.58 +1.02 +0.34 +0.63 +0.93 +0.35 +0.86 +1.56 +0.28 +0.58 +0.98 +0.35 +0.73 +1.29 +GSPRT +0.48 +0.74 +1.48 +0.21 +0.38 +0.76 +0.28 +0.48 +0.79 +0.33 +0.56 +1.24 +0.24 +0.49 +0.80 +0.31 +0.61 +1.07 +DJFR +0.33 +0.71 +1.72 +0.19 +0.38 +0.78 +0.25 +0.44 +0.79 +0.22 +0.50 +1.12 +0.20 +0.38 +0.76 +0.24 +0.45 +0.96 +PacNet +0.40 +0.82 +1.59 +0.22 +0.49 +0.84 +0.28 +0.53 +0.85 +0.28 +0.56 +1.08 +0.23 +0.44 +0.79 +0.29 +0.53 +1.00 +CUNet +0.47 +1.06 +2.34 +0.33 +0.63 +1.41 +0.40 +0.67 +1.27 +0.41 +0.80 +1.88 +0.29 +0.65 +1.12 +0.35 +0.69 +1.14 +PMBAN +0.28 +0.55 +1.11 +0.19 +0.30 +0.53 +0.23 +0.37 +0.64 +0.21 +0.36 +0.74 +0.18 +0.31 +0.57 +0.22 +0.39 +0.75 +DKN +0.25 +0.51 +1.22 +0.16 +0.30 +0.52 +0.21 +0.35 +0.61 +0.17 +0.34 +0.81 +0.16 +0.28 +0.54 +0.20 +0.38 +0.70 +AHMF +0.22 +0.50 +1.04 +0.14 +0.30 +0.50 +0.18 +0.35 +0.62 +0.15 +0.34 +0.73 +0.14 +0.28 +0.53 +0.18 +0.37 +0.64 +CTKT +0.25 +0.53 +1.44 +0.11 +0.26 +0.67 +0.16 +0.36 +0.65 +0.16 +0.36 +0.76 +0.13 +0.27 +0.69 +0.17 +0.35 +0.77 +RSAG +0.13 +0.23 +0.88 +0.09 +0.14 +0.50 +0.15 +0.20 +0.57 +0.10 +0.19 +0.58 +0.12 +0.17 +0.42 +0.13 +0.18 +0.52 +Table 1: Quantitative comparisons (in MAD) on Middlebury dataset. +Bicubic +TGV +DJF +FBS +DMSG +DJFR +GbFT +PacNet +FDKN +DKN +FDSR +CTKT +DCTNet +RSAG +×4 +8.16 +4.98 +3.54 +4.29 +3.02 +2.38 +3.35 +2.39 +1.86 +1.62 +1.61 +1.49 +1.59 +1.23 +×8 +14.22 +11.23 +6.20 +8.94 +2.99 +4.94 +5.73 +4.59 +3.58 +3.26 +3.18 +2.73 +3.16 +2.51 +×16 +22.32 +28.13 +10.21 +14.59 +9.17 +9.18 +9.01 +8.09 +6.96 +6.51 +5.86 +5.11 +5.84 +5.27 +Table 2: Quantitative comparisons (in RMSE (cm)) on NYU-v2 dataset. +Model +Lu +RGB-D-D +×4 +×8 +×16 +×4 +×8 +×16 +DJF +1.65 +3.96 +6.75 +3.41 +5.57 +8.15 +DJFR +1.15 +3.57 +6.77 +3.35 +5.57 +7.99 +FDKN +0.82 +2.10 +5.05 +1.18 +1.91 +3.41 +DKN +0.96 +2.16 +5.11 +1.30 +1.96 +3.42 +FDSR +0.81 +1.91 +4.64 +1.16 +1.82 +3.06 +RSAG +0.79 +1.67 +4.30 +1.14 +1.75 +2.96 +Table 3: Quantitative comparisons (in RMSE) on Lu dataset +and RGB-D-D dataset. +proposed method is implemented using PyTorch with one +RTX 2080Ti GPU. For simplicity, we name our Recurrent +Structure Attention Guided framework as RSAG. +Comparing to State-of-the-Arts +Quantitative Comparisons. +We first show the quantita- +tive evaluation results with SOTA methods under the same +conditions. Table 1 shows the results on Middlebury dataset +under three up-scaling factors. It can be observed that the +proposed RSAG outperforms the SOTA methods by signifi- +cant margins for all up-scaling factors. For example, RSAG +decreases the average MAD by 25%(×4), 48%(×8), and +30%(×16) compared to CTKT (Sun et al. 2021). We fur- +ther evaluate the proposed method on NYU-v2 dataset in +Table 2. The proposed method yields the best performance +for ×4 and ×8 DSR and comparable performance for ×16 +DSR. Compared with the second-best method, RSAG de- +creases the average RMSE by 17% for ×4 DSR. +To verify the generalization ability of our method on Lu +dataset and RGB-D-D dataset, we test RSAG for ×4, ×8, +and ×16 DSR, which is trained on NYU dataset. As shown +Model +Middlebury +NYU-v2 +baseline +0.26 +3.60 +baseline + DCN +0.24 +3.10 +baseline + DCN + HLF +0.23 +3.02 +baseline + DCN + HLF + SA +0.19 +2.51 +Table 4: Ablation studies of RSAG (in MAD) on Middlebury +dataset and (in RMSE) on NYU-v2 dataset for ×8 DSR. +in Table 3, we can see that RSAG performs the competi- +tive generalization results for all up-sampling cases, which +demonstrates the accuracy and effectiveness of our method. +Visual Comparisons. +We provide the visual comparisons +of the ×8 upsampled results on Middlebury dataset in Fig. 7. +It is worth noted that edges and luxuriant details are hard to +be reconstructed by interpolation or simple feature concate- +nation. Even though CUNet (Deng and Dragotti 2020) and +DKN (Kim, Ponce, and Ham 2021) can recover most bound- +aries, they fail to reconstruct some complex structures, such +as texture beside pencils in Art and boundaries of antlers in +Reindeer. In contrast, our results show sharper edges and +smaller errors with the ground truth. Fig. 8 shows ×8 re- +sults on NYU-v2 dataset. Boundaries and details generated +by RSAG are more accurate without introducing the texture +copying artifacts, which demonstrates that RSAG can well +recover both HF structures and LF content. +Furthermore, Fig. 9 demonstrates the good generalization +ability of the proposed method on Lu dataset for ×16 DSR. +Most methods generally tend to over-smooth the results and +fail to recover the depth details with low-light guidance im- +ages, while our method produces more convincing results. + +(g) GT +(f) Ours +(e) DKN +(d) CUNet +(c) DJF +(b) Bicubic +(a) GT and image +Figure 7: Visual comparisons of Art and Laundry on Middlebury dataset (×8 case). +(a) Image +(b) DKN +(c) FDSR +(d) Ours (e) GT +Figure 8: Visual comparisons on NYU-v2 dataset (×8 case). +(d) Ours +(b) DKN + (c) FDSR +(e) GT +(b) DKN + (c) FDSR +(d) Ours + (e) GT +(a) Images +(a) Images +Figure 9: Visual comparisons on Lu dataset (×16 case). +Ablation Study +Effect of DCN and HLF modules. Table 4 reports the abla- +tion studies on the DCN and HLF modules in our frame- +work. As shown in the first row of Table 4, the baseline +model uses a hand-designed operator for frequency-domain +decomposition and direct concatenation for cross-modality +feature fusion. The second row demonstrates that the pro- +posed DCN module, which selects HF component adap- +tively in a coarse-to-fine manner, can significantly improve +the performance over the baseline. When the HLF module is +added, the average RMSE of the NYU-v2 dataset shown in +the third row can be reduced from 3.60 to 3.02, which further +verifies the effectiveness of high-quality frequency-domain +separation and HF&LF feature fusion modules. +Effect of SA module. The last row in Table 4 demon- +strates the effectiveness of the SA module, which iteratively +utilizes the latest depth estimation to choose clear and con- +sistent image features. We can see that the SA module can +outperform them by a large margin. From the results of Ta- +2.50 +MAD +NYU-v2 +0(w/o RMA) 1 +2 + 3 +4 +RMSE +0(w/o RMA) 1 + 2 + 3 + 4 +Middlebury +Recurrent Steps +3.00 +3.50 +0.25 +0.20 +0.15 +Figure 10: Ablation studies of SA with different recursive +steps on Middlebury and NYU-v2 datasets (×8 case). +ble 4, it is observed that all the modules proposed in the +RSAG framework have made a positive contribution to the +ultimate success of our method. To further study the impact +of the recurrent steps of SA, we conduct experiments on +Middlebury and NYU-v2 datasets by varying the step from +0 (w/o SA) to 4, as illustrated in Fig. 10. It can be found that +the method achieves better performance when the recursion +steps increase, where 2 recurrent steps obtain the best trade- +off between speed and accuracy. It also proves that higher- +quality depth information can help obtain a more reliable +guidance structure for subsequent depth reconstruction. +Conclusion +In this paper, we proposed a novel recurrent structure atten- +tion guided (RSAG) framework for depth super-resolution. +In our framework, a deep contrastive network with multi- +scale filters (DCN) block was designed to adaptively de- +compose the high-quality HF and LF components by us- +ing contrastive networks from large kernels to small ones. +In addition, by leveraging the latest depth output and high- +resolution image as guidance, we introduced recurrent struc- +ture attention (SA) block, instead of the trivial feature con- +catenation, to select consistent and clear image features +for subsequent cross-modality fusion. Furthermore, we pre- +sented the HF&LF feature fusion block to refine the blurred +edges of the LF component. Extensive experiments on var- +ious benchmark datasets demonstrated the superiority and +effectiveness of the proposed framework. + +Acknowledgement +This work was supported by the National Science Fund of +China under Grant Nos. U1713208 and 62072242. +References +AlBahar, B.; and Huang, J.-B. 2019. Guided image-to-image +translation with bi-directional feature transformation. +In +ICCV, 9016–9025. +Barron, J. T.; and Poole, B. 2016. The fast bilateral solver. +In ECCV, 617–632. +Butler, D. J.; Wulff, J.; Stanley, G. B.; and Black, M. J. 2012. +A naturalistic open source movie for optical flow evaluation. +In ECCV, 611–625. +Caesar, H.; Bankiti, V.; Lang, A. H.; Vora, S.; Liong, V. E.; +Xu, Q.; Krishnan, A.; Pan, Y.; Baldan, G.; and Beijbom, O. +2020. nuscenes: A multimodal dataset for autonomous driv- +ing. In CVPR, 11621–11631. +Chen, B.; and Jung, C. 2018. +Single depth image super- +resolution using convolutional neural networks. In ICASSP, +1473–1477. +Chen, Y.; Fan, H.; Xu, B.; Yan, Z.; Kalantidis, Y.; Rohrbach, +M.; Yan, S.; and Feng, J. 2019. Drop an octave: Reducing +spatial redundancy in convolutional neural networks with +octave convolution. In ICCV, 3435–3444. +Deng, X.; and Dragotti, P. L. 2020. Deep convolutional neu- +ral network for multi-modal image restoration and fusion. +IEEE transactions on pattern analysis and machine intelli- +gence, PP(99): 1–1. +Ferstl, D.; Reinbacher, C.; Ranftl, R.; R¨uther, M.; and +Bischof, H. 2013. Image guided depth upsampling using +anisotropic total generalized variation. In ICCV, 993–1000. +Gu, S.; Zuo, W.; Guo, S.; Chen, Y.; Chen, C.; and Zhang, L. +2017. Learning dynamic guidance for depth image enhance- +ment. In CVPR, 3769–3778. +Guo, C.; Li, C.; Guo, J.; Cong, R.; Fu, H.; and Han, P. 2019. +Hierarchical Features Driven Residual Learning for Depth +Map Super-Resolution. IEEE Transactions on Image Pro- +cessing, 2545–2557. +Guo, Y.; Chen, J.; Wang, J.; Chen, Q.; Cao, J.; Deng, Z.; Xu, +Y.; and Tan, M. 2020. Closed-loop matters: Dual regression +networks for single image super-resolution. In CVPR, 5407– +5416. +Ham, B.; Cho, M.; and Ponce, J. 2017. Robust guided image +filtering using nonconvex potentials. IEEE transactions on +pattern analysis and machine intelligence, 40(1): 192–207. +Hao, X.; Lu, T.; Zhang, Y.; Wang, Z.; and Chen, H. 2019. +Multi-Source Deep Residual Fusion Network for Depth Im- +age Super-resolution. In RSVT, 62–67. +He, K.; Zhang, X.; Ren, S.; and Sun, J. 2015. Delving deep +into rectifiers: Surpassing human-level performance on ima- +genet classification. In ICCV, 1026–1034. +He, L.; Zhu, H.; Li, F.; Bai, H.; Cong, R.; Zhang, C.; Lin, +C.; Liu, M.; and Zhao, Y. 2021. Towards Fast and Accurate +Real-World Depth Super-Resolution: Benchmark Dataset +Baseline and. In CVPR, 9229–9238. +Hirschmuller, H.; and Scharstein, D. 2007. Evaluation of +cost functions for stereo matching. In CVPR, 1–8. +Hou, J.; Dai, A.; and Nießner, M. 2019. 3d-sis: 3d semantic +instance segmentation of rgb-d scans. In CVPR, 4421–4430. +Hui, T.-W.; Loy, C. C.; and Tang, X. 2016. +Depth Map +Super-Resolution by Deep Multi-Scale Guidance. In ECCV, +353–369. +Kim, B.; Ponce, J.; and Ham, B. 2021. Deformable kernel +networks for joint image filtering. International Journal of +Computer Vision, 129(2): 579–600. +Li, Y.; Huang, J.-B.; Ahuja, N.; and Yang, M.-H. 2016. Deep +joint image filtering. In ECCV, 154–169. +Li, Y.; Huang, J.-B.; Ahuja, N.; and Yang, M.-H. 2019. Joint +image filtering with deep convolutional networks. +IEEE +transactions on pattern analysis and machine intelligence, +41(8): 1909–1923. +Lu, S.; Ren, X.; and Liu, F. 2014. Depth enhancement via +low-rank matrix completion. In CVPR, 3390–3397. +Lutio, R. d.; D’aronco, S.; Wegner, J. D.; and Schindler, K. +2019. Guided super-resolution as pixel-to-pixel transforma- +tion. In ICCV, 8829–8837. +Makarov, I.; Aliev, V.; and Gerasimova, O. 2017. +Semi- +dense depth interpolation using deep convolutional neural +networks. In ACMMM, 1407–1415. +Meuleman, A.; Baek, S.-H.; Heide, F.; and Kim, M. H. 2020. +Single-shot monocular rgb-d imaging using uneven double +refraction. In CVPR, 2465–2474. +Pan, J.; Dong, J.; Ren, J. S.; Lin, L.; Tang, J.; and Yang, M.- +H. 2019. Spatially variant linear representation models for +joint filtering. In CVPR, 1702–1711. +Silberman, N.; Hoiem, D.; Kohli, P.; and Fergus, R. 2012. +Indoor segmentation and support inference from rgbd im- +ages. In ECCV, 746–760. +Song, X.; Dai, Y.; Zhou, D.; Liu, L.; Li, W.; Li, H.; and Yang, +R. 2020. Channel attention based iterative residual learning +for depth map super-resolution. In CVPR, 5631–5640. +Su, H.; Jampani, V.; Sun, D.; Gallo, O.; Learned-Miller, E.; +and Kautz, J. 2019. Pixel-adaptive convolutional neural net- +works. In CVPR, 11166–11175. +Sun, B.; Ye, X.; Li, B.; Li, H.; Wang, Z.; and Xu, R. 2021. +Learning Scene Structure Guidance via Cross-Task Knowl- +edge Transfer for Single Depth Super-Resolution. In CVPR, +7792–7801. +Tang, Q.; Cong, R.; Sheng, R.; He, L.; Zhang, D.; Zhao, Y.; +and Kwong, S. 2021. BridgeNet: A Joint Learning Network +of Depth Map Super-Resolution and Monocular Depth Esti- +mation. In ACMMM, 2148–2157. +Wang, K.; Zhang, Z.; Yan, Z.; Li, X.; Xu, B.; Li, J.; and +Yang, J. 2021. Regularizing nighttime weirdness: Efficient +self-supervised monocular depth estimation in the dark. In +ICCV, 16055–16064. +Wen, Y.; Sheng, B.; Li, P.; Lin, W.; and Feng, D. D. 2019. +Deep color guided coarse-to-fine convolutional network cas- +cade for depth image super-resolution. IEEE Transactions +on Image Processing, 28(2): 994–1006. + +Woo, S.; Park, J.; Lee, J.-Y.; and Kweon, I. S. 2018. Cbam: +Convolutional block attention module. In ECCV, 3–19. +Xiao, Y.; Cao, X.; Zhu, X.; Yang, R.; and Zheng, Y. 2018. +Joint convolutional neural pyramid for depth map super- +resolution. arXiv preprint arXiv:1801.00968. +Xu, K.; Yang, X.; Yin, B.; and Lau, R. W. 2020. Learn- +ing to restore low-light images via decomposition-and- +enhancement. In CVPR, 2281–2290. +Yan, Z.; Wang, K.; Li, X.; Zhang, Z.; Li, G.; Li, J.; and Yang, +J. 2022. Learning Complementary Correlations for Depth +Super-Resolution With Incomplete Data in Real World. +IEEE Transactions on Neural Networks and Learning Sys- +tems. +Yang, B.; Fan, X.; Zheng, Z.; Liu, X.; Zhang, K.; and Lei, +J. 2019. Depth Map Super-Resolution via Multilevel Recur- +sive Guidance and Progressive Supervision. IEEE Access, +7: 57616–57622. +Yang, J.; Lan, H.; Song, X.; and Li, K. 2017. Depth super- +resolution via fully edge-augmented guidance. In VCIP, 1– +4. +Ye, X.; Duan, X.; and Li, H. 2018. Depth super-resolution +with deep edge-inference network and edge-guided depth +filling. In ICASSP, 1398–1402. +Ye, X.; Sun, B.; Wang, Z.; Yang, J.; Xu, R.; Li, H.; and Li, B. +2020. Pmbanet: Progressive multi-branch aggregation net- +work for scene depth super-resolution. IEEE Transactions +on Image Processing, 29: 7427–7442. +Zhang, H.; Goodfellow, I.; Metaxas, D.; and Odena, A. +2019. +Self-attention generative adversarial networks. +In +ICML, 7354–7363. +Zhong, Z.; Liu, X.; Jiang, J.; Zhao, D.; Chen, Z.; and Ji, X. +2021. High-resolution Depth Maps Imaging via Attention- +based Hierarchical Multi-modal Fusion. +arXiv preprint +arXiv:2104.01530. +Zhu, J.; Zhai, W.; Cao, Y.; and Zha, Z.-J. 2018. Co-occurrent +structural edge detection for color-guided depth map super- +resolution. In MMM, 93–105. +Zuo, Y.; Fang, Y.; Yang, Y.; Shang, X.; and Wang, B. 2019a. +Residual dense network for intensity-guided depth map en- +hancement. Information Sciences, 495: 52–64. +Zuo, Y.; Wu, Q.; Fang, Y.; An, P.; Huang, L.; and Chen, +Z. 2019b. Multi-scale frequency reconstruction for guided +depth map super-resolution via deep residual network. IEEE +Transactions on Circuits and Systems for Video Technology, +30(2): 297–306. + diff --git a/1dFQT4oBgHgl3EQf1TYk/content/tmp_files/load_file.txt b/1dFQT4oBgHgl3EQf1TYk/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c471d782ca707187453c59b65009959368a61e9f --- /dev/null +++ b/1dFQT4oBgHgl3EQf1TYk/content/tmp_files/load_file.txt @@ -0,0 +1,1289 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf,len=1288 +page_content='Recurrent Structure Attention Guidance for Depth Super-Resolution Jiayi Yuan*, Haobo Jiang*, Xiang Li, Jianjun Qian†, Jun Li†, Jian Yang PCA Lab, Key Lab of Intelligent Perception and Systems for High-Dimensional Information of Ministry of Education Jiangsu Key Lab of Image and Video Understanding for Social Security School of Computer Science and Engineering, Nanjing University of Science and Technology, Nanjing, China {jiayiyuan, jiang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='hao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='bo, xiang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='implus, csjqian, junli, csjyang}@njust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='cn Abstract Image guidance is an effective strategy for depth super- resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Generally, most existing methods employ hand- crafted operators to decompose the high-frequency (HF) and low-frequency (LF) ingredients from low-resolution depth maps and guide the HF ingredients by directly concatenat- ing them with image features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' However, the hand-designed operators usually cause inferior HF maps (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=', distorted or structurally missing) due to the diverse appearance of com- plex depth maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Moreover, the direct concatenation often re- sults in weak guidance because not all image features have a positive effect on the HF maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' In this paper, we de- velop a recurrent structure attention guided (RSAG) frame- work, consisting of two important parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' First, we introduce a deep contrastive network with multi-scale filters for adaptive frequency-domain separation, which adopts contrastive net- works from large filters to small ones to calculate the pixel contrasts for adaptive high-quality HF predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Second, instead of the coarse concatenation guidance, we propose a recurrent structure attention block, which iteratively utilizes the latest depth estimation and the image features to jointly select clear patterns and boundaries, aiming at providing re- fined guidance for accurate depth recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' In addition, we fuse the features of HF maps to enhance the edge structures in the decomposed LF maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Extensive experiments show that our approach obtains superior performance compared with state-of-the-art depth super-resolution methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Introduction Depth super-resolution (DSR) is a fundamental low-level vi- sion topic in computer vision as it plays an important role in a variety of applications, such as 3D reconstruction (Hou, Dai, and Nießner 2019), autonomous driving (Caesar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2020), and virtual reality (Meuleman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Gener- ally, DSR is to recover a high-resolution depth map precisely from a given low-resolution depth map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Recently, an image guidance DSR framework becomes more and more popu- lar since it has demonstrated remarkable progress by bor- rowing the structures and boundaries in high-resolution im- age to improve the depth map (Hui, Loy, and Tang 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 1 (a), most efforts (Hui, Loy, and Tang These authors contributed equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' †corresponding authors Copyright © 2023, Association for the Advancement of Artificial Intelligence (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='aaai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' All rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='Hand-crafted HF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='& LF Separation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='RGB & HF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='Feature Fusion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='Adaptive HF & LF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='Separation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='Structure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='Attention ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='(a) Image Guided Residual Framework ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='(b) Recurrent Structure Attention Guided Framework ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='HF & LF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='Feature Fusion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='Bicubic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='Interpolation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='Recurrent ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='Figure 1: Image guided DSR framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' (a) Popular image guided residual DSR framework;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' (b) Our recurrent structure attention guided DSR framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Zuo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2019b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2019) usu- ally 1) use hand-crafted operators (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=', hand-designed fil- ters) to perform an early spectral decomposition of the low- resolution depth map (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=', high-frequency (HF) and low- frequency (LF)), 2) coarsely implement the image guidance by directly concatenating the image features into the HF maps, and 3) run a simple up-sampling like bicubic interpo- lation on the decomposed low-resolution LF map to a high- resolution one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' However, this framework still suffers from three challenging problems as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Firstly, the hand-designed operators often cause a weak spectral decomposition as they are difficult to handle the di- verse structures in the complex depth map, resulting in lost object structures in the HF map of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Secondly, the direct feature concatenation results in weak image guidance since the complex textures of the image usually produce in- ferior features based on our observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' For example, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2 (f-h) show clear features of the ceramic bottle (white box) and poor features of the poster (yellow box), corresponding to the complex and simple textures of the image in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2 (j), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Thirdly, the bicubic interpolation also results in blurred edges in the LF map of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2 (d), because it is unsuitable for up-sampling of all kinds of structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='13419v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='CV] 31 Jan 2023 (c) HF (Ours) (b) HF (hand-crafted) (d) LF (hand-crafted) (e) LF (Ours) (f) Fea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' map (DMSG) (i) Fea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' map (Ours) (j) Color image (a) LR depth (h) Fea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' map (Ours-direct) (g) Fea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' map (DJFR) Figure 2: Visualizations of the decomposed HF&LF and guidance feature maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' (b) and (d) show a weak frequency-domain separation using the hand-designed operators(Hui, Loy, and Tang 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' (f-h) show image guidance features with redundant textures and noise in DMSG (Hui, Loy, and Tang 2016), DJFR (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2019) and our network using direct concatenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Compared with them, our method produces better HF structure in (c), sharper LF boundaries in (e), and clearer guidance structure in (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' LR depth map and color image are plotted in (a) and (g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' To address these problems, we develop a novel recur- rent structure attention guided (RSAG) framework for high- quality DSR in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 1 (b) through three aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' First of all, we introduce a deep contrastive network with multi-scale fil- ters (DCN) to effectively decompose the HF and LF compo- nents of the input depth, instead of the hand-designed opera- tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' DCN is to subtly stack simple contrastive networks (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2020) three times from large filters to small ones for a coarse-to-fine HF prediction with contextual structures, and to calculate the LF component by subtracting the HF predic- tion from the input depth map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' To better guide the depth fea- tures, in addition, we propose a recurrent structure attention (SA) block to select the useful image features, instead of the direct concatenation guidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' The key step of SA is to add absolute values of contrastive attention features of the image and the latest depth prediction, and then calculate an atten- tion map by employing channel and spatial attention oper- ators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Finally, we present an HF&LF feature fusion (HLF) block to improve the blurred edges in the LF component by concatenating the HF feature produced by our SA block, as its contextual structure can enhance the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Overall, our RSAG framework has a significant improvement on the hand-crafted spectral decomposition and image guidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' In summary, our contributions are as follows: We introduce a deep contrastive network with multi-scale filters (DCN) for the robust HF and LF reconstruction, where the HF structure is implemented by stacking the pixel-wise contrast from large to small kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' We propose a novel recurrent structure attention (SA) block by combining the latest depth prediction with the image feature to select useful image guidance features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Extensive experiments on the benchmark datasets verify the superior effectiveness of the proposed framework and achieve state-of-the-art restoration performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Related work In this section, we mainly review the previous spectral de- composition and cross-modality fusion mechanisms used in depth map super-resolution (DSR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Spectral Decomposition in DSR Since the HF component of the depth map can provide suf- ficient structure information which coincides well with the image boundaries, most methods adopt early spectral de- composition for efficient DSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' A line of methods (Makarov, Aliev, and Gerasimova 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Xiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Zuo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2019b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2019) regard the interpolated depth input as the LF component and add a jump connection to transfer it to the end of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' This global residual learning forces the network to focus on recovering the HF details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Another line of methods adopt the hand-designed filters (Hui, Loy, and Tang 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2017) or edge- attention (Ye, Duan, and Li 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Chen and Jung 2018) blocks to extract HF information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' However, these methods require additional completion operation, since the HF out- puts always include broken edges and holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Recently, oc- tave convolution (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2019) is utilized for frequency division operation in DSR network (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2021), which is a plug-and-play convolutional unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' However, it separates the frequency domain in embedding space, which does not guar- antee that HF information is completely extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Instead, we propose a simple, fast, and adaptive separation method at the pixel level to provide reliable HF and LF maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Cross-modality Fusion Mechanism Multi-path/scale Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Previous methods (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Lutio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Chen and Jung 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Hao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Su et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2019) extract features in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='color space and depth space through two independent paths ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='Copy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='Add ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='SA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='Bicubic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='������������������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='DCN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='SA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='������������������������−������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='Recurrent ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='Figure 3: The pipeline of our RSAG framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' It consists of a green DCN module for the adaptive frequency-domain separa- tion, an orange recurrent SA module for the HF component recovery, and a blue module for the LF component recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' respectively, and transfer common structures through a joint branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' However, the multi-path methods may cause details missing since the cross-modality features are only fused in one specific layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' To handle the abovementioned problem, recent methods (Hui, Loy, and Tang 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Zuo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2019b,a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2022) adopt a multi-scale fusion strategy to merge the cross-modality fea- tures at different levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Although the multi-scale methods have achieved considerable performance, the coarse aggre- gation may cause texture copying and depth bleeding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Recursive Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' In order to generate higher-level details without introducing excessive parameters, recursive learning repeatedly applies similar modules for progres- sive image reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Existing recursive DSR meth- ods (Wen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2020) construct the depth map in a coarse-to-fine manner by re- garding the previous crude depth output as the input of the DSR network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Even though the multi-supervision and resid- ual learning avoid vanishing or exploding gradient problems to a certain extent, there still exists the risk of falling into a local optimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' However, we propose a recurrent guidance for DSR, which considers the previous depth prediction as the guidance information for the next recursion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' As the re- cursion progresses, the continuously refined guidance is a strong constraint for better choosing the image features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Attention Mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' In recent years, the attention mechanism (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2021) has achieved significant improvements in the low- level vision field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' In DSR task, Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' (Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2020) utilize the channel attention to focus on HF depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Mean- while, Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' (Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2021) also design an HF attention bridge to extract the useful HF information dur- ing the depth estimation process and input it into the re- construction network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Although these attention operations selectively highlight the HF information, they do not es- sentially solve the problem of texture copying and incon- sistent boundaries in guidance images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' The most related to our method is (Zhong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2021), which also aims to find the consistent structure with an attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' However, there are big differences between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 1) The proposed method uses contrastive networks to explore the cross-modality correlation in the HF layer since the HF modalities of the depth map and image are closer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2) Com- pared with single image guidance, we complement the guid- ance with progressively refined depth prediction in a recur- sive fashion to accurately mine the consistent structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Approach In this section, we introduce our recurrent structure attention guided (RSAG) framework for DSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 3, RSAG contains three modules, including a deep contrastive network with multi-scale filters (DCN), a recurrent structure attention module (SA), and an HF&LF feature fusion (HLF) module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' DCN adaptively learns the HF and LF decomposi- tion by cascading contrastive networks from large filters to small ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Then, by introducing the last depth prediction to complement the image guidance, recurrent SA jointly se- lects the useful and clear structure features of the image for accurate HF depth reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Furthermore, during the reconstruction process, HF features guided by recurrent SA are integrated with LF features to refine the LF edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Before presenting our method, we denote a high- resolution (HR) image by Y hr ∈ RH×W , where H and W are the sizes of the image, an HR depth map by Dhr ∈ RH×W , a low-resolution (LR) depth map by Dlr ∈ RpH×pW , where 0 < p ≤ 1 is the downscaling factor (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=', 1/4, 1/8, and 1/16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' For Dhr, Dlf ∈ RH×W and Dhf ∈ RH×W are denoted as its LF and HF components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Deep Contrastive Network with Multi-scale Filters As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 4 (a), we aim to explore a DCN network for high-quality frequency components, instead of the hand- designed operator for the frequency-domain decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Inspired by the contrast learning operator (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2020),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' which is designed for RGB image decomposition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' we stack ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='Conv7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='������������������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='(a) Deep Contrastive Network with Multi-scale Filters ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='Conv5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='Conv3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='Conv5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='Conv1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='Conv3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='LFE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='Channel Attention ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='Space Attention ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='������������������������−������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='(b) Structure Attention ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='Conv1 Conv1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='Conv3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='Conv1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='Conv3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='LFE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='Figure 4: The architectures of (a) DCN and (b) SA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' DCN aims to decompose HF and LF maps of a depth map by stacking three contrastive networks from large to small fil- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' SA tends to adaptively filter out unwanted textures and highlight the useful HF regions of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' it three times to a DCN with multi-scale filters for extracting high-quality HF components of the depth map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Specifically, given an LR depth map Dlr ∈ RpH×pW as input, we first upscale it to the desired resolution map Dbic ∈ RH×W by bicubic interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' We denote the number of layers of our DCN network as I, and the HF map Dhf is defined as a recursive formulation: Dhf = HI I , (1) HI i = Sigmoid � Convk(HI i−1) − Convk−2(HI i−1) � , (2) where HI 0 = Dbic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' HI i is the HF feature of the i-th layer in the DCN network with I layers (1 ≤ i ≤ I);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Convk(·) represents a k × k convolutional operation followed by PReLU (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2015) activation, k = 2(I −i)+3, and we set I = 3 in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Then, the LF map Dlf is calculated by subtracting the HF map Dhf from Dbic: Dlf = Dbic − Dhf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' (3) To better understand the DCN network with different layers (I = 1, 2, 3), Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 5 shows their HF features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 5 (a-c) plot the HF features H1 1, H2 1 and H2 2 of shallow DCN networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Compared to H1 1 and H2 2, the HF feature H3 3 of deeper DCN network are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 5 (f), which has the clearest and most complete edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' According to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 5 (d-f), it is worth noticing that deeper DCN is prone to weaken depth informa- tion and enhance structural information (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=', edge of plaster statue behind the teapot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Recurrent Structure Attention Removing textures while making full use of consistent boundaries in the image is a key challenge for guided DSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Instead of trivial cross-modality feature concatenation, we propose a novel recurrent structure attention (SA) mecha- nism to bridge the modality gap between depth input and image guidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 4 (b), we put our efforts into the following two aspects: (1) A cross-modality struc- ture feature attention is designed, where the consistent struc- tures are highlighted by contrast operators and the redundant features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=', textures and inconsistent boundaries) are sup- pressed in channel and space levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' (2) For better guiding depth details restoration, useful image features are selected with the progressively refined depth prediction recursively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Structure Attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Given the image Y hr ∈ RH×W and the same size depth map Dhr ∈ RH×W as input, we first use the learnable feature extractor to produce a set of hierarchical image and depth features, which match with the corresponding HF features in decoder path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Then, sharing the same spirit as contrastive networks used in DCN, we ex- ploit the contextual information under multiple-level recep- tive fields and calculate the high contrastive features as HF components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' We further sum the depth and image contrast maps and use absolute operations to enforce their consistent structures and prevent HF smoothing caused by positive and negative cancellations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' This process can be formulated as: J = |Fy1 − Fy2| + |Fd1 − Fd2|, (4) Fyi = Conv2i−1(LFE(Y hr)), (5) Fdi = Conv2i−1(LFE(Dhr)), i ∈ {1, 2} , (6) where LFE(·) denotes the learnable feature extractor for ini- tial hierarchical features learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Conv2i−1(·) are the con- volutions with kernel size 2i − 1 followed by PReLU (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2015) activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Fyi and Fdi are extracted image fea- tures and depth features under different receptive fields, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' J denotes the joint HF features, which are further fed into the channel and spatial attention blocks (Woo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Such a design encourages learning the interaction be- tween different channels and focusing on the important spa- tial locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' The features after the attention block denoted as structure-aware features Sa, can be formulated as: Sa = SpatA(CA(J)), (7) where SpatA(·) and CA(·) represent the spatial attention and the channel attention blocks, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' At last, Sa is added to the image features and combined with the depth features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' The SA process is formulated as follows: G =SA(Dhr, Y hr) =Cat(LFE(Dhr), Sa + γ ∗ LFE(Conv1(Y hr))), (8) where γ denotes a learnable parameter for controlling the degree of highlighting and Cat(·) means concatenation of features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Conv1(·) is a 1 × 1 convolutional kernel followed by PReLU (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2015) activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' G represents the fused guidance features for feeding into the decoder path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Recurrent Mechanism with Refined Depth Guidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' As mentioned above, compared to the single image guid- ance, the HR depth guidance owns the same modality as the LR depth input, which facilitates our attention module to lo- cate and select consistent edge structures in image guidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' More clear depth structures can achieve more accurate guid- ance information for better details restoration, thence we re- fine the depth guidance in a recursive manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' ������������1 1 (a) (b) (c) (d) (e) (f) ������������1 2 ������������2 2 ������������1 3 ������������2 3 ������������3 3 Figure 5: Visual HF features of our DCN network with dif- ferent layers (I = 1, 2, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Specifically, for the first recursion, the input depth guid- ance is the up-sampled version of LR depth map Dbic ∈ RH×W by bicubic interpolation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=', G0 = SA(Dbic, Y hr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' For the k-th recursion, the latest output of our DSR network is taken as the input of the attention module next time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' The recurrent SA can be formulated as follows: Gk =SA(HLF(Gk−1, Dlf, Dhf), Y hr), (9) where HLF(·) is the HF&LF feature fusion operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 6, the image feature map before being inputted into the SA module contains complex patterns and unclear boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' As the recursion progresses, complex textures are removed (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=', background pattern and cylinder label).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' HF&LF feature fusion module Different from previous methods directly up-sampling LF component by bicubic interpolation, we propose an HF&LF feature fusion (HLF) module to reconstruct the HF compo- nent and improve the blurred LF edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' The HF reconstruc- tion module is built upon the U-Net architecture, including an encoder path, an attention-based guidance branch, and a decoder branch (See orange blocks of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Rich hi- erarchical features extracted from the guidance branch and encoder-decoder structure are fused by using repeated resid- ual convolutional block attention modules (Woo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Then, the achieved contextual features in the decoder branch are concatenated with the LF features at multiple levels for the edges refining during the LF recon- struction (See blue blocks of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Loss Function We train our model by minimizing the smooth-L1 loss be- tween the network output Dhr of each recursion and the ground-truth depth map Dgt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' For the k-th recursion, the loss function Lk(·) is defined as below: Lk(Dhr k , Dgt) = N � i=1 smoothL1(Dhr k,i, Dgt i ), (10) where smoothL1(x) = � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='5x2, if |x| < 1 |x| − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='5, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Dhr k de- notes the network output of the k-th recursion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' N and i in- ������������ = 0 ������������ = 1 ������������ = 2 Figure 6: Visual image features calculated by the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' (8) when the recursive step is varied from k = 0 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' dicate the pixel number and the pixel index in the map, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' We can obtain K depth outputs and the overall loss is expressed as: Ls = K � k=1 λkLk, (11) where λk is the weight coefficient of the k-th loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Experiment Experimental Setting To evaluate the performance of our framework, we conduct sufficient experiments on five datasets: Middlebury (Hirschmuller and Scharstein 2007) & MPI Sintel (Butler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2012): Training dataset consists of 34 RGB/D pairs from Middlebury dataset and 58 RGB/D pairs from MPI Sintel dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Testing dataset includes 6 RGB/D pairs (Art, Books, Dolls, Laundry, Mobeius, Reindeer) from Middlebury 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' NYU-v2 (Silberman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2012): Following the widely used data splitting manner, we sample 1000 pairs for training and the rest 449 pairs for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Lu (Lu, Ren, and Liu 2014): We test 6 RGB/D pairs from this dataset with the training model on NYU-v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' RGB-D-D (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2021): Following FDSR (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2021), we use 405 RGB/D pairs for evaluation with the training model on NYU-v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' We compare our method with 3 traditional methods: TGV (Ferstl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2013), FBS (Barron and Poole 2016), SDF (Ham, Cho, and Ponce 2017), 3 classical methods: DJF (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2016), DMSG (Hui, Loy, and Tang 2016), DGDIE (Gu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2017) and 12 state-of-the-art (SOTA) methods: SVLRM (Pan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2019), GSPRT (Lutio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2019), DJFR (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2019), PacNet (Su et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2019), GbFT (AlBahar and Huang 2019), CUNet (Deng and Dragotti 2020), PM- BAN (Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2020), DKN (Kim, Ponce, and Ham 2021), FDKN (Kim, Ponce, and Ham 2021), FDSR (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2021), AHMF (Zhong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2021) and CTKT (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Mean Absolute Error (MAD) and Root Mean Squared Error (RMSE) are used to evaluate the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' During training, we randomly extract patches with stride = {96, 96, 128} for the scale = {4, 8, 16} respectively as ground truth and use bicubic interpolation to get LR in- puts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' The training and testing data are normalized to the range [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' To balance the training time and network perfor- mance, we set the recurrent steps of the SA blocks as k = 2 in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' The loss weights are set as λk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' The Model Art Books Dolls Laundry Mobeius Reindeer ×4 ×8 ×16 ×4 ×8 ×16 ×4 ×8 ×16 ×4 ×8 ×16 ×4 ×8 ×16 ×4 ×8 ×16 Bicbuic 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='72 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='44 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='51 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='86 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='11 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='84 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='27 Table 2: Quantitative comparisons (in RMSE (cm)) on NYU-v2 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Model Lu RGB-D-D ×4 ×8 ×16 ×4 ×8 ×16 DJF 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='65 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='96 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='75 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='41 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='57 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='15 DJFR 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='57 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='77 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='35 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='57 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='99 FDKN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='82 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='10 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='91 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='41 DKN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='96 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='16 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='30 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='96 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='42 FDSR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='81 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='91 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='64 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='82 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='06 RSAG 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='79 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='67 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='30 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='96 Table 3: Quantitative comparisons (in RMSE) on Lu dataset and RGB-D-D dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' proposed method is implemented using PyTorch with one RTX 2080Ti GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' For simplicity, we name our Recurrent Structure Attention Guided framework as RSAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Comparing to State-of-the-Arts Quantitative Comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' We first show the quantita- tive evaluation results with SOTA methods under the same conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Table 1 shows the results on Middlebury dataset under three up-scaling factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' It can be observed that the proposed RSAG outperforms the SOTA methods by signifi- cant margins for all up-scaling factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' For example, RSAG decreases the average MAD by 25%(×4), 48%(×8), and 30%(×16) compared to CTKT (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' We fur- ther evaluate the proposed method on NYU-v2 dataset in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' The proposed method yields the best performance for ×4 and ×8 DSR and comparable performance for ×16 DSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Compared with the second-best method, RSAG de- creases the average RMSE by 17% for ×4 DSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' To verify the generalization ability of our method on Lu dataset and RGB-D-D dataset, we test RSAG for ×4, ×8, and ×16 DSR, which is trained on NYU dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' As shown Model Middlebury NYU-v2 baseline 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='26 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='60 baseline + DCN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='24 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='10 baseline + DCN + HLF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='23 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='02 baseline + DCN + HLF + SA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='19 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='51 Table 4: Ablation studies of RSAG (in MAD) on Middlebury dataset and (in RMSE) on NYU-v2 dataset for ×8 DSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' in Table 3, we can see that RSAG performs the competi- tive generalization results for all up-sampling cases, which demonstrates the accuracy and effectiveness of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Visual Comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' We provide the visual comparisons of the ×8 upsampled results on Middlebury dataset in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' It is worth noted that edges and luxuriant details are hard to be reconstructed by interpolation or simple feature concate- nation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Even though CUNet (Deng and Dragotti 2020) and DKN (Kim, Ponce, and Ham 2021) can recover most bound- aries, they fail to reconstruct some complex structures, such as texture beside pencils in Art and boundaries of antlers in Reindeer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' In contrast, our results show sharper edges and smaller errors with the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 8 shows ×8 re- sults on NYU-v2 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Boundaries and details generated by RSAG are more accurate without introducing the texture copying artifacts, which demonstrates that RSAG can well recover both HF structures and LF content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Furthermore, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 9 demonstrates the good generalization ability of the proposed method on Lu dataset for ×16 DSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Most methods generally tend to over-smooth the results and fail to recover the depth details with low-light guidance im- ages, while our method produces more convincing results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' (g) GT (f) Ours (e) DKN (d) CUNet (c) DJF (b) Bicubic (a) GT and image Figure 7: Visual comparisons of Art and Laundry on Middlebury dataset (×8 case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' (a) Image (b) DKN (c) FDSR (d) Ours (e) GT Figure 8: Visual comparisons on NYU-v2 dataset (×8 case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' (d) Ours (b) DKN (c) FDSR (e) GT (b) DKN (c) FDSR (d) Ours (e) GT (a) Images (a) Images Figure 9: Visual comparisons on Lu dataset (×16 case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Ablation Study Effect of DCN and HLF modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Table 4 reports the abla- tion studies on the DCN and HLF modules in our frame- work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' As shown in the first row of Table 4, the baseline model uses a hand-designed operator for frequency-domain decomposition and direct concatenation for cross-modality feature fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' The second row demonstrates that the pro- posed DCN module, which selects HF component adap- tively in a coarse-to-fine manner, can significantly improve the performance over the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' When the HLF module is added, the average RMSE of the NYU-v2 dataset shown in the third row can be reduced from 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='60 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='02, which further verifies the effectiveness of high-quality frequency-domain separation and HF&LF feature fusion modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Effect of SA module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' The last row in Table 4 demon- strates the effectiveness of the SA module, which iteratively utilizes the latest depth estimation to choose clear and con- sistent image features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' We can see that the SA module can outperform them by a large margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' From the results of Ta- 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='50 MAD NYU-v2 0(w/o RMA) 1 2 3 4 RMSE 0(w/o RMA) 1 2 3 4 Middlebury Recurrent Steps 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='15 Figure 10: Ablation studies of SA with different recursive steps on Middlebury and NYU-v2 datasets (×8 case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' ble 4, it is observed that all the modules proposed in the RSAG framework have made a positive contribution to the ultimate success of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' To further study the impact of the recurrent steps of SA, we conduct experiments on Middlebury and NYU-v2 datasets by varying the step from 0 (w/o SA) to 4, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' It can be found that the method achieves better performance when the recursion steps increase, where 2 recurrent steps obtain the best trade- off between speed and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' It also proves that higher- quality depth information can help obtain a more reliable guidance structure for subsequent depth reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Conclusion In this paper, we proposed a novel recurrent structure atten- tion guided (RSAG) framework for depth super-resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' In our framework, a deep contrastive network with multi- scale filters (DCN) block was designed to adaptively de- compose the high-quality HF and LF components by us- ing contrastive networks from large kernels to small ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' In addition, by leveraging the latest depth output and high- resolution image as guidance, we introduced recurrent struc- ture attention (SA) block, instead of the trivial feature con- catenation, to select consistent and clear image features for subsequent cross-modality fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Furthermore, we pre- sented the HF&LF feature fusion block to refine the blurred edges of the LF component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Extensive experiments on var- ious benchmark datasets demonstrated the superiority and effectiveness of the proposed framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Acknowledgement This work was supported by the National Science Fund of China under Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' U1713208 and 62072242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' References AlBahar, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' and Huang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Guided image-to-image translation with bi-directional feature transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' In ICCV, 9016–9025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Barron, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' and Poole, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' The fast bilateral solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' In ECCV, 617–632.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Butler, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Wulff, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Stanley, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' and Black, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' A naturalistic open source movie for optical flow evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' In ECCV, 611–625.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Caesar, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Bankiti, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Lang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Vora, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Liong, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Xu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Krishnan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Pan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Baldan, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' and Beijbom, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' nuscenes: A multimodal dataset for autonomous driv- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' In CVPR, 11621–11631.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Chen, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' and Jung, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Single depth image super- resolution using convolutional neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' In ICASSP, 1473–1477.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Fan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Xu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Yan, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Kalantidis, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Rohrbach, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Yan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' and Feng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Drop an octave: Reducing spatial redundancy in convolutional neural networks with octave convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' In ICCV, 3435–3444.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Deng, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' and Dragotti, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Deep convolutional neu- ral network for multi-modal image restoration and fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' IEEE transactions on pattern analysis and machine intelli- gence, PP(99): 1–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Ferstl, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Reinbacher, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Ranftl, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' R¨uther, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' and Bischof, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Image guided depth upsampling using anisotropic total generalized variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' In ICCV, 993–1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Gu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Zuo, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Guo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Chen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' and Zhang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Learning dynamic guidance for depth image enhance- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' In CVPR, 3769–3778.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Guo, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Li, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Guo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Cong, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Fu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' and Han, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Hierarchical Features Driven Residual Learning for Depth Map Super-Resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' IEEE Transactions on Image Pro- cessing, 2545–2557.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Guo, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Chen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Chen, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Cao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Deng, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Xu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' and Tan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Closed-loop matters: Dual regression networks for single image super-resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' In CVPR, 5407– 5416.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Ham, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Cho, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' and Ponce, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Robust guided image filtering using nonconvex potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' IEEE transactions on pattern analysis and machine intelligence, 40(1): 192–207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Hao, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Lu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Wang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' and Chen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Multi-Source Deep Residual Fusion Network for Depth Im- age Super-resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' In RSVT, 62–67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' He, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Ren, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' and Sun, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Delving deep into rectifiers: Surpassing human-level performance on ima- genet classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' In ICCV, 1026–1034.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' He, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Zhu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Li, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Bai, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Cong, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Zhang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Lin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Liu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' and Zhao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Towards Fast and Accurate Real-World Depth Super-Resolution: Benchmark Dataset Baseline and.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' In CVPR, 9229–9238.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Hirschmuller, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' and Scharstein, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Evaluation of cost functions for stereo matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' In CVPR, 1–8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Hou, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Dai, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' and Nießner, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 3d-sis: 3d semantic instance segmentation of rgb-d scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' In CVPR, 4421–4430.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Hui, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Loy, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' and Tang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Depth Map Super-Resolution by Deep Multi-Scale Guidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' In ECCV, 353–369.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Kim, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Ponce, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' and Ham, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Deformable kernel networks for joint image filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' International Journal of Computer Vision, 129(2): 579–600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Huang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Ahuja, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' and Yang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Deep joint image filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' In ECCV, 154–169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Huang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Ahuja, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' and Yang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Joint image filtering with deep convolutional networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' IEEE transactions on pattern analysis and machine intelligence, 41(8): 1909–1923.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Lu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Ren, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' and Liu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Depth enhancement via low-rank matrix completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' In CVPR, 3390–3397.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Lutio, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' D’aronco, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Wegner, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' and Schindler, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Guided super-resolution as pixel-to-pixel transforma- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' In ICCV, 8829–8837.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Makarov, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Aliev, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' and Gerasimova, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Semi- dense depth interpolation using deep convolutional neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' In ACMMM, 1407–1415.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Meuleman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Baek, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Heide, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' and Kim, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Single-shot monocular rgb-d imaging using uneven double refraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' In CVPR, 2465–2474.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Pan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Dong, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Ren, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Lin, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Tang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' and Yang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='- H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Spatially variant linear representation models for joint filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' In CVPR, 1702–1711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Silberman, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Hoiem, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Kohli, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' and Fergus, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Indoor segmentation and support inference from rgbd im- ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' In ECCV, 746–760.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Song, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Dai, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Zhou, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Liu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Li, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' and Yang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Channel attention based iterative residual learning for depth map super-resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' In CVPR, 5631–5640.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Su, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Jampani, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Sun, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Gallo, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Learned-Miller, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' and Kautz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Pixel-adaptive convolutional neural net- works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' In CVPR, 11166–11175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Sun, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Ye, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Li, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Wang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' and Xu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Learning Scene Structure Guidance via Cross-Task Knowl- edge Transfer for Single Depth Super-Resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' In CVPR, 7792–7801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Tang, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Cong, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Sheng, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' He, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Zhang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Zhao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' and Kwong, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' BridgeNet: A Joint Learning Network of Depth Map Super-Resolution and Monocular Depth Esti- mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' In ACMMM, 2148–2157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Wang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Yan, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Xu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' and Yang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Regularizing nighttime weirdness: Efficient self-supervised monocular depth estimation in the dark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' In ICCV, 16055–16064.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Wen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Sheng, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Li, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Lin, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' and Feng, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Deep color guided coarse-to-fine convolutional network cas- cade for depth image super-resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' IEEE Transactions on Image Processing, 28(2): 994–1006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Woo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Park, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Lee, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' and Kweon, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Cbam: Convolutional block attention module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' In ECCV, 3–19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Xiao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Cao, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Zhu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Yang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' and Zheng, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Joint convolutional neural pyramid for depth map super- resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' arXiv preprint arXiv:1801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='00968.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Xu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Yang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Yin, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' and Lau, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Learn- ing to restore low-light images via decomposition-and- enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' In CVPR, 2281–2290.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Yan, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Wang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Li, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' and Yang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Learning Complementary Correlations for Depth Super-Resolution With Incomplete Data in Real World.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' IEEE Transactions on Neural Networks and Learning Sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Yang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Fan, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Zheng, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Liu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Zhang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' and Lei, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Depth Map Super-Resolution via Multilevel Recur- sive Guidance and Progressive Supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' IEEE Access, 7: 57616–57622.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Yang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Lan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Song, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' and Li, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Depth super- resolution via fully edge-augmented guidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' In VCIP, 1– 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Ye, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Duan, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' and Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Depth super-resolution with deep edge-inference network and edge-guided depth filling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' In ICASSP, 1398–1402.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Ye, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Sun, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Wang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Yang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Xu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' and Li, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Pmbanet: Progressive multi-branch aggregation net- work for scene depth super-resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' IEEE Transactions on Image Processing, 29: 7427–7442.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Goodfellow, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Metaxas, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' and Odena, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Self-attention generative adversarial networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' In ICML, 7354–7363.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Zhong, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Liu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Jiang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Zhao, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Chen, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' and Ji, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' High-resolution Depth Maps Imaging via Attention- based Hierarchical Multi-modal Fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' arXiv preprint arXiv:2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='01530.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Zhu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Zhai, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Cao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' and Zha, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Co-occurrent structural edge detection for color-guided depth map super- resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' In MMM, 93–105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Zuo, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Fang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Yang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Shang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' and Wang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2019a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Residual dense network for intensity-guided depth map en- hancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Information Sciences, 495: 52–64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Zuo, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Wu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Fang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' An, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Huang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' and Chen, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' 2019b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' Multi-scale frequency reconstruction for guided depth map super-resolution via deep residual network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} +page_content=' IEEE Transactions on Circuits and Systems for Video Technology, 30(2): 297–306.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQf1TYk/content/2301.13419v1.pdf'} diff --git a/2dFKT4oBgHgl3EQfPy1W/content/tmp_files/2301.11764v1.pdf.txt b/2dFKT4oBgHgl3EQfPy1W/content/tmp_files/2301.11764v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a16467b857393f43cbaa487bef8a6a4b4bff20a7 --- /dev/null +++ b/2dFKT4oBgHgl3EQfPy1W/content/tmp_files/2301.11764v1.pdf.txt @@ -0,0 +1,1955 @@ +3D GENOME RECONSTRUCTION FROM PARTIALLY +PHASED HI-C DATA +DIEGO CIFUENTES, JAN DRAISMA, OSKAR HENRIKSSON, +ANNACHIARA KORCHMAROS, AND KAIE KUBJAS +Abstract. The 3-dimensional (3D) structure of the genome is of significant importance for +many cellular processes. In this paper, we study the problem of reconstructing the 3D struc- +ture of chromosomes from Hi-C data of diploid organisms, which poses additional challenges +compared to the better-studied haploid setting. With the help of techniques from algebraic +geometry, we prove that a small amount of phased data is sufficient to ensure finite identifi- +ability, both for noiseless and noisy data. In the light of these results, we propose a new 3D +reconstruction method based on semidefinite programming, paired with numerical algebraic ge- +ometry and local optimization. The performance of this method is tested on several simulated +datasets under different noise levels and with different amounts of phased data. We also apply +it to a real dataset from mouse X chromosomes, and we are then able to recover previously +known structural features. +1. Introduction +The eukaryotic chromatin has a three-dimensional (3D) structure in the cell nucleus which +has been shown to be important in regulating basic cellular functions, including gene regulation, +transcription, replication, recombination, and DNA repair [41, 43]. The 3D DNA organization is +also associated to brain development and function; in particular, it is shown to be misregulated +in schizophrenia [32, 34] and Alzheimer’s disease [28]. +All genetic material is stored in chromosomes which interact in the cell nucleus, and the 3D +chromatin structure influences the frequencies of such interactions. A benchmark tool to measure +such frequencies is high-throughput chromosome conformation capture (Hi-C) [16]. Hi-C first +crosslinks cell genomes, which “freezes” contacts between DNA segments. Then the genome is +cut in fragments, the fragments are ligated together and then are associated to equally-sized +segments of the genome using high-throughput sequencing [33]. These segments of the genome +are called loci and their size is known as resolution (e.g., bins of size 1Mb or 50Kb). The result of +Hi-C is stored in a matrix called contact matrix whose elements are the contact counts between +pairs of loci. +According to the structure they generate, computational methods for inferring the 3D chro- +matin structure from a contact matrix fall into two classes: ensemble and consensus methods. +In a haploid setting (organisms having a single set of chromosomes), ensemble models such as +MCMC5C [35], BACH-MIX [11] and Chrom3D [30], try to account for structure variations on +the genome across cells by inferring a population of 3D structures. On the other hand, consensus +methods aim at reconstructing one single 3D structure which may be used as a model for fur- +ther analysis. In this category, probability-based methods such as PASTIS [42, 4] model contact +counts as Poisson random variables of the Euclidean distances between loci, and distance-based +methods such as ChromSDE [46] and ShRec3D [17] model contact counts as functions of the +Euclidean distances. An extensive overview of different 3D genome reconstruction techniques is +given in [29]. +Date: January 30, 2023. +1 +arXiv:2301.11764v1 [q-bio.GN] 27 Jan 2023 + +2 +D. CIFUENTES, J. DRAISMA, O. HENRIKSSON, A. KORCHMAROS, AND K. KUBJAS +Most of the methods for 3D genome reconstructions from Hi-C data are for haploid organisms. +However, humans like most mammals are diploid organisms, in which the genetic information is +stored in pairs of chromosomes called homologs. Homologous chromosomes are almost identical +besides some single nucleotide polymorphisms (SNPs) [18]. In the case of diploid organisms, +the Hi-C data does not generally differentiate between homologous chromosomes. If we model +each chromosome as a string of beads, then we associate two beads to each locus i ∈ {1, . . . , n}, +one bead for each homolog. Therefore, each observed contact count ci,j between loci i and j +represents aggregated contacts of four different types of interactions, more precisely one of the +two homologous beads associated to locus i gets in contact with one of the two homologous +beads associated to locus j, see Figure 1. This means that the Hi-C data is unphased. Phased +Hi-C data that distinguishes contacts for homologs is rare. In our setting, we assume that the +data is partially phased, i.e., some of the contact counts can be associated with a homolog. For +example, in the (mouse) Patski (BL6xSpretus) [6, 45] cell line, 35.6% of the contact counts are +phased; while this value is as low as 0.14% in the human GM12878 cell line [33, 45]. Therefore, +methods for inferring diploid 3D chromatin structure need to take into account the ambiguity +of diploid Hi-C data to avoid inaccurate reconstructions. +Figure 1. Ambiguity of phased data. +Each entry ci,j of the Hi-C matrix corresponds to four +different contacts between the two pairs (xi, yi) for locus i and (xj, yj) for locus j. +Methods for 3D genome reconstruction in diploid organisms have been studied in [40, 4, 23, 2, +22, 37]. One approach is to phase Hi-C data [40, 23, 22], for example by assigning haplotypes to +contacts based on assignments at neighboring contacts [40, 22]. Cauer et al. [4] models contact +counts as Poisson random variables. To find the optimal 3D chromatin structure, the associated +likelihood function combined with two structural constraints is maximized. The first constraint +imposes that the distances between neighboring beads are similar and the second one requires +that homologous chromosomes are located in different regions of the cell nucleus. Belyaeva et +al. [2] shows identifiability of the 3D structure when the Euclidean distances between neighboring +beads and higher-order contact counts between three or more loci simultaneously are given. +Under these assumptions, the 3D reconstruction is obtained by combining distance geometry +with semidefinite programming. Segal [37] applies recently developed imaging technology, in +situ genome sequencing (IGS) [31], to point out issues in the assumptions made in [40, 4, 2], and +suggests as alternative assumptions that intra-homolog distances are smaller than corresponding +inter-homolog distances and intra-homolog distances are similar for homologous chromosomes. + +ci +Reference Genome +Homologous Chromosomes +HiC-matrix3D GENOME RECONSTRUCTION FROM PARTIALLY PHASED HI-C DATA +3 +IGS [31] provides yet another method for inferring the 3D structure of the genome, however, at +present the resolution and availability of IGS data is limited. +Contributions. In this work, we focus on a distance-based approach for partially phased Hi-C +data. In particular, we assume that contacts only for some loci are phased. In the string of beads +model, the locations of the pair of beads associated to i-th loci are denoted by xi, yi ∈ R3. Then +homologs are represented by two sequences x1, x2, . . . , xn and y1, x2, . . . , yn in R3; see Figure 1. +Inferring the 3D chromatin structure corresponds to estimating the bead coordinates. Based +on Lieberman-Aiden et al. [21], we assume the power law dependency ci,j = γdα +i,j, where α is +a negative conversion factor, between the distance di,j and contact count ci,j of loci i and j. +Following Cauer et al. [4], we assume that a contact count between loci is given by the sum of +all possible contact counts between the corresponding beads. We call a bead unambiguous if +the contacts for the corresponding locus are phased; otherwise we call a bead ambiguous. +Our first main contribution is to show that for negative rational conversion factors α, knowing +the locations of six unambiguous beads ensures that there are generically finitely many possible +locations for the other beads, both in the noiseless (Theorem 3.1) and noisy (Corollary 3.5) +setting. Moreover, we prove finite identifiability also in the fully ambiguous setting when α = −2 +and the number of loci is at least 13 (Theorem 3.6). Note that the identifiability does not hold +for α = 2 as shown in [2]. +Our second main contribution is to provide a reconstruction method when α = −2, based +on semidefinite programming combined with numerical algebraic geometry and local optimiza- +tion (section 4). The general idea is the following: We first estimate the coordinates of the +unambiguous beads using only the unambiguous contact counts (which precisely corresponds +to the haploid setting) using the SDP-based solver implemented in ChromSDE [46]. We then +exploit our theoretical result on finite identifiability to estimate the coordinates of the ambigu- +ous beads, one by one, by solving several polynomial systems numerically. These estimates are +then improved by a local estimation step that take into account all contact counts. Finally, a +clustering algorithm is used to overcome the symmetry (xi, yi) �→ (yi, xi) in the estimation for +the ambiguous beads. +The paper is organized as follows. In section 2, we introduce our mathematical model for +the 3D genome reconstruction problem. In section 3, we recall identifiability results in the un- +ambigous setting (section 3.1), and then prove identifiability results in the partially ambiguous +setting (section 3.2) and in the fully ambiguous setting (section 3.3). We describe our recon- +struction method in section 4. We test the performance of our method on synthetic datasets +and on a real dataset from the mouse X chromosomes in section 5. We conclude with a dis- +cussion about future research directions in section 6. The code for computations and exper- +iments is available at https://github.com/kaiekubjas/3D-genome-reconstruction-from- +partially-phased-HiC-data. +2. Mathematical model for 3D genome reconstruction +In this section we introduce the distance-based model under which we study 3D genome re- +construction. In section 2.1 we give the background on contact count matrices. In section 2.2 we +describe a power-law between contacts and distances, which allows to translate the information +about contacts into distances. +2.1. Contact count matrices. We model the genome as a string of 2n beads, corresponding +to n pairs of homologous beads. The positions of the beads are recorded by a matrix +Z = [x1, . . . , xn, y1, . . . , yn]T ∈ R2n×3. + +4 +D. CIFUENTES, J. DRAISMA, O. HENRIKSSON, A. KORCHMAROS, AND K. KUBJAS +The positions xi and yi correspond to homologous beads. When convenient, we use the notation +z1 := x1, . . . , zn := xn, zn+1 := y1, . . . , z2n := yn. In this notation, +Z = [z1, . . . , zn, zn+1, . . . , z2n]T ∈ R2n×3. +Let U be the subset of pairs that are unambiguous, i.e., beads in the pair can be distinguished, +and let A be the subset of pairs that are ambiguous, i.e., beads in the pair cannot be distin- +guished. The sets U and A form a partition of [n]. +A Hi-C matrix C is a matrix with each row and column corresponding to a genomic locus. +Following Cauer et al. [4], we call these contact counts ambiguous and denote the corresponding +contact count matrix by CA. If parental genotypes are available, then one can use SNPs to +map some reads to each haplotype [6, 24, 33]. +If both ends of a read contains SNPs that +can be associated to a single parent, then the contact count is called unambiguous and the +corresponding contact count matrix is denoted by CU. Finally, if only one of the genomic loci +present in an interaction can be mapped to one of the homologous chromosomes, then the count +is called partially ambiguous and the contact count matrix is denoted by CP . +The unambiguous count matrix CU is a 2n×2n matrix with the first n indices corresponding +to x1, . . . , xn and the last n indices corresponding to y1, . . . , yn. The ambiguous count matrix +CA is an n×n matrix and we assume that each ambiguous count is the sum of four unambiguous +counts: +cA +i,j = cU +i,j + cU +i,j+n + cU +i+n,j + cU +i+n,j+n. +The partially ambiguous count matrix CP is a 2n×n matrix and each partially ambiguous count +is the sum of two unambiguous counts: +cP +i,j = cU +i,j + cU +i,j+n. +xi +xj +yi +yj +(a) cA +i,j for i, j ∈ A +xi +xj +yi +yj +(b) cP +i,j for i ∈ U, j ∈ A +xi +xj +yi +yj +(c) cP +i+n,j for i ∈ U, j ∈ A +xi +xj +yi +yj +(d) cU +i,j for i, j ∈ U +xi +xj +yi +yj +(e) cU +i,j+n for i, j ∈ U +xi +xj +yi +yj +(f) cU +i+n,j for i, j ∈ U +xi +xj +yi +yj +(g) cU +i+n,j+n for i, j ∈ U +Figure 2. Seven different types of contacts between the ith and jth locus. +2.2. Contacts and distances. Denoting the distance ∥zi − zj∥ between zi and zj by di,j, the +power law dependency observed by Lieberman-Aiden et al. [21] can be written as +cU +i,j = γdα +i,j, +(2.1) +where α < 0 is a conversion factor and γ > 0 is a scaling factor. This relationship between +contact counts and distances is assumed in [2, 46], while in [4, 42] the contact counts ci,j are +modeled as Poisson random variables with the Poisson parameter being βdα +i,j. + +3D GENOME RECONSTRUCTION FROM PARTIALLY PHASED HI-C DATA +5 +In our paper, we assume that contact counts are related to distances by (2.1). Similarly to [2], +we set γ = 1 and in parts of the article α = −2. In general, the conversion factor α depends on +a dataset and its estimation can be part of the reconstruction problem [42, 46]. Setting γ = 1 +means that we recover the configuration up to a scaling factor. In practice, the configuration +can be rescaled using biological knowledge, e.g., the radius of the nucleus. +Our approach to 3D genome reconstruction builds on the power law dependency between +contacts and distances between unambiguous beads. We convert the empirical contact counts +to Euclidean distances and then aim to reconstruct the positions of beads from the distances. +This leads us to the following system of equations: +� +� +� +� +� +� +� +� +� +cA +i,j = ∥xi − xj∥α + ∥xi − yj∥α + ∥yi − xj∥α + ∥yi − yj∥α +∀i, j ∈ A +cP +i,j = ∥xi − xj∥α + ∥xi − yj∥α, +cP +i+n,j = ∥yi − xj∥α + ∥yi − yj∥α +∀i ∈ U, j ∈ A +cU +i,j = ∥xi − xj∥α, +cU +i,j+n = ∥xi − yj∥α, +cU +i+n,j = ∥yi − xj∥α, +cU +i+n,j+n = ∥yi − yj∥α +∀i, j ∈ U +(2.2) +If α is an even integer, then (2.2) is a system of rational equations. +Determining the points xi, yi, where i ∈ U, is the classical Euclidean distance problem: We +know the (noisy) pairwise distances between points and would like to construct the locations of +points, see section 3.1 for details. Hence after section 3.1 we assume that we have estimated the +locations of points xi, yi, where i ∈ U, and we would like to determine the points xi, yi, where +i ∈ A. +3. Identifiability +In this section, we study the uniqueness of the solutions of the system (2.2) up to rigid +transformations (translations, rotations and reflections), or in other words, the identifiability of +the locations of beads. We study the unambiguous, partially ambiguous and ambiguous settings +in sections 3.1, 3.2 and 3.3, respectively. +3.1. Unambiguous setting and Euclidean distance geometry. If all pairs are unambigu- +ous, i.e., U = [n], then constructing the original points translates to a classical problem in +Euclidean distance geometry. The principal task in Euclidean distance geometry is to construct +original points from pairwise distances between them. In the rest of the subsection, we will recall +how to solve this problem. Since pairwise distances are invariant under translations, rotations +and reflections (rigid transformations), then the original points can be reconstructed up to rigid +transformations. For an overview of distance geometry and Euclidean distance matrices, we +refer the reader to [7, 15, 20, 26]. +The Gram matrix of the points z1, . . . , z2n is defined as +G = ZZT = [z1, . . . , z2n]T · [z1, . . . , z2n] ∈ R2n×2n. +Let z = +1 +2n +�2n +i=1 zi and ˜zi = zi − z for i = 1, . . . , 2n. The matrix ˜Z = [˜z1, . . . , ˜z2n]T gives the +locations of points after centering them around the origin. Let ˜G denote the Gram matrix of +the centered point configuration ˜z1, . . . , ˜z2n. +Let Di,j = ∥zi−zj∥2 denote the squared Euclidean distance between the points zi and zj. The +Euclidean distance matrix of the points z1, . . . , z2n is defined as D = (Di,j)1≤i,j≤2n ∈ R2n×2n. +To express the centered Gram matrix in terms of the Euclidean distance matrix, we define the +geometric centering matrix +J = I2n − 1 +2n11T , + +6 +D. CIFUENTES, J. DRAISMA, O. HENRIKSSON, A. KORCHMAROS, AND K. KUBJAS +where I2n is the 2n × 2n identity matrix and 1 is the vector of ones. The linear relationship +between ˜G and D is given by +˜G = −1 +2JDJ. +Therefore, given the Euclidean distance matrix, we can construct the centered Gram matrix for +the points z1, . . . , z2n. +The centered points up to rigid transformations are extracted from the centered Gram matrix +˜G using the eigendecomposition ˜G = QΛQ−1, where Q is orthonormal and Λ is a diagonal +matrix with entries ordered in decreasing order λ1 ≥ λ2 ≥ . . . ≥ λ2n ≥ 0. We define Λ1/2 +3 +:= +[diag(√λ1, √λ2, √λ3), 03×(2n−3)]T and set ˆZ = QΛ1/2 +3 +. In the case of noiseless distance matrix +D, the Gram matrix ˜G has rank three and the diagonal matrix Λ has precisely three non-zero +entries. Hence we could obtain ˆZ also from QΛ1/2 by truncating zero columns. Using Λ1/2 +3 +has +the advantage that it gives an approximation for the points also for a noisy distance matrix D. +The uniqueness of ˆZ up to rotations and reflections follows from [14, Proposition 3.2], which +states that AAT = BBT if and only if A = BQ for some orthogonal matrix Q. +The procedure that transforms the distance matrix to origin centered Gram matrix and then +uses eigendecomposition for constructing original points is called classical multidimensional scal- +ing (cMDS) [5]. Although cMDS is widely used in practice, it does not always find the distance +matrix that minimizes the Frobenius norm to the empirical noisy distance matrix [39]. Other +approaches to solving the Euclidean distance and Euclidean completion problems include non- +convex [9, 25] as well semidefinite formulations [1, 10, 27, 44, 46, 47]. +3.2. Partially ambiguous setting. The next theorem establishes the uniqueness of the solu- +tions of the system (2.2) in the presence of ambiguous pairs. In particular, it states that there +are finitely many possible locations for beads in one ambiguous pair given the locations of six +unambiguous beads. The identifiability results in this subsection hold for all negative rational +numbers α. In the rest of the paper, we denote the true but unknown coordinates by x∗ and the +symbol x stands for a variable that we want to solve for. We write ∥ · ∥ for the standard inner +product on R3. +Theorem 3.1. Let α be a negative rational number. Then for a∗, b∗, . . . , f∗, x∗, y∗ ∈ R3 suffi- +ciently general, the system of six equations +∥x − t∗∥α + ∥y − t∗∥α = ∥x∗ − t∗∥α + ∥y∗ − t∗∥α for t∗ = a∗, b∗, . . . , f∗ +(3.1) +in the six unknowns x1, x2, x3, y1, y2, y3 ∈ R has only finitely many solutions. +Remark 3.2. The proof will show that this system has only finitely many solutions over the +complex numbers. +We believe that the theorem holds for general nonzero rational α. +Indeed, our argument +works, with a minor modification, also for α > 2, but for α in the range (0, 2] a refinement of +the argument is needed. +Proof. First write Q(x) := x2 +1 + x2 +2 + x2 +3, so that ∥x∥ = +� +Q(x) for x ∈ R3. The advantage of Q +over ∥x∥ is that it is well-defined on C3. +Write α +2 = +m +n with m, n integers, m ̸= 0, and n > 0. +Consider the affine variety X ⊆ +(C3)8 × (C2)6 consisting of all tuples +((a∗, . . . , f∗, x∗, y∗), (rt∗, st∗)t∗=a∗,...,f∗) +such that +Q(x∗ − t∗)m = rn +t∗ ̸= 0 and Q(y∗ − t∗)m = sn +t∗ ̸= 0 for t∗ = a∗, . . . , f∗. + +3D GENOME RECONSTRUCTION FROM PARTIALLY PHASED HI-C DATA +7 +Note that, if x∗, t∗ are real, then it follows that +Q(x∗ − t∗)m = (∥x∗ − t∗∥α)n, +and similarly for Q(y∗ − t∗). Hence if a∗, . . . , y∗ are all real, then the point +((a∗, . . . , f∗, x∗, y∗), (∥x∗ − t∗∥α, ∥y∗ − t∗∥α)t∗) +(3.2) +is a point in X with real-valued coordinates. +The projection π from X to the open affine subset U ⊆ (C3)8 where all Q(x∗−t∗) and Q(y∗−t∗) +are nonzero is a finite morphism with fibres of cardinality n12; to see this cardinality note that +there are n possible choices for each of the numbers rt∗, st∗. Each irreducible component of X +is a smooth variety of dimension 24. +Consider the map ψ : X → (C3 × C1)6 defined by +((a∗, . . . , f∗, x∗, y∗), (rt∗, st∗)t∗) �→ ((t∗, rt∗ + st∗))t∗ +We claim that for q in some open dense subset of X, the derivative dqψ has full rank 24. For +this, it suffices to find one point p ∈ U such that dqψ has rank 24 at each of the n12 points +q ∈ π−1(p). We take a real-valued point p := (a∗, b∗, . . . , f∗, x∗, y∗) ∈ (R3)8 to be specified later +on. Let q ∈ π−1(p). Then, near q, the map ψ factorises via π and the unique algebraic map +ψ′ : U → (C3 × C1)6 (defined near p) which on a neighbourhood of p in U ∩ (R3)8 equals +ψ′(a, . . . , f, x, y) = ((t, ξt∗ · Q(x − t)α/2 + ηt∗ · Q(y − t)α/2))t=a,...,f ∈ (C3 × C1)6 +where ξt∗ and ζt∗ are n-th roots of unity in C depending on which q is chosen among the n12 +points in π−1(p). The situation is summarised in the following diagram: +(X, q) +π +� +ψ +� +(U, p) +ψ′ +� ((C3 × C1)6, ψ(q)). +Now, dqψ = dpψ′ ◦ dqπ, and since dqπ is a linear isomorphism, it suffices to prove that dpψ′ +is a linear isomorphism. Suppose that (a′, . . . , f′, x′, y′) ∈ ker dpψ′. Then, since the map ψ′ +remembers a, . . . , f, it follows immediately that a′ = . . . = f′ = 0. On the other hand, by +differentiating we find that, for each t∗ ∈ {a∗, . . . , f∗}, +ξt∗ · (α/2) · Q(x∗ − t∗)α/2−1 · 2 · ⟨x′, x∗ − t∗⟩ ++ηt∗ · (α/2) · Q(y∗ − t∗)α/2−1 · 2 · ⟨y′, y∗ − t∗⟩ = 0, +where ⟨·, ·⟩ stands for the standard bilinear form on C3. In other words, the vector (x′, y′) ∈ C6 +is in the kernel of the 6 × 6-matrix +M := +� +�� +∥x∗ − a∗∥α−2 · ξa∗ · (x∗ − a∗) +∥y∗ − a∗∥α−2 · ηa∗ · (y∗ − a∗) +... +... +∥x∗ − f∗∥α−2 · ξf∗ · (x∗ − f∗) +∥y∗ − f∗∥α−2 · ηf∗ · (y∗ − f∗) +� +�� +where we have interpreted a∗, . . . , f∗, x∗, y∗ as row vectors. It suffices to show that, for some +specific choice of p = (a∗, . . . , f∗, x∗, y∗) ∈ (R3)8, this matrix is nonsingular for all n12 choices +of ((ξt∗, ηt∗))t∗. +We choose a∗, . . . , f∗, x∗, y∗ as the vertices of the unit cube, as follows: +a∗ = (1, 0, 0) +b∗ = (0, 1, 0) +c∗ = (0, 0, 1) +c∗ = (0, 1, 1) +d∗ = (1, 0, 1) +f∗ = (1, 1, 0) +x∗ = (0, 0, 0) +y∗ = (1, 1, 1). + +8 +D. CIFUENTES, J. DRAISMA, O. HENRIKSSON, A. KORCHMAROS, AND K. KUBJAS +Then the matrix M becomes, with β = α − 2: +� +��������� +−ξa∗ +0 +0 +0 +2 +β +2 · ηa∗ +2 +β +2 · ηa∗ +0 +−ξb∗ +0 +2 +β +2 · ηb∗ +0 +2 +β +2 · ηb∗ +0 +0 +−ξc∗ +2 +β +2 · ηc∗ +2 +β +2 · ηc∗ +0 +0 +−(2 +β +2 · ξd∗) +−(2 +β +2 · ξd∗) +ηd∗ +0 +0 +−(2 +β +2 · ξe∗) +0 +−(2 +β +2 · ξe∗) +0 +ηe∗ +0 +−(2 +β +2 · ξf∗) +−(2 +β +2 · ξf∗) +0 +0 +0 +ηf∗ +� +��������� +. +Now, det(M) equals +− ξa∗ · ξb∗ · ξc∗ · ηd∗ · ηe∗ · ηf∗ + 22+3β · ηa∗ · ηb∗ · ηc∗ · ξd∗ · ξe∗ · ξf∗ + 22β · R +(3.3) +where R is a sum of (products of) roots of unity. Now α < 0 implies that β < −2, so that +2 + 3β < 2β < 0. Since roots of unity have 2-adic valuation 0, the second term in the expression +above is the unique term with minimal 2-adic valuation. Hence det(M) ̸= 0, as desired. +It follows that ψ is a dominant morphism from each irreducible component of X into (C3 × +C1)6, and hence for all q in an open dense subset of X, the fibre ψ−1(ψ(q)) is finite. This then +holds, in particular, for q in an open dense subset of the real points as in (3.2). This proves the +theorem. +□ +Remark 3.3. If α > 2, then β > 0, and hence the unique term with minimal 2-adic valuation in +(3.3) is the first term. This can be used to show that the theorem holds then, as well. The only +subtlety is that for positive α, solutions where x or y equal one of the points a∗, . . . , f∗ are not +automatically excluded, and these are not seen by the variety X. But a straightforward argument +shows that such solutions do not exist for sufficiently general choices of a∗, . . . , f∗, x∗, y∗. +We now consider the setting when we know locations of seven unambiguous beads. In the +special case when α = −2, we construct the ideal generated by the polynomials obtained from +rational equations (3.1) for seven unambiguous beads after moving all terms to one side and +clearing the denominators. Based on symbolic computations in Macaulay2 for the degree of +this ideal, we conjecture that the location of a seventh unambiguous bead guarantees unique +identifiability of an ambiguous pair of beads: +Conjecture 3.4. Let a∗, b∗, c∗, d∗, e∗, f∗, g∗, x∗, y∗ ∈ R3 be sufficiently general. The system of +rational equations +1 +∥t∗ − x∗∥2 + +1 +∥t∗ − y∗∥2 = +1 +∥t∗ − x∥2 + +1 +∥t∗ − y∥2 for t∗ = a∗, b∗, c∗, d∗, e∗, f∗, g∗ +(3.4) +has precisely two solutions (x∗, y∗) and (y∗, x∗). +In practice, we only have noisy estimates a, b, . . . , f ∈ R3 of the true positions of unambiguous +beads a∗, b∗, . . . , f∗ ∈ R3, and we have noisy observations ct of the true contact counts c∗ +t := +∥x∗ − t∗∥α + ∥y∗ − t∗∥α. We aim to find x, y ∈ R3 such that +∥x − t∥α + ∥y − t∥α = ct for t = a, b, . . . , f. +We may write ct = ∥x∗ − t∥α + ∥y∗ − t∥α + ϵt for some ϵt that depends on the noise level. Hence, +the above system of equations can be rephrased as +∥x − t∥α + ∥y − t∥α = ∥x∗ − t∥α + ∥y∗ − t∥α + ϵt for t = a, b, . . . , f. +(3.5) +In the following corollary we show that this system has generically finitely many solutions. +Corollary 3.5. Let α be a negative rational number. +Then for a, b, . . . , f, x∗, y∗ ∈ R3 and +ϵa, ϵb, . . . , ϵf ∈ R sufficiently general, the system of six equations +∥x − t∥α + ∥y − t∥α = ∥x∗ − t∥α + ∥y∗ − t∥α + ϵt for t = a, b, . . . , f +(3.6) + +3D GENOME RECONSTRUCTION FROM PARTIALLY PHASED HI-C DATA +9 +in the six unknowns x1, x2, x3, y1, y2, y3 ∈ R has only finitely many solutions. +Proof. Recall the map ψ : X → (C3 × C1)6 from the proof of Theorem 3.1 defined by +((a, . . . , f, x∗, y∗), (rx∗,t, sy∗,t)t) �→ ((t, rx∗,t + sy∗,t))t. +We showed that ψ is a dominant morphism from each irreducible component of X into (C3×C1)6, +and that each irreducible component of X is 24-dimensional. Every solution to (3.6) is the (x, y)- +component of a point in the fibre +ψ−1((t, ||x∗ − t||α + ||y∗ − t||α + ϵt))t. +Since this is a fibre over a sufficiently general point, the fibre is finite. +□ +Corollary 3.5 will be the basis of a numerical algebraic geometric based reconstruction method +in section 4. +3.3. Ambiguous setting. Finally we consider the ambiguous setting, where one would like to +reconstruct the locations of beads only from ambiguous contact counts. It is shown in [2] that +for α = 2, one does not have finite identifiability no matter how many pairs of ambiguous beads +one considers. We show finite identifiability for the locations of beads given contact counts for +13 pairs of ambiguous beads for α = −2. We believe that the result might be true for further +conversion factors α’s, however our proof technique does not directly generalize. +Theorem 3.6. Let α = −2. Then for x∗ +1, y∗ +1, . . . , x∗ +12, y∗ +12 ∈ R3 sufficiently general, the system +of 66 equations +∥xi − xj∥α + ∥xi − yj∥α + ∥yi − xj∥α + ∥yi − yj∥α = +∥x∗ +i − x∗ +j∥α + ∥x∗ +i − y∗ +j ∥α + ∥y∗ +i − x∗ +j∥α + ∥y∗ +i − y∗ +j ∥α for 1 ≤ i < j ≤ 12 +(3.7) +in the 72 unknowns x1,1, x1,2, x1,3, y1,1, y1,2, y1,3, . . . , x12,1, x12,2, x12,3, y12,1, y12,2, y12,3 ∈ R has +only finitely many solutions up to rigid transformations. +Proof. As before, we write Q(x) := x2 +1 + x2 +2 + x2 +3, so that ∥x∥ = +� +Q(x) for x ∈ R3. Consider +the affine open subset X ⊆ (C3)24 consisting of all tuples (x∗ +1, y∗ +1, . . . , x∗ +12, y∗ +12) such that +Q(x∗ +i − x∗ +j) ̸= 0, Q(x∗ +i − y∗ +j ) ̸= 0, Q(y∗ +i − x∗ +j) ̸= 0 and Q(y∗ +i − y∗ +j ) ̸= 0 for 1 ≤ i < j ≤ 12. +Consider also the map ψ : X → C66 defined by +(x∗ +1, y∗ +1, . . . , x∗ +12, y∗ +12) �→ (Q(x∗ +i − x∗ +j)−1 + Q(x∗ +i − y∗ +j )−1 + Q(y∗ +i − x∗ +j)−1 + Q(y∗ +i − y∗ +j )−1)i0, b∈R3 +� +� +� +� 1 +2n +n +� +i=1 +� +∥(sRxi + b) − x∗ +i ∥2 + ∥(sRyi + b) − y∗ +i ∥2 +� +. + +14 +D. CIFUENTES, J. DRAISMA, O. HENRIKSSON, A. KORCHMAROS, AND K. KUBJAS +This can be seen as a version of the classical Procrustes problem solved in [36], which is imple- +mented in Matlab as the function procrustes. +Specific examples of reconstructions of the Brownian motion and helix-shaped chromosomes +obtained with SNLC at varying noise levels and 50% of ambiguous beads are shown in Figure 3. +For low noise levels the reconstructions by SNLC and the original structure highly overlap. For +higher noise levels the general region occupied by the reconstructions overlaps with the original +structure, while the local features become less aligned. Analogous reconstructions obtained with +SNLC without the local optimization step are shown in Figure S1. +A comparison of how the quality of the reconstruction depends on the noise level and pro- +portion of ambiguous beads for SNLC and PASTIS is done in Figure 4. We measure the RMSD +value between the reconstructed and original 3D structure for different noise levels over 20 runs. +The RMSD values obtained by SNLC are consistently lower than the ones obtained by PASTIS. +The difference is specially large for low to medium noise levels. While our method outperforms +PASTIS in the setting considered in this paper, it is worth mentioning that PASTIS works also +in a more general setting, where there might be contacts of all three types (ambiguous, partially +ambiguous and unambiguous) between every pair of loci. +-5 +0 +5 +10 +-4 +-2 +0 +2 +4 +6 +8 +RMSD = 0.17757 +X true +X estimated +Y true +Y estimated +Start +Unambiguous +(a) ε = 0.10 +-5 +0 +5 +10 +-4 +-2 +0 +2 +4 +6 +8 +RMSD = 0.5478 +X true +X estimated +Y true +Y estimated +Start +Unambiguous +(b) ε = 0.50 +-5 +0 +5 +10 +-4 +-2 +0 +2 +4 +6 +8 +RMSD = 0.9856 +X true +X estimated +Y true +Y estimated +Start +Unambiguous +(c) ε = 0.90 +0.5 +RMSD = 0.052324 +0 +-1 +-0.5 +-0.5 +0 +0.5 +-2 +1 +1.5 +0 +2 +2.5 +2 +3 +-1 +4 +6 +X true +X estimated +Y true +Y estimated +Start +Unambiguous +(d) ε = 0.10 +1 +RMSD = 0.19914 +0 +-1 +-0.5 +0 +0.5 +-2 +1 +1.5 +0 +2 +2.5 +2 +-1 +3 +4 +6 +X true +X estimated +Y true +Y estimated +Start +Unambiguous +(e) ε = 0.50 +1 +RMSD = 0.54979 +0 +-1 +-1 +-0.5 +0 +0.5 +-2 +1 +1.5 +0 +2 +2.5 +2 +3 +-2 +4 +6 +X true +X estimated +Y true +Y estimated +Start +Unambiguous +(f) ε = 0.90 +Figure 3. Examples of reconstructions for varying noise levels, for a chromosome pair with 60 loci, out +of which 50% are ambiguous. Subfigures (a)–(c) show chromosomes simulated with Brownian motion +(projected onto the xy-plane), whereas figure (d)–(e) show helix-shaped chromosomes. +5.2. Experimentally obtained data. We compute SNLC reconstructions based on the real +dataset explored in [4], which is obtained from Hi-C experiments on the X chromosomes in the + +3D GENOME RECONSTRUCTION FROM PARTIALLY PHASED HI-C DATA +15 +0 +0.2 +0.4 +0.6 +0.8 +1 +epsilon +-0.5 +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +RMSD +PASTIS +SNLC +(a) 25% ambiguous loci +0 +0.2 +0.4 +0.6 +0.8 +1 +epsilon +-0.5 +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +RMSD +PASTIS +SNLC +(b) 50% ambiguous loci +Figure 4. Comparison between our reconstruction method and PASTIS. The values are the average +over 20 runs, with the error bars showing the standard deviation. +All experiments took place with +60 loci, with varying levels of noise, as well as varying number of ambiguous loci, uniformly randomly +distributed over the chromosomes. +Patski (BL6xSpretus) cell line. The data has been recorded at a resolution of 500 kb, which +corresponds to 343 bead pairs in our model. +For some of these pairs, no or only very low contact counts have been recorded. Since such low +contact counts are susceptible to high uncertainty and can be assumed to be a consequence of +experimental errors, we exclude the 47 loci with the lowest total contact counts from the analysis. +To select the cutoff, the loci are sorted according to the total contact counts (see Figure S2 (a)), +and the ratios between the total contact counts for consecutive loci are computed. A peak for +these ratios is observed at the 47th contact count, as shown in Figure S2 (b). After applying +this filter, we obtain a dataset with 296 loci. Out of these, we consider as ambiguous all loci i +for which less than 40% of the total contact count comes from contacts where xi and yi were +not distinguishable. These proportions for all loci are shown in Figure S2 (c). For the Patski +dataset, we obtain 46 ambiguous loci and 250 unambiguous loci in this way. +In the PASTIS dataset, a locus can simultaneously participate in unambiguous, partially am- +biguous and ambiguous contacts. To obtain the setting of our paper where loci are partitioned +into unambiguous or ambiguous, we reassign the contacts according to whether a locus is unam- +biguous or ambiguous. Our reassignment method is motivated by the assignment of haplotype +to unphased Hi-C reads in [22]. The exact formulas are given in Supplementary Material. +The reconstruction obtained via SNLC can be found in Figure 5 (a). +The logarithmic +heatmaps for contact count matrices for original data and the SNLC reconstruction are shown +in Figure S3. +It was discovered in [6] that the inactive homolog in the Patski X chromosome pair has a +bipartite structure, consisting of two superdomains with frequent intra-chromosome contacts +within the superdomains and a boundary region between the two superdomains. The active +homolog does not exhibit the same behaviour. The boundary region on the inactive X chromo- +some is centered at 72.8-72.9 MB [6] which at the 500 kB resolution corresponds to the bead +146 [4]. We show in Figure 5 (b) that the two chromosomes reconstructed using SNLC exhibit +this structure by computing the bipartite index for the respective homologs as in [4, 6]. We +recall that, in the setting of a single chromosome with beads z1, . . . , zn ∈ R3, the bipartite index +is defined as the ratio of intra-superdomain to inter-superdomain contacts in the reconstruction: +BI(h) = +1 +h2 +�h +i=1 +�h +j=1 +1 +∥zi−zj∥2 + +1 +(n−h)2 +�n +i=h+1 +�n +j=h+1 +1 +∥zi−zj∥2 +2 +h(n−h) +�h +i=1 +�n +j=h+1 +1 +∥zi−zj∥2 +. + +16 +D. CIFUENTES, J. DRAISMA, O. HENRIKSSON, A. KORCHMAROS, AND K. KUBJAS +-0.5 +0 +0.5 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +Inactive homolog +Active homolog +Ambiguous +(a) +0 +50 +100 +150 +200 +250 +300 +350 +h +0 +5 +10 +15 +20 +25 +Bipartite index +Inactive homolog +Active homolog +(b) +Figure 5. (a) Reconstruction from a real dataset using our reconstruction method. +A dashed line +between two beads is used to indicate that there is one or more beads between them, for which we +have not given an estimation (due to low contact counts). (b) Bipartite index for the reconstructed +chromosomes. The dashed vertical line indicates the known hinge point at locus 146. +6. Discussion +In this article we study the finite identifiability of 3D genome reconstruction from contact +counts under the model where the distances di,j and contact counts ci,j between two beads i +and j follow the power law dependency ci,j = dα +i,j for a conversion factor α < 0. We show that +if at least six beads are unambiguous, then the locations of the rest of the beads can be finitely +identified from partially ambiguous contact counts for rational α satisfying α < 0 or α > 2. +In the fully ambiguous setting, we prove finite identifiability for α = −2, given ambiguous +contact counts for at least 12 pairs of beads. From [2] it is known that finite identifiability +does not hold in the fully ambiguous setting for α = 2. It is an open question whether finite +identifiability of 3D genome reconstruction holds for other α ∈ R\{−2, 2} in the fully ambiguous +setting and for rational α ∈ (0, 2] in the partially ambiguous setting. We conjecture that in the +partially ambiguous setting seven unambiguous loci guarantee unique identifiability of the 3D +reconstruction for rational α < 0 or α > 2. When α = −2, then one approach to studying the +unique identifiability might be via the degree of a parametrized family of algebraic varieties. +After establishing the identifiability, we suggest a reconstruction method for the partially am- +biguous setting with α = −2 that combines semidefinite programming, homotopy continuation +in numerical algebraic geometry, local optimization and clustering. To speed up the homotopy +continuation based part, we observe that the parametrized system of polynomial equations cor- +responding to six unambiguous beads has 40 pairs of complex solutions and we trace one path +for each orbit. It is an open question to prove that for sufficiently general parameters the sys- +tem has 40 pairs of complex solution. This question again reduces to studying the degree of a +family of algebraic varieties. While our goal is to highlight the potential of our method, one +could further regularize its output and use interpolation for the beads that are far away from +the neighboring beads. A future research direction is to explore whether numerical algebraic +geometry or semidefinite programming based methods can be proposed also for other conversion +factors α < 0. +Acknowledgements +Oskar Henriksson and Kaie Kubjas were partially supported by the Academy of Finland Grant +No. 323416. We thank Anastasiya Belyaeva, Gesine Cauer, AmirHossein Sadegemanesh, Luca +Sodomaco, and Caroline Uhler for very helpful discussions and answers to our questions. + +3D GENOME RECONSTRUCTION FROM PARTIALLY PHASED HI-C DATA +17 +References +[1] Abdo Y Alfakih, Amir Khandani, and Henry Wolkowicz. Solving euclidean distance matrix completion +problems via semidefinite programming. Computational optimization and applications, 12(1):13–30, 1999. +Cited on page 6. +[2] Anastasiya Belyaeva, Kaie Kubjas, Lawrence J Sun, and Caroline Uhler. Identifying 3D genome organiza- +tion in diploid organisms via Euclidean distance geometry. SIAM J. Math. Data Sci., 4(1):204–228, 2022. +Cited on pages 2, 3, 4, 5, 9, 10, 13, and 16. +[3] Paul Breiding and Sascha Timme. Homotopycontinuation.jl: A package for homotopy continuation in julia. +In James H. Davenport, Manuel Kauers, George Labahn, and Josef Urban, editors, Mathematical Software +– ICMS 2018, pages 458–465, Cham, 2018. Springer International Publishing. Cited on pages 11 and 13. +[4] Alexandra Gesine Cauer, G¨urkan Yardimci, Jean-Philippe Vert, Nelle Varoquaux, and William Stafford No- +ble. Inferring diploid 3D chromatin structures from Hi-C data. In 19th International Workshop on Algorithms +in Bioinformatics (WABI 2019), 2019. Cited on pages 1, 2, 3, 4, 13, 14, and 15. +[5] Michael AA Cox and Trevor F Cox. Multidimensional scaling. In Handbook of data visualization, pages +315–347. Springer, 2008. Cited on page 6. +[6] Xinxian Deng, Wenxiu Ma, Vijay Ramani, Andrew Hill, Fan Yang, Ferhat Ay, Joel B Berletch, Carl Anthony +Blau, Jay Shendure, Zhijun Duan, et al. Bipartite structure of the inactive mouse X chromosome. Genome +Biol., 16(1):1–21, 2015. Cited on pages 2, 4, and 15. +[7] Ivan Dokmanic, Reza Parhizkar, Juri Ranieri, and Martin Vetterli. Euclidean distance matrices: Essential +theory, algorithms, and applications. IEEE Signal Process. Mag., 32(6):12–30, 2015. Cited on page 5. +[8] Kyle P Eagen. Principles of chromosome architecture revealed by Hi-C. Trends Biochem. Sci., 43(6):469–478, +2018. Cited on page 12. +[9] Haw-ren Fang and Dianne P O’Leary. Euclidean distance matrix completion problems. Optim. Methods +Softw., 27(4-5):695–717, 2012. Cited on page 6. +[10] Maryam Fazel, Haitham Hindi, and Stephen P Boyd. Log-det heuristic for matrix rank minimization with +applications to Hankel and Euclidean distance matrices. In Proceedings of the 2003 American Control Con- +ference, 2003., volume 3, pages 2156–2162. IEEE, 2003. Cited on page 6. +[11] Ming Hu, Ke Deng, Zhaohui Qin, Jesse Dixon, Siddarth Selvaraj, Jennifer Fang, Bing Ren, and Jun S +Liu. Bayesian inference of spatial organizations of chromosomes. PLoS Comput. Biol., 9(1):e1002893, 2013. +Cited on page 1. +[12] Birkett Huber and Bernd Sturmfels. A polyhedral method for solving sparse polynomial systems. Math. +Comput., 64(212):1541–1555, 1995. Cited on page 11. +[13] Kaifeng Jiang, Defeng Sun, and Kim-Chuan Toh. A partial proximal point algorithm for nuclear norm +regularized matrix least squares problems. Math. Program. Comput., 6, 09 2014. Cited on page 10. +[14] Nathan Krislock. Semidefinite Facial Reduction for Low-Rank Euclidean Distance Matrix Completion. PhD +thesis, University of Waterloo, 2010. Cited on page 6. +[15] Nathan Krislock and Henry Wolkowicz. Euclidean distance matrices and applications. In Handbook on semi- +definite, conic and polynomial optimization, pages 879–914. Springer, 2012. Cited on page 5. +[16] Denis L Lafontaine, Liyan Yang, Job Dekker, and Johan H Gibcus. Hi-C 3.0: Improved protocol for genome- +wide chromosome conformation capture. Curr. Protoc., 1(7):e198, 2021. Cited on page 1. +[17] Annick Lesne, Julien Riposo, Paul Roger, Axel Cournac, and Julien Mozziconacci. 3D genome reconstruction +from chromosomal contacts. Nat. methods, 11(11):1141–1143, 2014. Cited on page 1. +[18] Jing Li, Yu Lin, Qianzi Tang, and Mingzhou Li. Understanding three-dimensional chromatin organization in +diploid genomes. Comput. Struct. Biotechnol. J., 2021. Cited on page 2. +[19] Tien-Yien Li and Xiaoshen Wang. The BKK root count in Cn. Math. Comput., 65(216):1477–1484, 1996. +Cited on page 11. +[20] Leo Liberti, Carlile Lavor, Nelson Maculan, and Antonio Mucherino. Euclidean distance geometry and ap- +plications. SIAM Rev., 56(1):3–69, 2014. Cited on page 5. +[21] Erez Lieberman-Aiden, Nynke L Van Berkum, Louise Williams, Maxim Imakaev, Tobias Ragoczy, Agnes +Telling, Ido Amit, Bryan R Lajoie, Peter J Sabo, Michael O Dorschner, et al. Comprehensive mapping of + +18 +D. CIFUENTES, J. DRAISMA, O. HENRIKSSON, A. KORCHMAROS, AND K. KUBJAS +long-range interactions reveals folding principles of the human genome. Science, 326(5950):289–293, 2009. +Cited on pages 3 and 4. +[22] Stephen Lindsly, Wenlong Jia, Haiming Chen, Sijia Liu, Scott Ronquist, Can Chen, Xingzhao Wen, Cooper +Stansbury, Gabrielle A Dotson, Charles Ryan, et al. Functional organization of the maternal and paternal +human 4D nucleome. IScience, 24(12):103452, 2021. Cited on pages 2 and 15. +[23] Han Luo, Xinxin Li, Haitao Fu, and Cheng Peng. Hichap: a package to correct and analyze the diploid hi-c +data. BMC Genomics, 21(1):1–13, 2020. Cited on page 2. +[24] Anand Minajigi, John E Froberg, Chunyao Wei, Hongjae Sunwoo, Barry Kesner, David Colognori, Derek +Lessing, Bernhard Payer, Myriam Boukhali, Wilhelm Haas, et al. A comprehensive Xist interactome reveals +cohesin repulsion and an RNA-directed chromosome conformation. Science, 349(6245), 2015. Cited on page 4. +[25] Bamdev Mishra, Gilles Meyer, and Rodolphe Sepulchre. Low-rank optimization for distance matrix com- +pletion. In 2011 50th IEEE Conference on Decision and Control and European Control Conference, pages +4455–4460. IEEE, 2011. Cited on page 6. +[26] Antonio Mucherino, Carlile Lavor, Leo Liberti, and Nelson Maculan. Distance geometry: theory, methods, +and applications. Springer Science & Business Media, 2012. Cited on page 5. +[27] Jiawang Nie. Sum of squares method for sensor network localization. Comput. Optim. Appl., 43(2):151–179, +2009. Cited on page 6. +[28] Alexi Nott, Inge R Holtman, Nicole G Coufal, Johannes CM Schlachetzki, Miao Yu, Rong Hu, Claudia Z +Han, Monique Pena, Jiayang Xiao, Yin Wu, et al. Brain cell type–specific enhancer–promoter interactome +maps and disease-risk association. Science, 366(6469):1134–1139, 2019. Cited on page 1. +[29] Oluwatosin Oluwadare, Max Highsmith, and Jianlin Cheng. An overview of methods for reconstructing 3-d +chromosome and genome structures from hi-c data. Biol. Proced. Online, 21(1):1–20, 2019. Cited on page 1. +[30] Jonas Paulsen, Monika Sekelja, Anja R Oldenburg, Alice Barateau, Nolwenn Briand, Erwan Delbarre, Akshay +Shah, Anita L Sørensen, Corinne Vigouroux, Brigitte Buendia, et al. Chrom3D: Three-dimensional genome +modeling from Hi-C and nuclear lamin-genome contacts. Genome Biol., 18(1):1–15, 2017. Cited on page 1. +[31] Andrew C Payne, Zachary D Chiang, Paul L Reginato, Sarah M Mangiameli, Evan M Murray, Chun- +Chen Yao, Styliani Markoulaki, Andrew S Earl, Ajay S Labade, Rudolf Jaenisch, et al. In situ genome +sequencing resolves DNA sequence and structure in intact biological samples. Science, 371(6532):eaay3446, +2021. Cited on pages 2 and 3. +[32] Prashanth Rajarajan, Tyler Borrman, Will Liao, Nadine Schrode, Erin Flaherty, Charlize Casi˜no, Samuel +Powell, Chittampalli Yashaswini, Elizabeth A LaMarca, Bibi Kassim, et al. Neuron-specific signatures in the +chromosomal connectome associated with schizophrenia risk. Science, 362(6420), 2018. Cited on page 1. +[33] Suhas SP Rao, Miriam H Huntley, Neva C Durand, Elena K Stamenova, Ivan D Bochkov, James T Robinson, +Adrian L Sanborn, Ido Machol, Arina D Omer, Eric S Lander, et al. A 3D map of the human genome at +kilobase resolution reveals principles of chromatin looping. Cell, 159(7):1665–1680, 2014. Cited on pages 1, +2, and 4. +[34] Suhn K Rhie, Shannon Schreiner, Heather Witt, Chris Armoskus, Fides D Lay, Adrian Camarena, Valeria N +Spitsyna, Yu Guo, Benjamin P Berman, Oleg V Evgrafov, et al. Using 3D epigenomic maps of primary +olfactory neuronal cells from living individuals to understand gene regulation. Sci. Adv., 4(12):eaav8550, +2018. Cited on page 1. +[35] Mathieu Rousseau, James Fraser, Maria A Ferraiuolo, Jos´ee Dostie, and Mathieu Blanchette. Three- +dimensional modeling of chromatin structure from interaction frequency data using markov chain monte +carlo sampling. Bioinform., 12(1):414, 2011. Cited on page 1. +[36] Peter H Sch¨onemann. A generalized solution of the orthogonal Procrustes problem. Psychometrika, 31(1):1– +10, 1966. Cited on page 14. +[37] Mark R Segal. Can 3D diploid genome reconstruction from unphased Hi-C data be salvaged? NAR Genomics +and Bioinformatics, 4(2):lqac038, 2022. Cited on page 2. +[38] Andrew J. Sommese and Charles W. Wampler. Numerical Solution Of Systems Of Polynomials Arising In +Engineering And Science. World Scientific Publishing Company, 2005. Cited on page 11. +[39] Rishi Sonthalia, Greg Van Buskirk, Benjamin Raichel, and Anna Gilbert. How can classical multidimensional +scaling go wrong? Adv. Neural Inf. Process. Syst, 34:12304–12315, 2021. Cited on page 6. + +3D GENOME RECONSTRUCTION FROM PARTIALLY PHASED HI-C DATA +19 +[40] Longzhi Tan, Dong Xing, Chi-Han Chang, Heng Li, and X Sunney Xie. Three-dimensional genome structures +of single diploid human cells. Science, 361(6405):924–928, 2018. Cited on page 2. +[41] Caroline Uhler and GV Shivashankar. Regulation of genome organization and gene expression by nuclear +mechanotransduction. Nat. Rev. Mol. Cell Biol., 18(12):717–727, 2017. Cited on page 1. +[42] Nelle Varoquaux, Ferhat Ay, William Stafford Noble, and Jean-Philippe Vert. A statistical approach for +inferring the 3D structure of the genome. Bioinformatics, 30(12):i26–i33, 2014. Cited on pages 1, 4, and 5. +[43] Haifeng Wang, Xiaoshu Xu, Cindy M Nguyen, Yanxia Liu, Yuchen Gao, Xueqiu Lin, Timothy Daley, +Nathan H Kipniss, Marie La Russa, and Lei S Qi. CRISPR-mediated programmable 3D genome positioning +and nuclear organization. Cell, 175(5):1405–1417, 2018. Cited on page 1. +[44] Kilian Q Weinberger, Fei Sha, Qihui Zhu, and Lawrence K Saul. Graph Laplacian regularization for large- +scale semidefinite programming. In Advances in neural information processing systems, pages 1489–1496, +2007. Cited on page 6. +[45] Tiantian Ye and Wenxiu Ma. Ashic: Hierarchical Bayesian modeling of diploid chromatin contacts and +structures. Nucleic Acids Res., 48(21):e123–e123, 2020. Cited on page 2. +[46] ZhiZhuo Zhang, Guoliang Li, Kim-Chuan Toh, and Wing-Kin Sung. Inference of spatial organizations of +chromosomes using semi-definite embedding approach and Hi-C data. In Annual international conference +on research in computational molecular biology, pages 317–332. Springer, 2013. Cited on pages 1, 3, 4, 5, 6, +and 10. +[47] Shenglong Zhou, Naihua Xiu, and Hou-Duo Qi. Robust Euclidean embedding via EDM optimization. Math. +Program. Comput., 12(3):337–387, 2020. Cited on page 6. +Authors’ addresses: +Diego Cifuentes, Georgia Institute of Technology +diego.cifuentes@isye.gatech.edu +Jan Draisma, University of Bern +jan.draisma@math.unibe.ch +Oskar Henriksson, University of Copenhagen +oskar.henriksson@math.ku.dk +Annachiara Korchmaros, University of Leipzig +annachiara@bioinf.uni-leipzig.de +Kaie Kubjas, Aalto University +kaie.kubjas@aalto.fi +Supplementary Material +In this part of the paper, we include additional details and figures for the experiments in +section 5. +Figure S1 shows reconstructions of the same chromosomes as displayed in Figure 3 but with- +out the local optimization step, indicating that semidefinite programming, numerical algebraic +geometry and clustering alone can recover the main features of the 3D structure. +Figure S2 illustrates the preprocessing steps of the real dataset where loci with low contact +counts are removed and the rest of the loci are partitioned into unambiguous and ambiguous. +The total contact count for the ith locus is defined as the sum of all contacts where it participates: +T(i) = +� +j∈[n] +� +cA(i, j) + cP (i, j) + cP (i + n, j) +� ++ +� +j∈[2n] +� +cP (j, i) + cU(i, j) + cU(i + n, j) +� +. +Similarly, we define the unambiguity quotient as the proportion of T(i) that consists of contacts +where xi and yi could be distinguished: +UQ(i) = +1 +T(i) +� +� � +j∈[n] +� +cP (i, j) + cP (i + n, j) +� ++ +� +j∈[2n] +� +cU(i, j) + cU(i + n, j) +� +� +� . +To obtain the setting of our paper where loci are partitioned into unambiguous or ambiguous, +we reassign the contact counts of ˜CU ˜CP and ˜CA of the Patski dataset according to whether a + +20 +D. CIFUENTES, J. DRAISMA, O. HENRIKSSON, A. KORCHMAROS, AND K. KUBJAS +-5 +0 +5 +10 +-4 +-2 +0 +2 +4 +6 +8 +RMSD = 0.58575 +X true +X estimated +Y true +Y estimated +Start +Unambiguous +(a) ε = 0.10 +-6 +-4 +-2 +0 +2 +4 +6 +8 +10 +-4 +-2 +0 +2 +4 +6 +8 +RMSD = 0.86406 +X true +X estimated +Y true +Y estimated +Start +Unambiguous +(b) ε = 0.50 +-6 +-4 +-2 +0 +2 +4 +6 +8 +10 +-4 +-2 +0 +2 +4 +6 +8 +RMSD = 1.2564 +X true +X estimated +Y true +Y estimated +Start +Unambiguous +(c) ε = 0.90 +Figure S1. SNLC reconstructions, without the local optimization step. +0 +50 +100 +150 +200 +250 +300 +350 +i +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +5 +104 +The i-th smallest total contact count +(a) +0 +50 +100 +150 +200 +250 +300 +350 +i +1 +1.2 +1.4 +1.6 +1.8 +2 +2.2 +2.4 +2.6 +2.8 +3 Ratio between the i-th and (i+1)-th smallest total contact count +(b) +0 +50 +100 +150 +200 +250 +300 +i +0.25 +0.3 +0.35 +0.4 +0.45 +0.5 +0.55 +0.6 +The i-th smallest unambiguity quotient +(c) +Figure S2. (a) Total contact counts sorted in increasing order. (b) Ratios between total contact counts. +The peak corresponding to the ratio between the 48th and the 47th smallest count is used as a motivation +for excluding the 47 loci with smallest total contact from the analysis. (c) Unambiguity quotients for +each of the remaining 296 loci, sorted in increasing order. We consider a locus as ambiguous if this ratio +is less than 0.4; otherwise, we consider it as unambiguous. +locus is unambiguous or ambiguous. For i, j ∈ U, we define +cU +i,j = ˜cU +i,j + ˜cP +i,j +˜cU +i,j +˜cU +i,j + ˜cU +i,j+n ++ ˜cP +j,i +˜cU +i,j +˜cU +i,j + ˜cU +i+n,j ++ ˜cA +i,j +˜cU +i,j +˜cU +i,j + ˜cU +i,j+n + ˜cU +i+n,j + ˜cU +i+n,j+n +, +cU +i,j+n = ˜cU +i,j+n + ˜cP +i,j +˜cU +i,j+n +˜cU +i,j + ˜cU +i,j+n ++ ˜cP +j+n,i +˜cU +i,j+n +˜cU +i,j+n + ˜cU +i+n,j+n ++ ˜cA +i,j +˜cU +i,j+n +˜cU +i,j + ˜cU +i,j+n + ˜cU +i+n,j + ˜cU +i+n,j+n +, +cU +i+n,j = ˜cU +i+n,j + ˜cP +i+n,j +˜cU +i+n,j +˜cU +i+n,j + ˜cU +i+n,j+n ++ ˜cP +j,i +˜cU +i+n,j +˜cU +i,j + ˜cU +i+n,j ++ ˜cA +i,j +˜cU +i+n,j +˜cU +i,j + ˜cU +i,j+n + ˜cU +i+n,j + ˜cU +i+n,j+n +, +cU +i+n,j+n = ˜cU +i+n,j+n + ˜cP +i+n,j +˜cU +i+n,j+n +˜cU +i+n,j + ˜cU +i+n,j+n ++ ˜cP +j+n,i +˜cU +i+n,j+n +˜cU +i,j+n + ˜cU +i+n,j+n ++ ˜cA +i,j +˜cU +i+n,j+n +˜cU +i,j + ˜cU +i,j+n + ˜cU +i+n,j + ˜cU +i+n,j+n +. +For i ∈ U, j ∈ A, we define +cP +i,j = ˜cU +i,j + ˜cU +i,j+n + ˜cP +i,j + ˜cP +j,i +˜cU +i,j +˜cU +i,j + ˜cU +i+n,j ++ ˜cP +j+n,i +˜cU +i,j+n +˜cU +i,j+n + ˜cU +i+n,j+n ++ ˜cA +i,j +˜cP +i,j +˜cP +i,j + ˜cP +i+n,j +, + +3D GENOME RECONSTRUCTION FROM PARTIALLY PHASED HI-C DATA +21 +cP +i+n,j = ˜cU +i+n,j + ˜cU +i+n,j+n + ˜cP +i+n,j + ˜cP +j,i +˜cU +i+n,j +˜cU +i,j + ˜cU +i+n,j ++ ˜cP +j+n,i +˜cU +i+n,j+n +˜cU +i,j+n + ˜cU +i+n,j+n ++ ˜cA +i,j +˜cP +i+n,j +˜cP +i,j + ˜cP +i+n,j +. +Finally, for i, j ∈ A, we define +cA +i,j = ˜cU +i,j + ˜cU +i,j+n + ˜cU +i+n,j + ˜cU +i+n,j+n + ˜cP +i,j + ˜cP +i+n,j + ˜cP +j,i + ˜cP +j+n,i + ˜cA +i,j. +In Figure S3, the experimental contact counts from the Patski dataset are compared with the +contact counts from the SNLC reconstruction. + +22 +D. CIFUENTES, J. DRAISMA, O. HENRIKSSON, A. KORCHMAROS, AND K. KUBJAS +Patski data +100 +200 +300 +400 +500 +50 +100 +150 +200 +250 +300 +350 +400 +450 +500 +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +(a) +SNLC reconstruction +100 +200 +300 +400 +500 +50 +100 +150 +200 +250 +300 +350 +400 +450 +500 +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +(b) +Patski data +5 +10 +15 +20 +25 +30 +35 +40 +45 +50 +100 +150 +200 +250 +300 +350 +400 +450 +500 +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +(c) +SNLC reconstruction +5 +10 +15 +20 +25 +30 +35 +40 +45 +50 +100 +150 +200 +250 +300 +350 +400 +450 +500 +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +(d) +Patski data +5 +10 +15 +20 +25 +30 +35 +40 +45 +5 +10 +15 +20 +25 +30 +35 +40 +45 +0 +0.5 +1 +1.5 +2 +2.5 +3 +(e) +SNLC reconstruction +5 +10 +15 +20 +25 +30 +35 +40 +45 +5 +10 +15 +20 +25 +30 +35 +40 +45 +0 +0.5 +1 +1.5 +2 +2.5 +3 +(f) +Figure S3. Logarithmic heat maps for the reassigned contact count matrices obtained from the original +Patski dataset and from the SNLC reconstruction: (a) and (b) CU; (c) and (d) CP ; (e) and (f) CA. +The axis labels correspond to the 500 unambiguous beads, and the 46 ambiguous loci. + diff --git a/2dFKT4oBgHgl3EQfPy1W/content/tmp_files/load_file.txt b/2dFKT4oBgHgl3EQfPy1W/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..11b2e37ff0f85ef40ce5c436c82e05f7643bd8ba --- /dev/null +++ b/2dFKT4oBgHgl3EQfPy1W/content/tmp_files/load_file.txt @@ -0,0 +1,1238 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf,len=1237 +page_content='3D GENOME RECONSTRUCTION FROM PARTIALLY PHASED HI-C DATA DIEGO CIFUENTES, JAN DRAISMA, OSKAR HENRIKSSON, ANNACHIARA KORCHMAROS, AND KAIE KUBJAS Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' The 3-dimensional (3D) structure of the genome is of significant importance for many cellular processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' In this paper, we study the problem of reconstructing the 3D struc- ture of chromosomes from Hi-C data of diploid organisms, which poses additional challenges compared to the better-studied haploid setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' With the help of techniques from algebraic geometry, we prove that a small amount of phased data is sufficient to ensure finite identifi- ability, both for noiseless and noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' In the light of these results, we propose a new 3D reconstruction method based on semidefinite programming, paired with numerical algebraic ge- ometry and local optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' The performance of this method is tested on several simulated datasets under different noise levels and with different amounts of phased data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' We also apply it to a real dataset from mouse X chromosomes, and we are then able to recover previously known structural features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Introduction The eukaryotic chromatin has a three-dimensional (3D) structure in the cell nucleus which has been shown to be important in regulating basic cellular functions, including gene regulation, transcription, replication, recombination, and DNA repair [41, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' The 3D DNA organization is also associated to brain development and function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' in particular, it is shown to be misregulated in schizophrenia [32, 34] and Alzheimer’s disease [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' All genetic material is stored in chromosomes which interact in the cell nucleus, and the 3D chromatin structure influences the frequencies of such interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' A benchmark tool to measure such frequencies is high-throughput chromosome conformation capture (Hi-C) [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Hi-C first crosslinks cell genomes, which “freezes” contacts between DNA segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Then the genome is cut in fragments, the fragments are ligated together and then are associated to equally-sized segments of the genome using high-throughput sequencing [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' These segments of the genome are called loci and their size is known as resolution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=', bins of size 1Mb or 50Kb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' The result of Hi-C is stored in a matrix called contact matrix whose elements are the contact counts between pairs of loci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' According to the structure they generate, computational methods for inferring the 3D chro- matin structure from a contact matrix fall into two classes: ensemble and consensus methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' In a haploid setting (organisms having a single set of chromosomes), ensemble models such as MCMC5C [35], BACH-MIX [11] and Chrom3D [30], try to account for structure variations on the genome across cells by inferring a population of 3D structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' On the other hand, consensus methods aim at reconstructing one single 3D structure which may be used as a model for fur- ther analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' In this category, probability-based methods such as PASTIS [42, 4] model contact counts as Poisson random variables of the Euclidean distances between loci, and distance-based methods such as ChromSDE [46] and ShRec3D [17] model contact counts as functions of the Euclidean distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' An extensive overview of different 3D genome reconstruction techniques is given in [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Date: January 30, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='11764v1 [q-bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='GN] 27 Jan 2023 2 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' CIFUENTES, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' DRAISMA, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' HENRIKSSON, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' KORCHMAROS, AND K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' KUBJAS Most of the methods for 3D genome reconstructions from Hi-C data are for haploid organisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' However, humans like most mammals are diploid organisms, in which the genetic information is stored in pairs of chromosomes called homologs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Homologous chromosomes are almost identical besides some single nucleotide polymorphisms (SNPs) [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' In the case of diploid organisms, the Hi-C data does not generally differentiate between homologous chromosomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' If we model each chromosome as a string of beads, then we associate two beads to each locus i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' , n}, one bead for each homolog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Therefore, each observed contact count ci,j between loci i and j represents aggregated contacts of four different types of interactions, more precisely one of the two homologous beads associated to locus i gets in contact with one of the two homologous beads associated to locus j, see Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' This means that the Hi-C data is unphased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Phased Hi-C data that distinguishes contacts for homologs is rare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' In our setting, we assume that the data is partially phased, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=', some of the contact counts can be associated with a homolog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' For example, in the (mouse) Patski (BL6xSpretus) [6, 45] cell line, 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='6% of the contact counts are phased;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' while this value is as low as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='14% in the human GM12878 cell line [33, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Therefore, methods for inferring diploid 3D chromatin structure need to take into account the ambiguity of diploid Hi-C data to avoid inaccurate reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Ambiguity of phased data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Each entry ci,j of the Hi-C matrix corresponds to four different contacts between the two pairs (xi, yi) for locus i and (xj, yj) for locus j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Methods for 3D genome reconstruction in diploid organisms have been studied in [40, 4, 23, 2, 22, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' One approach is to phase Hi-C data [40, 23, 22], for example by assigning haplotypes to contacts based on assignments at neighboring contacts [40, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Cauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' [4] models contact counts as Poisson random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' To find the optimal 3D chromatin structure, the associated likelihood function combined with two structural constraints is maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' The first constraint imposes that the distances between neighboring beads are similar and the second one requires that homologous chromosomes are located in different regions of the cell nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Belyaeva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' [2] shows identifiability of the 3D structure when the Euclidean distances between neighboring beads and higher-order contact counts between three or more loci simultaneously are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Under these assumptions, the 3D reconstruction is obtained by combining distance geometry with semidefinite programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Segal [37] applies recently developed imaging technology, in situ genome sequencing (IGS) [31], to point out issues in the assumptions made in [40, 4, 2], and suggests as alternative assumptions that intra-homolog distances are smaller than corresponding inter-homolog distances and intra-homolog distances are similar for homologous chromosomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' ci Reference Genome Homologous Chromosomes HiC-matrix3D GENOME RECONSTRUCTION FROM PARTIALLY PHASED HI-C DATA 3 IGS [31] provides yet another method for inferring the 3D structure of the genome, however, at present the resolution and availability of IGS data is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' In this work, we focus on a distance-based approach for partially phased Hi-C data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' In particular, we assume that contacts only for some loci are phased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' In the string of beads model, the locations of the pair of beads associated to i-th loci are denoted by xi, yi ∈ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Then homologs are represented by two sequences x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' , xn and y1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' , yn in R3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' see Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Inferring the 3D chromatin structure corresponds to estimating the bead coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Based on Lieberman-Aiden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' [21], we assume the power law dependency ci,j = γdα i,j, where α is a negative conversion factor, between the distance di,j and contact count ci,j of loci i and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Following Cauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' [4], we assume that a contact count between loci is given by the sum of all possible contact counts between the corresponding beads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' We call a bead unambiguous if the contacts for the corresponding locus are phased;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' otherwise we call a bead ambiguous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Our first main contribution is to show that for negative rational conversion factors α, knowing the locations of six unambiguous beads ensures that there are generically finitely many possible locations for the other beads, both in the noiseless (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='1) and noisy (Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='5) setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Moreover, we prove finite identifiability also in the fully ambiguous setting when α = −2 and the number of loci is at least 13 (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Note that the identifiability does not hold for α = 2 as shown in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Our second main contribution is to provide a reconstruction method when α = −2, based on semidefinite programming combined with numerical algebraic geometry and local optimiza- tion (section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' The general idea is the following: We first estimate the coordinates of the unambiguous beads using only the unambiguous contact counts (which precisely corresponds to the haploid setting) using the SDP-based solver implemented in ChromSDE [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' We then exploit our theoretical result on finite identifiability to estimate the coordinates of the ambigu- ous beads, one by one, by solving several polynomial systems numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' These estimates are then improved by a local estimation step that take into account all contact counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Finally, a clustering algorithm is used to overcome the symmetry (xi, yi) �→ (yi, xi) in the estimation for the ambiguous beads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' In section 2, we introduce our mathematical model for the 3D genome reconstruction problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' In section 3, we recall identifiability results in the un- ambigous setting (section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='1), and then prove identifiability results in the partially ambiguous setting (section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='2) and in the fully ambiguous setting (section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' We describe our recon- struction method in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' We test the performance of our method on synthetic datasets and on a real dataset from the mouse X chromosomes in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' We conclude with a dis- cussion about future research directions in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' The code for computations and exper- iments is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='com/kaiekubjas/3D-genome-reconstruction-from- partially-phased-HiC-data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Mathematical model for 3D genome reconstruction In this section we introduce the distance-based model under which we study 3D genome re- construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' In section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='1 we give the background on contact count matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' In section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='2 we describe a power-law between contacts and distances, which allows to translate the information about contacts into distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Contact count matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' We model the genome as a string of 2n beads, corresponding to n pairs of homologous beads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' The positions of the beads are recorded by a matrix Z = [x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' , xn, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' , yn]T ∈ R2n×3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' 4 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' CIFUENTES, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' DRAISMA, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' HENRIKSSON, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' KORCHMAROS, AND K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' KUBJAS The positions xi and yi correspond to homologous beads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' When convenient, we use the notation z1 := x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' , zn := xn, zn+1 := y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' , z2n := yn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' In this notation, Z = [z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' , zn, zn+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' , z2n]T ∈ R2n×3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Let U be the subset of pairs that are unambiguous, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=', beads in the pair can be distinguished, and let A be the subset of pairs that are ambiguous, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=', beads in the pair cannot be distin- guished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' The sets U and A form a partition of [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' A Hi-C matrix C is a matrix with each row and column corresponding to a genomic locus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Following Cauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' [4], we call these contact counts ambiguous and denote the corresponding contact count matrix by CA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' If parental genotypes are available, then one can use SNPs to map some reads to each haplotype [6, 24, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' If both ends of a read contains SNPs that can be associated to a single parent, then the contact count is called unambiguous and the corresponding contact count matrix is denoted by CU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Finally, if only one of the genomic loci present in an interaction can be mapped to one of the homologous chromosomes, then the count is called partially ambiguous and the contact count matrix is denoted by CP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' The unambiguous count matrix CU is a 2n×2n matrix with the first n indices corresponding to x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' , xn and the last n indices corresponding to y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' , yn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' The ambiguous count matrix CA is an n×n matrix and we assume that each ambiguous count is the sum of four unambiguous counts: cA i,j = cU i,j + cU i,j+n + cU i+n,j + cU i+n,j+n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' The partially ambiguous count matrix CP is a 2n×n matrix and each partially ambiguous count is the sum of two unambiguous counts: cP i,j = cU i,j + cU i,j+n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' xi xj yi yj (a) cA i,j for i, j ∈ A xi xj yi yj (b) cP i,j for i ∈ U, j ∈ A xi xj yi yj (c) cP i+n,j for i ∈ U, j ∈ A xi xj yi yj (d) cU i,j for i, j ∈ U xi xj yi yj (e) cU i,j+n for i, j ∈ U xi xj yi yj (f) cU i+n,j for i, j ∈ U xi xj yi yj (g) cU i+n,j+n for i, j ∈ U Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Seven different types of contacts between the ith and jth locus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Contacts and distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Denoting the distance ∥zi − zj∥ between zi and zj by di,j, the power law dependency observed by Lieberman-Aiden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' [21] can be written as cU i,j = γdα i,j, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='1) where α < 0 is a conversion factor and γ > 0 is a scaling factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' This relationship between contact counts and distances is assumed in [2, 46], while in [4, 42] the contact counts ci,j are modeled as Poisson random variables with the Poisson parameter being βdα i,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' 3D GENOME RECONSTRUCTION FROM PARTIALLY PHASED HI-C DATA 5 In our paper, we assume that contact counts are related to distances by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Similarly to [2], we set γ = 1 and in parts of the article α = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' In general, the conversion factor α depends on a dataset and its estimation can be part of the reconstruction problem [42, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Setting γ = 1 means that we recover the configuration up to a scaling factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' In practice, the configuration can be rescaled using biological knowledge, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=', the radius of the nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Our approach to 3D genome reconstruction builds on the power law dependency between contacts and distances between unambiguous beads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' We convert the empirical contact counts to Euclidean distances and then aim to reconstruct the positions of beads from the distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' This leads us to the following system of equations: � � � � � � � � � cA i,j = ∥xi − xj∥α + ∥xi − yj∥α + ∥yi − xj∥α + ∥yi − yj∥α ∀i, j ∈ A cP i,j = ∥xi − xj∥α + ∥xi − yj∥α, cP i+n,j = ∥yi − xj∥α + ∥yi − yj∥α ∀i ∈ U, j ∈ A cU i,j = ∥xi − xj∥α, cU i,j+n = ∥xi − yj∥α, cU i+n,j = ∥yi − xj∥α, cU i+n,j+n = ∥yi − yj∥α ∀i, j ∈ U (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='2) If α is an even integer, then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='2) is a system of rational equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Determining the points xi, yi, where i ∈ U, is the classical Euclidean distance problem: We know the (noisy) pairwise distances between points and would like to construct the locations of points, see section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='1 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Hence after section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='1 we assume that we have estimated the locations of points xi, yi, where i ∈ U, and we would like to determine the points xi, yi, where i ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Identifiability In this section, we study the uniqueness of the solutions of the system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='2) up to rigid transformations (translations, rotations and reflections), or in other words, the identifiability of the locations of beads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' We study the unambiguous, partially ambiguous and ambiguous settings in sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Unambiguous setting and Euclidean distance geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' If all pairs are unambigu- ous, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=', U = [n], then constructing the original points translates to a classical problem in Euclidean distance geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' The principal task in Euclidean distance geometry is to construct original points from pairwise distances between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' In the rest of the subsection, we will recall how to solve this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Since pairwise distances are invariant under translations, rotations and reflections (rigid transformations), then the original points can be reconstructed up to rigid transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' For an overview of distance geometry and Euclidean distance matrices, we refer the reader to [7, 15, 20, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' The Gram matrix of the points z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' , z2n is defined as G = ZZT = [z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' , z2n]T · [z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' , z2n] ∈ R2n×2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Let z = 1 2n �2n i=1 zi and ˜zi = zi − z for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' , 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' The matrix ˜Z = [˜z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' , ˜z2n]T gives the locations of points after centering them around the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Let ˜G denote the Gram matrix of the centered point configuration ˜z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' , ˜z2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Let Di,j = ∥zi−zj∥2 denote the squared Euclidean distance between the points zi and zj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' The Euclidean distance matrix of the points z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' , z2n is defined as D = (Di,j)1≤i,j≤2n ∈ R2n×2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' To express the centered Gram matrix in terms of the Euclidean distance matrix, we define the geometric centering matrix J = I2n − 1 2n11T , 6 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' CIFUENTES, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' DRAISMA, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' HENRIKSSON, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' KORCHMAROS, AND K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' KUBJAS where I2n is the 2n × 2n identity matrix and 1 is the vector of ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' The linear relationship between ˜G and D is given by ˜G = −1 2JDJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Therefore, given the Euclidean distance matrix, we can construct the centered Gram matrix for the points z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' , z2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' The centered points up to rigid transformations are extracted from the centered Gram matrix ˜G using the eigendecomposition ˜G = QΛQ−1, where Q is orthonormal and Λ is a diagonal matrix with entries ordered in decreasing order λ1 ≥ λ2 ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' ≥ λ2n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' We define Λ1/2 3 := [diag(√λ1, √λ2, √λ3), 03×(2n−3)]T and set ˆZ = QΛ1/2 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' In the case of noiseless distance matrix D, the Gram matrix ˜G has rank three and the diagonal matrix Λ has precisely three non-zero entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Hence we could obtain ˆZ also from QΛ1/2 by truncating zero columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Using Λ1/2 3 has the advantage that it gives an approximation for the points also for a noisy distance matrix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' The uniqueness of ˆZ up to rotations and reflections follows from [14, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='2], which states that AAT = BBT if and only if A = BQ for some orthogonal matrix Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' The procedure that transforms the distance matrix to origin centered Gram matrix and then uses eigendecomposition for constructing original points is called classical multidimensional scal- ing (cMDS) [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Although cMDS is widely used in practice, it does not always find the distance matrix that minimizes the Frobenius norm to the empirical noisy distance matrix [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Other approaches to solving the Euclidean distance and Euclidean completion problems include non- convex [9, 25] as well semidefinite formulations [1, 10, 27, 44, 46, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Partially ambiguous setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' The next theorem establishes the uniqueness of the solu- tions of the system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='2) in the presence of ambiguous pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' In particular, it states that there are finitely many possible locations for beads in one ambiguous pair given the locations of six unambiguous beads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' The identifiability results in this subsection hold for all negative rational numbers α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' In the rest of the paper, we denote the true but unknown coordinates by x∗ and the symbol x stands for a variable that we want to solve for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' We write ∥ · ∥ for the standard inner product on R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Let α be a negative rational number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Then for a∗, b∗, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' , f∗, x∗, y∗ ∈ R3 suffi- ciently general, the system of six equations ∥x − t∗∥α + ∥y − t∗∥α = ∥x∗ − t∗∥α + ∥y∗ − t∗∥α for t∗ = a∗, b∗, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' , f∗ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='1) in the six unknowns x1, x2, x3, y1, y2, y3 ∈ R has only finitely many solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' The proof will show that this system has only finitely many solutions over the complex numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' We believe that the theorem holds for general nonzero rational α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Indeed, our argument works, with a minor modification, also for α > 2, but for α in the range (0, 2] a refinement of the argument is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' First write Q(x) := x2 1 + x2 2 + x2 3, so that ∥x∥ = � Q(x) for x ∈ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' The advantage of Q over ∥x∥ is that it is well-defined on C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Write α 2 = m n with m, n integers, m ̸= 0, and n > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Consider the affine variety X ⊆ (C3)8 × (C2)6 consisting of all tuples ((a∗, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' , f∗, x∗, y∗), (rt∗, st∗)t∗=a∗,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=',f∗) such that Q(x∗ − t∗)m = rn t∗ ̸= 0 and Q(y∗ − t∗)m = sn t∗ ̸= 0 for t∗ = a∗, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' , f∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' 3D GENOME RECONSTRUCTION FROM PARTIALLY PHASED HI-C DATA 7 Note that, if x∗, t∗ are real, then it follows that Q(x∗ − t∗)m = (∥x∗ − t∗∥α)n, and similarly for Q(y∗ − t∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Hence if a∗, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' , y∗ are all real, then the point ((a∗, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' , f∗, x∗, y∗), (∥x∗ − t∗∥α, ∥y∗ − t∗∥α)t∗) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='2) is a point in X with real-valued coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' The projection π from X to the open affine subset U ⊆ (C3)8 where all Q(x∗−t∗) and Q(y∗−t∗) are nonzero is a finite morphism with fibres of cardinality n12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' to see this cardinality note that there are n possible choices for each of the numbers rt∗, st∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Each irreducible component of X is a smooth variety of dimension 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Consider the map ψ : X → (C3 × C1)6 defined by ((a∗, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' , f∗, x∗, y∗), (rt∗, st∗)t∗) �→ ((t∗, rt∗ + st∗))t∗ We claim that for q in some open dense subset of X, the derivative dqψ has full rank 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' For this, it suffices to find one point p ∈ U such that dqψ has rank 24 at each of the n12 points q ∈ π−1(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' We take a real-valued point p := (a∗, b∗, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' , f∗, x∗, y∗) ∈ (R3)8 to be specified later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Let q ∈ π−1(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Then, near q, the map ψ factorises via π and the unique algebraic map ψ′ : U → (C3 × C1)6 (defined near p) which on a neighbourhood of p in U ∩ (R3)8 equals ψ′(a, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' , f, x, y) = ((t, ξt∗ · Q(x − t)α/2 + ηt∗ · Q(y − t)α/2))t=a,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=',f ∈ (C3 × C1)6 where ξt∗ and ζt∗ are n-th roots of unity in C depending on which q is chosen among the n12 points in π−1(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' The situation is summarised in the following diagram: (X, q) π � ψ � (U, p) ψ′ � ((C3 × C1)6, ψ(q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Now, dqψ = dpψ′ ◦ dqπ, and since dqπ is a linear isomorphism, it suffices to prove that dpψ′ is a linear isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Suppose that (a′, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' , f′, x′, y′) ∈ ker dpψ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Then, since the map ψ′ remembers a, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' , f, it follows immediately that a′ = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' = f′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' On the other hand, by differentiating we find that, for each t∗ ∈ {a∗, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' , f∗}, ξt∗ · (α/2) · Q(x∗ − t∗)α/2−1 · 2 · ⟨x′, x∗ − t∗⟩ +ηt∗ · (α/2) · Q(y∗ − t∗)α/2−1 · 2 · ⟨y′, y∗ − t∗⟩ = 0, where ⟨·, ·⟩ stands for the standard bilinear form on C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' In other words, the vector (x′, y′) ∈ C6 is in the kernel of the 6 × 6-matrix M := � �� ∥x∗ − a∗∥α−2 · ξa∗ · (x∗ − a∗) ∥y∗ − a∗∥α−2 · ηa∗ · (y∗ − a∗) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' ∥x∗ − f∗∥α−2 · ξf∗ · (x∗ − f∗) ∥y∗ − f∗∥α−2 · ηf∗ · (y∗ − f∗) � �� where we have interpreted a∗, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' , f∗, x∗, y∗ as row vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' It suffices to show that, for some specific choice of p = (a∗, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' , f∗, x∗, y∗) ∈ (R3)8, this matrix is nonsingular for all n12 choices of ((ξt∗, ηt∗))t∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' We choose a∗, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' , f∗, x∗, y∗ as the vertices of the unit cube, as follows: a∗ = (1, 0, 0) b∗ = (0, 1, 0) c∗ = (0, 0, 1) c∗ = (0, 1, 1) d∗ = (1, 0, 1) f∗ = (1, 1, 0) x∗ = (0, 0, 0) y∗ = (1, 1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' 8 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' CIFUENTES, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' DRAISMA, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' HENRIKSSON, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' KORCHMAROS, AND K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' KUBJAS Then the matrix M becomes, with β = α − 2: � ��������� −ξa∗ 0 0 0 2 β 2 · ηa∗ 2 β 2 · ηa∗ 0 −ξb∗ 0 2 β 2 · ηb∗ 0 2 β 2 · ηb∗ 0 0 −ξc∗ 2 β 2 · ηc∗ 2 β 2 · ηc∗ 0 0 −(2 β 2 · ξd∗) −(2 β 2 · ξd∗) ηd∗ 0 0 −(2 β 2 · ξe∗) 0 −(2 β 2 · ξe∗) 0 ηe∗ 0 −(2 β 2 · ξf∗) −(2 β 2 · ξf∗) 0 0 0 ηf∗ � ��������� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Now, det(M) equals − ξa∗ · ξb∗ · ξc∗ · ηd∗ · ηe∗ · ηf∗ + 22+3β · ηa∗ · ηb∗ · ηc∗ · ξd∗ · ξe∗ · ξf∗ + 22β · R (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='3) where R is a sum of (products of) roots of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Now α < 0 implies that β < −2, so that 2 + 3β < 2β < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Since roots of unity have 2-adic valuation 0, the second term in the expression above is the unique term with minimal 2-adic valuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Hence det(M) ̸= 0, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' It follows that ψ is a dominant morphism from each irreducible component of X into (C3 × C1)6, and hence for all q in an open dense subset of X, the fibre ψ−1(ψ(q)) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' This then holds, in particular, for q in an open dense subset of the real points as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' This proves the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' If α > 2, then β > 0, and hence the unique term with minimal 2-adic valuation in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='3) is the first term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' This can be used to show that the theorem holds then, as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' The only subtlety is that for positive α, solutions where x or y equal one of the points a∗, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' , f∗ are not automatically excluded, and these are not seen by the variety X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' But a straightforward argument shows that such solutions do not exist for sufficiently general choices of a∗, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' , f∗, x∗, y∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' We now consider the setting when we know locations of seven unambiguous beads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' In the special case when α = −2, we construct the ideal generated by the polynomials obtained from rational equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='1) for seven unambiguous beads after moving all terms to one side and clearing the denominators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Based on symbolic computations in Macaulay2 for the degree of this ideal, we conjecture that the location of a seventh unambiguous bead guarantees unique identifiability of an ambiguous pair of beads: Conjecture 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Let a∗, b∗, c∗, d∗, e∗, f∗, g∗, x∗, y∗ ∈ R3 be sufficiently general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' The system of rational equations 1 ∥t∗ − x∗∥2 + 1 ∥t∗ − y∗∥2 = 1 ∥t∗ − x∥2 + 1 ∥t∗ − y∥2 for t∗ = a∗, b∗, c∗, d∗, e∗, f∗, g∗ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='4) has precisely two solutions (x∗, y∗) and (y∗, x∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' In practice, we only have noisy estimates a, b, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' , f ∈ R3 of the true positions of unambiguous beads a∗, b∗, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' , f∗ ∈ R3, and we have noisy observations ct of the true contact counts c∗ t := ∥x∗ − t∗∥α + ∥y∗ − t∗∥α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' We aim to find x, y ∈ R3 such that ∥x − t∥α + ∥y − t∥α = ct for t = a, b, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' , f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' We may write ct = ∥x∗ − t∥α + ∥y∗ − t∥α + ϵt for some ϵt that depends on the noise level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Hence, the above system of equations can be rephrased as ∥x − t∥α + ∥y − t∥α = ∥x∗ − t∥α + ∥y∗ − t∥α + ϵt for t = a, b, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' , f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='5) In the following corollary we show that this system has generically finitely many solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Let α be a negative rational number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Then for a, b, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' , f, x∗, y∗ ∈ R3 and ϵa, ϵb, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' , ϵf ∈ R sufficiently general, the system of six equations ∥x − t∥α + ∥y − t∥α = ∥x∗ − t∥α + ∥y∗ − t∥α + ϵt for t = a, b, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' , f (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='6) 3D GENOME RECONSTRUCTION FROM PARTIALLY PHASED HI-C DATA 9 in the six unknowns x1, x2, x3, y1, y2, y3 ∈ R has only finitely many solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Recall the map ψ : X → (C3 × C1)6 from the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='1 defined by ((a, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' , f, x∗, y∗), (rx∗,t, sy∗,t)t) �→ ((t, rx∗,t + sy∗,t))t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' We showed that ψ is a dominant morphism from each irreducible component of X into (C3×C1)6, and that each irreducible component of X is 24-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Every solution to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='6) is the (x, y)- component of a point in the fibre ψ−1((t, ||x∗ − t||α + ||y∗ − t||α + ϵt))t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Since this is a fibre over a sufficiently general point, the fibre is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' □ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='5 will be the basis of a numerical algebraic geometric based reconstruction method in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Ambiguous setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Finally we consider the ambiguous setting, where one would like to reconstruct the locations of beads only from ambiguous contact counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' It is shown in [2] that for α = 2, one does not have finite identifiability no matter how many pairs of ambiguous beads one considers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' We show finite identifiability for the locations of beads given contact counts for 13 pairs of ambiguous beads for α = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' We believe that the result might be true for further conversion factors α’s, however our proof technique does not directly generalize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Let α = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Then for x∗ 1, y∗ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' , x∗ 12, y∗ 12 ∈ R3 sufficiently general, the system of 66 equations ∥xi − xj∥α + ∥xi − yj∥α + ∥yi − xj∥α + ∥yi − yj∥α = ∥x∗ i − x∗ j∥α + ∥x∗ i − y∗ j ∥α + ∥y∗ i − x∗ j∥α + ∥y∗ i − y∗ j ∥α for 1 ≤ i < j ≤ 12 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content='7) in the 72 unknowns x1,1, x1,2, x1,3, y1,1, y1,2, y1,3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' , x12,1, x12,2, x12,3, y12,1, y12,2, y12,3 ∈ R has only finitely many solutions up to rigid transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' As before, we write Q(x) := x2 1 + x2 2 + x2 3, so that ∥x∥ = � Q(x) for x ∈ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Consider the affine open subset X ⊆ (C3)24 consisting of all tuples (x∗ 1, y∗ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' , x∗ 12, y∗ 12) such that Q(x∗ i − x∗ j) ̸= 0, Q(x∗ i − y∗ j ) ̸= 0, Q(y∗ i − x∗ j) ̸= 0 and Q(y∗ i − y∗ j ) ̸= 0 for 1 ≤ i < j ≤ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' Consider also the map ψ : X → C66 defined by (x∗ 1, y∗ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFKT4oBgHgl3EQfPy1W/content/2301.11764v1.pdf'} +page_content=' , x∗ 12, y∗ 12) �→ (Q(x∗ i − x∗ j)−1 + Q(x∗ i − y∗ j )−1 + Q(y∗ i − x∗ j)−1 + Q(y∗ i − y∗ j )−1)i +PS +(/+<)PS +Input +■ +SM +BS +c +Atom +S +CMR1 +CMT1 +Output +SM, +Zone 0 +Zone 1spontaneous emission decay rate and transition frequency of the atomic transition between |𝑒⟩  +and |↑⟩, respectively. The coupling constant between the cavity mode with frequency 𝜔� and the  +atomic transition is 𝑔. The atom‐cavity detuning is Δ � 𝜔� � 𝜔�. Moreover, we set 𝜅���� as the  +cavity field decay rate into the external light field on the 𝐶𝑀���� side. Considering that the atom  +is hardly excited  in  our scheme, as long as the  condition of slowly varying light intensities is  +satisfied [37,54,57], 𝑆𝑆𝐶� can be well described by the input‐output theory [58,59]. Suppose that  +|𝛼�,�↑⟩ is the incident coherent light field from 𝐶𝑀�� side when the atom is in |↑⟩. The cavity  +reflection |𝛼�,�↑⟩ satisfies [56]  +𝛼�,�↑ � �1 � +2𝜅��𝑖𝛥 � 𝛾� +𝜅�𝑖𝛥 � 𝛾� � 𝑔�� 𝛼�,�↑ � �𝜂�,�↑�𝑒���,�↑𝛼�,�↑, +�2�  +where 𝜅 � 𝜅� � 𝜅�, �𝜂�,�↑� +� is the reflectivity and 𝛽�,�↑ describes the  phase  of the reflection.  +Similarly, for the transmission of the cavity �𝛼�,�↑�, we have   +𝛼�,�↑ � � 2�𝑖𝛥 � 𝛾�√𝜅�𝜅� +𝜅�𝑖𝛥 � 𝛾� � 𝑔� 𝛼�,�↑. +�3�  +Regarding the scattering field �𝛼�,�↑� due to the atomic spontaneous emission, we have  +𝛼�,�↑ � +2𝑔√𝜅�𝛾 +𝜅�𝑖𝛥 � 𝛾� � 𝑔� 𝑎�,�↑. +�4�  +As for the situation that the atom is in |↓⟩, we still assume that the atom is completely unaffected  +by the cavity mode due to the large detuning. Therefore, 𝑆𝑆𝐶� in such case can be treated the  +same as the empty cavity 𝑆𝑆𝐶�. By setting 𝑔 � 0 in Eqs. (2)‐(4), we can immediately obtain the  +corresponding reflection and transmission. As for the scattering light field, it is obviously 0.  +Based on the above mathematical description of 𝑆𝑆𝐶� and 𝑆𝑆𝐶�, we can numerically simulate  +the dynamic evolution process of the input coherent pulse |𝛼⟩ and the fidelity of the output. We  +suppose that the target state is |𝜓�⟩ � �|𝛼⟩|↑⟩ � |�𝛼⟩|↓⟩� √2 +⁄ +, and the final state of the whole  +system  after  𝑀  cycles  is  �𝜓�� � �|𝐶�↑⟩|𝑙𝑜𝑠𝑠↑⟩|↑⟩ � |𝐶�↓⟩|𝑙𝑜𝑠𝑠↓⟩|↓⟩� √2 +⁄ +with  �𝑙𝑜𝑠𝑠↑�↓�� � +�𝐶�↑�↓�� ⊗ ∏ +�𝛼�,�↑�↓� +��� +� �𝛼�,�↑�↓� +��� +� �𝛼�,�↑�↓� +��� +� �𝛼�,�↑�↓� +��� +� +� +��� +. Here state |𝐶�↑�↓�⟩ (𝑧 � 0,1) denotes the  +outputs  appearing  at 𝑆𝑀� side  when  the  atom  is  in  state |↑ �↓�⟩,  and   �𝛼�,�↑�↓� +��� +�  ��𝛼�,�↑�↓� +��� +��  +denotes the transmission (scattering) field generated by 𝑆𝑆𝐶�  in 𝑚‐th cycle. Therefore, �𝑙𝑜𝑠𝑠↑�↓��  +includes  all  optical  losses,  while  the  fidelity  is  obtained  by  tracing  �𝑙𝑜𝑠𝑠↑�↓�� ,  i.e.,  𝐹 � +𝑇𝑟�����⟨𝜓�|𝜓�〉�𝜓��𝜓�〉� � �|⟨𝛼|𝐶�↑⟩|� � |⟨�𝛼|𝐶�↓⟩|� � 2Re�⟨𝛼|𝐶�↑〉⟨𝐶�↓|�𝛼⟩⟨𝑙𝑜𝑠𝑠↓|𝑙𝑜𝑠𝑠↑⟩��/4.  +As a comparison, we also consider the single reflection model in Ref.[37]. More specifically, the  +input  |𝛼⟩  is  directly  reflected  by  𝑆𝑆𝐶� ,  and  the  corresponding  output  state  is  +��𝛼�,�↑�|𝑙𝑜𝑠𝑠↑⟩|↑⟩ � �𝛼�,�↓�|𝑙𝑜𝑠𝑠↓⟩|↓⟩� √2 +⁄ + with   �𝑙𝑜𝑠𝑠↑�↓�� � �𝛼�,�↑�↓���𝛼�,�↑�↓��.  In  this  model,  +the constraints on the atomic parameters 𝛾 and Δ can be directly obtained from Eq. (2). For the  +empty  cavity  case  (atom  is  in |↓⟩),  as  long  as 𝜅� � 0,  the  ideal  reflection 𝛼� � �𝛼�  can  be  +obtained. As for the case where the atom is in |↑⟩, the condition for ideal reflection 𝛼� � 𝛼� is  +Δ � 𝛾 � 𝜅� � 0. If only 𝛾 is non‐zero, we can see that the ideal reflection can be approximately  +achieved  when  𝛾 ≪ 𝑔�/𝜅� .  As  𝛾  increases,  the  cavity  reflectivity  �𝜂�,�↑� +� decreases  +monotonically until it drops to 0 when 𝛾 � 𝑔�/𝜅�. If we focus on Δ, however, it only affects 𝛽�,�↑  + +when 𝛾 � 𝜅� � 0, since �𝜂�,�↑� +� � 1. As Δ varies from �∞ to ∞, 𝛽�,↑ decreases monotonically  +from 𝜋 to �𝜋.  In order to ensure that 𝛽�,�↑ is close to 0, the constraint Δ ≪ 𝑔�/𝜅� is required.   +  +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +=2x3.0MHz +max +Color: + ||2=4 + ||2=10 + ||2=16 +Style: + M=5 + M=20 + M=100 + max +Multiple reflection model +Fidelity + + ||2=4 +Single reflection model +  +Fig.2.  Fidelity 𝐹 versus  dimensionless 𝛾� � 𝜅�𝛾/𝑔�with 𝑔 � 2𝜋 � 7.8𝑀𝐻𝑧, 𝜅� � 𝜅 � 2𝜋 � 2.3𝑀𝐻𝑧 and  +𝛥 � 0. The dashed double doted pink curve is for the single reflection case. Other curves are for the multiple  +reflection case. Different colors represent different |𝛼|�. Different styles represent different 𝑀, except that  +the dotted curves are plotted for 𝑣��� with 𝑀 � 20, which is the maximum value of the average photon  +number reaching 𝑆𝑆𝐶� in each cycle when the atom is in |↑⟩.  +  +In our multiple reflection scheme, however, the above constraints are relaxed. In the following,  +we show that our scheme can be insensitive to atomic parameters 𝛾 and Δ, thus the fidelity of  +the CS depends only on the quality of the linear optical system.  +In  order  to  analyze  the  effect  of  𝛾 ,  we  plot  the  fidelity  against  𝛾� � 𝜅�𝛾/𝑔�  with  𝑔 � +2𝜋 � 7.8𝑀𝐻𝑧 , 𝜅� � 𝜅 � 2𝜋 � 2.3𝑀𝐻𝑧  and  Δ � 0  in  Fig.  2.  The  pink  dot‐dot‐dash  curve  is  +plotted for the single reflection model with |𝛼|� � 4, which has almost reached the upper limit  +of  such  model  [37].  Other  curves  are  plotted  for  the  multiple  reflection  model.  The  color  +black/red/blue represents |𝛼|� � 4 10 16 +⁄ +⁄ +. The curve style dash/solid/dot‐dash denotes 𝑀 � +5 20 100 +⁄ +⁄ +, while the dotted curves are drawn for 𝑣��� with 𝑀 � 20 instead of fidelity, where  +𝑣��� � max ��𝛼�,�↑ +��� � +� +, �𝛼�,�↑ +��� � +� +, … , �𝛼�,�↑ +���� +� +… �  is  the  maximum  value  of  the  average  photon  +number reaching 𝑆𝑆𝐶� in each cycle when the atom is in |↑⟩. As shown in the figure,  𝑣��� is  +always less than 1 (For other 𝑀, the situation is similar), which validates the low atomic excitation  +probability condition, hence Eqs. (2)‐(4) are valid for simulations.   +By  comparison,  we  can  see  that  the  multiple  reflection  scheme  outperforms  the  single  +reflection scheme. In our scheme, it is evident that fidelity increases as 𝑀 increases. Whereas for  +larger |𝛼|�, larger 𝑀 is required to achieve the same fidelity. More importantly, for 𝛾 much larger  +than 2𝜋 � 3.0𝑀𝐻𝑧 (This value is taken from the experiment in Ref. [37]. It corresponds to 𝛾� � +0.11 and has been marked in the figure), our scheme can still provide large 𝐹. To better explain  + +the result, we consider the extreme case when 𝛾� � 1, which means all photons reaching 𝑆𝑆𝐶� in  +a single cycle are lost when the atom is in |↑⟩. Under such conditions, the interference between  +Zone 0 and Zone 1 is continuously interrupted, resulting in the output light field state in Zone 0  +becomes |𝛼 cos� 𝜃� cos��� 2 𝜃�⟩ [50,51]. Since cos� 𝜃� cos��� 2 𝜃� � 1 � 𝜋� 2𝑀 +⁄ + tends to 1  +as 𝑀 tends to infinity, this implies that the light field is frozen in its initial state. Such result is  +exactly what we look for, and the mechanism is called the quantum Zeno effect [49,50]. In practice,  +𝑀 is finite, hence, the quantum Zeno effect is inevitably accompanied by photon loss, which is  +proportional to |𝛼|�, but tends to 0 as 𝑀 increases. Subsequently, this can explain that in Fig.2,  +the larger 𝑀 and the smaller |𝛼|�, the better the fidelity. So far, our discussion is about 𝛾� � 1. As  +for the case of 0 � 𝛾� � 1, the situation is similar. There is a mixture of two physical mechanisms.  +The first is to maintain the initial state by phase modulation, which does not bring any photon  +loss. The second is the quantum Zeno effect. It is worth  mentioning that the fidelity in Fig.2  +decreases monotonically as 𝛾� increases, which implies that the upper limit of the total photon  +loss of our scheme is determined by the quantum Zeno effect, i.e., 𝑀 and |𝛼| only. Together, the  +two mechanisms ensure that our scheme has higher fidelity and higher tolerance to 𝛾 than the  +single reflection scheme as 𝑀 increases. In addition, since the condition 𝛾 ≪ 𝑔�/𝜅� is relaxed, it  +implies that our scheme does not require strong coupling between atom and cavity.  +Following the analysis of 𝛾, we discuss the impact of Δ. We have shown that by interrupting the  +interference, the transmission of the light field from Zone 0 to Zone 1 can be suppressed. Note  +that the phase mismatch between the two Zones also interrupts the interference, we expect that  +our scheme can have high tolerance of Δ as well. In Fig. 3, we plot the fidelity against Δ� � 𝜅�Δ/𝑔�.  +Solid curves are for 𝛾 � 0. Dotted dashed curves are for 𝛾 � 2𝜋 � 3.0𝑀𝐻𝑧. The values of 𝑔, 𝜅�  +and 𝜅� are the same as in Fig. 2. In addition, the pink curves are plotted for the single reflection  +model with |𝛼|� � 4. As for the multiple reflection model, the black curves are for |𝛼|� � 4, 𝑀 � +5, the red curves are for |𝛼|� � 10, 𝑀 � 20 and the blue curves are for |𝛼|� � 16, 𝑀 � 100,  +respectively.  As  shown  in  Fig.3,  even  for  large |𝛼|,  as  long  as 𝑀 is  large,  our  scheme  can  be  +insensitive to Δ.  +  +  + +-1.0 +-0.5 +0.0 +0.5 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Style: + =0 + =2x3.0MHz +Color: + Single reflection model + ||2=4, M=5 + + ||2=16, M=100 + ||2=10, M=20 +Fidelity + +  +Fig.3. Fidelity 𝐹 versus dimensionless Δ� � 𝜅�Δ/𝑔� with 𝑔 � 2𝜋 � 7.8𝑀𝐻𝑧 and 𝜅� � 𝜅 � 2𝜋 � 2.3𝑀𝐻𝑧.  +The solid curves are for 𝛾 � 0, and the dotted dashed curves are for 𝛾 � 2𝜋 � 3.0𝑀𝐻𝑧. The pink curves  +are for the single reflection model with |𝛼|� � 4, and other curves are for the multiple reflection cases.   +  +In  the  above  analyses,  we  ignore  the  influence  of  the  cavity  parameter 𝜅�,  which  will  be  +discussed  below.  According  to  Eq.  (2),  the  reflectivity  of  an  empty  cavity  is   +|�𝜅� � 𝜅�� �𝜅� � 𝜅� +⁄ +�|� .  In  Ref.  [37],  𝜅� � 2𝜋 � 0.2 𝑀𝐻𝑧  and  𝜅� � 2𝜋 � 2.3 𝑀𝐻𝑧 ,  which  +results in a reflectivity of only about 0.7 for single reflection, while after a few reflections, almost  +all photons are lost. Therefore, the cavity employed in Ref. [37] is unfortunately not suitable for  +our scheme. To increase reflectivity, one needs either decrease 𝜅�, or increase 𝜅�. The latter is  +simpler  in  practice.  However,  although  increasing 𝜅� can  reduce  the  photon  loss  during  the  +interference of two empty cavities (The atom is in state |↓⟩), it also increases the photon loss in  +the presence of atom‐cavity coupling (The atom is in state |↑⟩). To verify this, we plot effective  +fidelity 𝐹�� � 𝑇𝑟������𝜓���𝜓�⟩ � against 𝜅� with |𝛼|� � 8, 𝑀 � 10, 𝛾 � 2𝜋 � 3.0𝑀𝐻𝑧 and Δ � 0  +in Fig.4. Here, the target state is set as �𝜓��� � ��𝛼���|↑⟩ � ��𝛼���|↓⟩� √2 +⁄ + with 𝛼�� � �𝐶�↓ � +𝛼 ��𝜅� � 𝜅�� �𝜅� � 𝜅�� +⁄ +�� .  Note  that  |𝐶�↓⟩  is  the  output  when  the  atom  is  in  |↓⟩ ,  where  +interference  occurs  between  the  two  empty  cavities.  If  the  optical  parameters  of  these  two  +cavities are the same, only intensity of the output is affected and reduced from |𝛼|� to �𝛼��� +�. In  +the  figure,  the  solid  (dashed)  curves  are  plotted  for  𝜅� � 2𝜋 � 0.02 �0.002�𝑀𝐻𝑧 .  The  +black/red/blue curves are plotted for 𝑔 � 2𝜋 � 7.8 15 30 +⁄ +⁄ +𝑀𝐻𝑧. In addition, the pink curves are  +plotted for �𝛼��� +�. We can see that 𝐹�� can be significantly improved as 𝜅� decreases. As for 𝜅�,  +when it increases at the beginning, �𝛼��� +� rapidly rises to its maximum value 8, which causes 𝐹��  +to increase. Subsequently, photon loss due to atom‐cavity coupling plays a major role, resulting  +in the decrease of 𝐹��. Particularly, we note that for the black curve, when 𝐹�� starts to decrease,  +its corresponding �𝛼��� +� is not close to 8. The reason is that 𝜅� is approaching to the limit 𝑔� 𝛾 +⁄ .  +Under such limit, the photon loss of a single reflection on 𝑆𝑆𝐶� when the atom is in |↑⟩ is almost  + +100%. Therefore, we plot for larger 𝑔 in order to increase the limit so that �𝛼��� +� can get closer  +to the maximum value 8. We can see that the maximum value of 𝐹�� increases as 𝑔 increases.   +Moreover, 𝜅� maintains wide range of high fidelity (see blue curve). This is because the constraint  +𝑔� ≫ 𝜅�𝛾 in  our  scheme  is  relaxed.  However,  we  must  emphasize  that  the  larger 𝑔 is  not  +necessary  for  high  fidelity.  By  decreasing 𝜅�,  we  can  achieve  the  same  purpose.  In  fact,  the  +motivation of this work is to reduce the influence of the atom, and to show that the performance  +of  our  protocol  can  be  improved  by  just  upgrading  the  linear  optical  system,  such  as  the  +parameters 𝑀 and 𝜅�.  +Besides the atomic parameters (𝛾, Δ) and linear optical system parameters (𝜅�, 𝑀), next we  +provide a discussion about the influence of the decoherence between the atomic states |↓⟩ and  +|↑⟩. Obviously, our scheme requires the atom to remain in superposition at least until the end of  +𝑀 cycles. Nevertheless, we need to mention that the multiple reflection processes hardly affect  +the atomic decoherence. When the atom is in |↑⟩, the low atomic excitation probability can be  +satisfied. As for the atom in |↓⟩, it is not coupled to the light field. While the atomic superposition  +state has been reported to last about 400𝜇𝑠 [60,61]. The full‐width at half‐maximum of the light  +pulse that is employed in the experiment of single reflection model is 2.3𝜇𝑠 [37]. Therefore, it is  +possible for our scheme to be completed before the decoherence.   +10 +20 +30 +40 +50 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +10 +20 +30 +40 +50 +0 +2 +4 +6 +8 +Effective Fidelity, Fef +Color : + Fef , g=2x7.8MHz + Fef , g=2x15MHz + + Fef , g=2x30MHz +Style : + =2x0.02MHz + =2x0.002MHz +Multiple reflection model ||2=8 M=10 +|ef|2 +R(x2MHz) + |ef|2 +0.5 +  +Fig.4.  Effective  fidelity 𝐹��  and  the  output  intensity �𝛼��� +�  versus 𝜅�  with  different 𝑔  and 𝜅�  for  the  +multiple reflection model. In addition, |𝛼|� � 8, 𝑀 � 10, 𝛾 � 2𝜋 � 3.0𝑀𝐻𝑧 and 𝛥 � 0. The initial state of  +the system is  �|𝛼⟩|↑⟩ � |�𝛼⟩| ↓⟩� √2 +⁄ + and the target state is ��𝛼���|↑⟩ � ��𝛼���� ↓�� √2 +⁄ +.  +  +DISCUSSION  +Advantages of multiple reflection scheme  +Compared  with  the  single  reflection  model,  our  multiple  reflection  scheme  has  two  main  +advantages.   + +First, our scheme provides the single atom with the means to manipulate a strong coherent  +light field. When the atom is in |↓ �↑�⟩, the light field |𝛼⟩ evolves to | � 𝛼⟩�|𝛼⟩�. We emphasizes  +that in the single reflection model [37], the above phase manipulation can only be realized when  +|𝛼|� is small. As |𝛼|� increases, the single atom can no longer prevent the light field from entering  +the cavity (In this case, regardless of the state of the atom, the reflected light field carries a 𝜋  +phase shift just like the empty cavity case), causing the atom to be excited from state |↑⟩ to |𝑒⟩.  +As a result, Eq. (2) is no longer valid. In our scheme, however, when the atom is in |↑⟩, only a small  +fraction  of  light  touches  𝑆𝑆𝐶�  in  each  cycle.  Its  average  photon  number  is  |𝛼 sin 𝜃�|� .  By  +adjusting the transmittance of 𝐵𝑆, this value can be far less than 1, thus preventing the atom from  +being excited. Consequently, even after a large number of cycles, the phase of a strong coherent  +light field still can be manipulated by the single atom. This result illustrates that our multiple  +reflection scheme provides a single qubit with the ability to control large amplitude light field,  +even at macroscopic level.  +Second, our scheme does not require a high‐quality atom‐cavity coupling system, and it has a  +high  tolerance  for  atomic  parameters  (𝛾 and Δ ).  In  the  single  reflection  model,  the  phase  +manipulation depends on the interaction between the atom and the cavity. Hence, the constraint  +𝑔� ≫ 𝜅�𝛾  is  necessary.  In  our  multiple  reflection  model,  however,  the  phase  manipulation  +depends  on  the  interference  of  light  between  Zones  0  and  1.  If  the  interference  continues  +uninterrupted,  the  light  field  eventually  carries  a  𝜋  phase  shift  ( |𝛼⟩ → | � 𝛼⟩ ),  whereas  if  +interrupted, the phase remains unchanged (|𝛼⟩ → |𝛼⟩). It is worth noting that in the process of  +generating 𝜋 phase, the interference occurs only between two empty cavities and the atom is not  +involved. Unlike the interference case, the interruption of the interference is more likely to occur,  +bearing  in  mind  that  atom‐cavity  system  from  Ref.  [37]  is  not  the  only  way  to  realize  the  +interruption. For example, if we replace 𝑆𝑆𝐶�(𝑆𝑆𝐶�) by a photon‐absorbing object (mirror), the  +scheme in Fig.1 becomes a typical interaction‐free measurement scheme based on quantum Zeno  +effect [49] (the difference from Ref. [49] is that here we use a Michelson interferometer instead  +of  a  chain  of  Mach‐Zehnder  interferometers,  and  a  coherent  light  source  instead  of  a  single  +photon source). Since the photons entering Zone 1 are absorbed in each cycle, the light field is  +suppressed in Zone 0, maintaining its initial state |𝛼⟩. In fact, some studies have further shown  +that even if the object causes only a partial loss of light, it still can interrupt the interference  +process and prevent the evolution of the light field [62], which is consistent with our numerical  +analysis results. Note that in our CS preparation scheme, the main role of atom‐cavity system is  +just to interrupt the interference. Therefore, our scheme does not require a high‐quality atom‐ +cavity coupling. Even if the atom‐cavity system has imperfections such as photon scattering by  +the atom, CS can be still prepared.   +  +Scalability of multiple reflection scheme  +From the above analysis, we can see that the atom‐cavity system from Ref. [37] is not necessary  +to accomplish our CS preparation. It can certainly be replaced by any quantum object that is in a  +superposition  of  passing/absorbing  photons  such  as  Rydberg  blockade  [43‐45]  and  photon  +blockade  [46].  Moreover,  Fig.  3  implies  that  if  the  object  adds  an  additional  phase  to  those  +photons passing through it instead of absorbing them, it also leads to freezing the evolution of  +the  initial  state |𝛼⟩.  This  suggests  that  the  three‐level  atomic  model,  used  in  nondemolition  +measurement [47,48], can also be used to replace the atom‐cavity system. Therefore, in our CS  + +preparation method, the multiple reflection model is more indispensable. In addition, our scheme  +can be used beyond the preparation of CS, and realize the entangled coherent state required in  +Ref. [20,21]. To do so, we just need to turn off 𝑆𝑀 when 𝑚 � 𝑀/2 instead of 𝑚 � 𝑀,  so that Eq.  +(2) becomes  �|𝛼, 0⟩|↑⟩ � |0, 𝛼⟩|↓⟩� √2 +⁄ +. Last but not least, we focus on the optical platform so far,  +nevertheless, our method also works for other platforms such as superconducting microwave  +resonator [63,64].  +  +In  summary,  we  have  proposed  a  deterministic  method  to  entangle  an  atom  to  a  large‐ +amplitude  coherent  pulse,  thus  realizing  the  preparation  of  a  large‐amplitude  optical  CS.  A  +multiple reflection scheme is used, which brings two advantages. First, in each reflection, the  +actual number of photons manipulated by the atom is very small, which ensures that the single  +atom can properly control the phase of the reflected field. Second, due to quantum Zeno effect,  +our  scheme  becomes  insensitive  to  atomic  parameters 𝛾 and Δ.  The  sensitivity  continues  to  +decrease as the number of reflections 𝑀 increases. This allows our scheme to improve the fidelity  +of the output CS only by improving the linear optical system.  +  +Data availability:  +Data sharing not applicable to this article as no datasets were generated or analyzed during the  +current study.  +  +Code availability:  +The code generated to analyze the protocol is available from the corresponding author upon  +reasonable request.  +  +Acknowledgements:  +This work is supported by a grant from the King Abdulaziz City for Science and Technology (KACST),  +and Project No. NPRP 13S‐0205‐200258 of the Qatar National Research Fund (QNRF).  +  +Author contributions:   +The theory was conceived by Z‐H.L. Numerical calculations were performed by Z‐Y.L. and F.Y.  +under the supervision of Z‐H.L. The project was supervised by M.A. and M.S.Z. All the authors  +participated in the manuscript preparation, discussions, and checks of the results.  +  +Competing interests:  +The authors declare no competing interests.  +  +Additional information:  +Correspondence and requests for materials should be addressed to Z‐H.L.  +  +  +Reference:  +[1] E. Schrödinger, “Die gegenwärtige Situation in der Quantenmechanik”, Naturwissenschaften  +23, 807 (1935).  + +[2]  S.  Haroche,  “Nobel  Lecture:  Controlling  photons  in  a  box  and  exploring  the  quantum  to  +classical boundary”, Rev. Mod. Phys. 85, 1083 (2013).  +[3] D. J. Wineland, “Nobel Lecture: Superposition, entanglement, and raising Schrödinger’s cat”,  +Rev. Mod. Phys. 85, 1103 (2013).  +[4] L. Li, C.‐L. Zou, V. V. Albert, S. Muralidharan, S. M. Girvin, and L. Jiang, “Cat codes with optimal  +decoherence suppression for a lossy bosonic channel”, Phys. Rev. Lett. 119, 030502 (2017).  +[5] S.‐W. Lee, and H. Jeong, “Near‐deterministic quantum teleportation and resource efficient  +quantum computation using linear optics and hybrid qubits”, Phys. Rev. A 87, 022326 (2013).  +[6]  N.  Sangouard,  C.  Simon,  N.  Gisin,  J.  Laurat,  R.  Tualle‐Brouri,  and  P.  Grangier,  “Quantum  +repeaters with entangled coherent states”, J. Opt. Soc. Am. B 27, A137 (2010).  +[7] J. B. Brask, I. Rigas, E. S. Polzik, U. L. Andersen, and A. S. Sørensen, “Hybrid long distance  +entanglement distribution protocol”, Phys. Rev. Lett. 105, 160501 (2010).  +[8] H. Jeong and M. S. Kim, “Efficient quantum computation using coherent states”, Phys. Rev. A  +65, 042305 (2002).  +[9] T. C. Ralph, A. Gilchrist, G. J. Milburn, W. J. Munro, and S. Glancy, “Quantum computation with  +optical coherent states”, Phys. Rev. A 68, 042319 (2003).   +[10] A. P. Lund, T. C. Ralph, and H. L. Haselgrove, “ Fault‐tolerant linear optical quantum computing  +with small‐amplitude coherent states”, Phys. Rev. Lett. 100, 030503 (2008).   +[11] M. Mirrahimi, Z. Leghtas, V. V. Albert, S. Touzard, R. J. Schoelkopf, L. Jiang, and M. H. Devoret,  +“Dynamically protected cat‐qubits: a new paradigm for universal quantum computation”, New  +Journal of Physics 16, 045014 (2014).    +[12]  D.  Su,  I.  Dhand,  and  T.  C.  Ralph,  “Universal  quantum  computation  with  optical  four‐ +component cat qubits”, arXiv:2109.12278 (2021).  +[13]  J.  Hastrup  and  U.  L.  Andersen,  “All‐optical  catcode  quantum  error  correction”,  +arXiv:2108.12225 (2021).  +[14]  P.  T.  Cochrane,  G.  J.  Milburn,  and  W.  J.  Munro,  “Macroscopically  distinct  quantum‐ +superposition states as a bosonic code for amplitude damping”, Phys. Rev. A 59, 2631 (1999).  +[15] Z. Leghtas, G. Kirchmair, B. Vlastakis, R. J. Schoelkopf, M. H. Devoret, and M. Mirrahimi,  +“Hardware‐efficient  autonomous  quantum  memory  protection”,  Phys.  Rev.  Lett.  111,  120501  +(2013).  +[16]  M.  Bergmann  and  P.  van  Loock,  “Quantum  error  correction  against  photon  loss  using  +multicomponent cat states”, Phys. Rev. A 94, 042332 (2016).   +[17] N. Ofek, A. Petrenko, R. Heeres, P. Reinhold, Z. Leghtas, B. Vlastakis, Y. Liu, L. Frunzio, S. M.  +Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, Extending the lifetime of a  +quantum bit with error correction in superconducting circuits, Nature 536, 441 (2016).  +[18] W. J. Munro, K. Nemoto, G. J. Milburn, and S. L. Braunstein, “Weak‐force detection with  +superposed coherent states”, Phys. Rev. A 66, 023819 (2002).  +[19] A. Gilchrist, K. Nemoto, W. J. Munro, T. C. Ralph, S. Glancy, S. L. Braunstein, and G. J. Milburn,  +“Schrödinger cats and their power for quantum information processing”, J. Opt. B 6, S828 (2004).   +[20] J. Joo, W. J. Munro, and T. P. Spiller, “Quantum metrology with entangled coherent states”,  +Phys. Rev. Lett. 107, 083601 (2011).  +[21] Y. M. Zhang, X. W. Li, W. Yang, and G. R. Jin, “Quantum Fisher information of entangled  +coherent states in the presence of photon loss”, Phys. Rev. A 88, 043832 (2013).  + +[22] S. Ghosh, R. Sharma, U. Roy, and P. K. Panigrahi, “Mesoscopic quantum superposition of the  +generalized cat state: A diffraction limit”, Phys. Rev. A 92, 053819 (2015).   +[23] A. Facon, E.‐K. Dietsche, D. Grosso, S. Haroche, J.‐M. Raimond, M. Brune, and S. Gleyzes, “A  +sensitive electrometer based on a Rydberg atom in a Schrödinger‐cat state”, Nature 535, 262– +265 (2016).  +[24]  E.  Polino,  M.  Valeri,  N.  Spagnolo,  and  F.  Sciarrino,  “Photonic  quantum  metrology”,  AVS  +Quantum Science 2, 024703 (2020).  +[25] C. Monroe, D. Meekhof, B. King, and D. J. Wineland, “A “Schrödinger cat” superposition state  +of an atom”, Science 272, 1131 (1996).  +[26] D. Kienzler, C. Flühmann, V. Negnevitsky, H.‐Y. Lo, M. Marinelli, D. Nadlinger, and J. P. Home,  +“Observation of quantum interference between separated mechanical oscillator wave packets”,  +Phys. Rev. Lett. 116, 140402 (2016).   +[27]  S.  Deleglise,  I.  Dotsenko,  C.  Sayrin,  J.  Bernu,  M.  Brune,  J.‐M.  Raimond,  and  S.  Haroche,  +“Reconstruction of non‐classical cavity field states with snapshots of their decoherence”, Nature  +455, 510 (2008).  +[28] B. Vlastakis, G. Kirchmair, Z. Leghtas, S. E. Nigg, L. Frunzio, S. M. Girvin, M. Mirrahimi, M. H.  +Devoret,  and  R.  J.  Schoelkopf,  “Deterministically  encoding  quantum  information  using  100‐ +photon Schrödinger cat states”, Science 342, 607 (2013).   +[29] J. S. Neergaard‐Nielsen, B. M. Nielsen, C. Hettich, K. Mølmer, and E. S. Polzik, “Generation of  +a superposition of odd photon number states for quantum information networks”, Phys. Rev. Lett.  +97, 083604 (2006).  +[30] A. Ourjoumtsev, R. Tualle‐Brouri, J. Laurat, and P. Grangier, “Generating optical Schrödinger  +kittens for quantum information processing”, Science 312, 83 (2006).  +[31] T. Gerrits, S. Glancy, T. S. Clement, B. Calkins, A. E. Lita, A. J. Miller, A. L. Migdall, S. W. Nam,  +R. P. Mirin, and E. Knill, “Generation of optical coherent‐state superpositions by number‐resolved  +photon subtraction from the squeezed vacuum”, Phys. Rev. A 82, 031802(R) (2010).  +[32] H. Takahashi, K. Wakui, S. Suzuki, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki,  +“Generation  of  Large‐Amplitude  Coherent‐State  Superposition  Via  Ancilla‐Assisted  Photon  +Subtraction”, Phys. Rev. Lett. 101, 233605 (2008).  +[33] K. Huang, H. Le Jeannic, J. Ruaudel, V. B. Verma, M. D. Shaw, F. Marsili, S.W. Nam, E Wu, H.  +Zeng, Y.‐C. Jeong, R. Filip, O. Morin, and J. Laurat, “Optical Synthesis of Large‐Amplitude Squeezed  +Coherent‐State Superpositions with Minimal Resources”, Phys. Rev. Lett. 115, 023602 (2015).  +[34]  A.  Ourjoumtsev,  H.  Jeong,  R.  Tualle‐Brouri,  and  P.  Grangier,  “Generation  of  optical  +‘Schrödinger cats’ from photon number states”, Nature (London) 448, 784 (2007).  +[35] A. E. Ulanov, I. A. Fedorov, D. Sychev, P. Grangier, and A. I. Lvovsky, “Loss‐tolerant state  +engineering  for  quantum  enhanced  metrology  via  the  reverse  Hong‐Ou‐Mandel  effect”,  Nat.  +Commun. 7, 11925 (2016).  +[36] D. V. Sychev, A. E. Ulanov, A. A. Pushkina, M. W. Richards, I. A. Fedorov, and A. I. Lvovsky,  +“Enlargement of optical Schrödinger's cat states”, Nature Photonics 11, 379 (2017).  +[37] B. Hacker, S. Welte, S. Daiss, A. Shaukat, S. Ritter, L. Li, and G. Rempe, “Deterministic creation  +of entangled atom–light Schrödinger‐cat states”, Nature Photonics 13, 110 (2019).   +[38] M. Dakna, T. Anhut, T. Opatrný, L. Knöll, and D.‐G. Welsch, “Generating Schrödinger‐cat‐like  +states by means of conditional measurements on a beam splitter”, Phys. Rev. A 55, 3184 (1997).  +[39] K. Takase, J.‐i. Yoshikawa, W. Asavanant, M. Endo, and A. Furusawa, “Generation of optical  + +Schrödinger cat states by generalized photon subtraction”, Phys. Rev. A 103, 013710 (2021).  +[40] A. P. Lund, H. Jeong, T. C. Ralph, and M. S. Kim, “Conditional production of superpositions of  +coherent states with inefficient photon detection”, Phys. Rev. A 70, 020101(R) (2004). +[41] A. Laghaout, J. S. Neergaard‐Nielsen, I. Rigas, C. Kragh, A. Tipsmark, and U. L. Andersen,  +“Amplification of realistic Schrödinger‐cat‐state‐like states by homodyne heralding”, Phys. Rev. A  +87, 043826 (2013).  +[42] B. Wang and L.‐M. Duan, “Engineering superpositions of coherent states in coherent optical  +pulses through cavity‐assisted interaction”, Phys. Rev. A 72, 022320 (2005).  +[43] E. Urban, T. A. Johnson, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker, and M. Saffman,  +“Observation of Rydberg blockade between two atoms”, Nat. Phys. 5, 110 (2009).  +[44] T. A. Johnson, E. Urban, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker, and M. Saffman,  +“Rabi Oscillations between Ground and Rydberg States with Dipole‐Dipole Atomic Interactions”,  +Phys. Rev. Lett. 100, 113003 (2008).  +[45] Q. Guo, L. Y. Cheng, L. Chen, H. F. Wang, and S. Zhang, “Counterfactual quantum‐information  +transfer without transmitting any physical particles”, Sci. Rep. 5, 8416 (2015).  +[46] K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon  +blockade in an optical cavity with one trapped atom”, Nature (London) 436, 87 (2005)  +[47] G. Nogues, A. Rauschenbeutel, S. Osnaghi, M. Brune, J. M. Raimond, and S. Haroche, “Seeing  +a single photon without destroying it”, Nature (London) 400, 239 (1999).  +[48] M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University Press, Cambridge,  +1997).  +[49]  P.  G.  Kwiat,  H.  Weinfurter,  T.  Herzog,  A.  Zeilinger,  and  M.  A.  Kasevich,  “Interaction‐free  +measurement”, Phys. Rev. Lett. 74, 4763 (1995).  +[50] H. Salih, Z.‐H. Li, M. Al‐Amri, and M. S. Zubairy, “Protocol for direct counterfactual quantum  +communication”, Phys. Rev. Lett. 110, 170502 (2013).  +[51]  Z.‐H.  Li,  S.‐Y.  Feng,  M.  Al‐Amri,  and  M.  S.  Zubairy,  “Direct  counterfactual  quantum  +communication protocol beyond single photon source”, Phys. Rev. A 106, 032610 (2022).  +[52]  L.‐M.  Duan,  and  H.  J.  Kimble,  “Scalable  Photonic  Quantum  Computation  through  Cavity‐ +Assisted Interactions”, Phys. Rev. Lett. 92, 127902 (2004).  +[53] A. Reiserer, S. Ritter, and G. Rempe, “Nondestructive Detection of an Optical Photon”, Science  +342, 1349 (2013).  +[54] A. Reiserer, and G. Rempe, “Cavity‐ based quantum networks with single atoms and optical  +photons”, Rev. Mod. Phys. 87, 1379 (2015).  +[55] Y. Liu, L. Ju, X.‐L. Liang, S.‐B. Tang, G.‐L. Shen Tu, L. Zhou, C.‐Z. Peng, K. Chen, T.‐Y. Chen, Z.‐B.  +Chen, and J.‐W. Pan, “Experimental Demonstration of Counterfactual Quantum Communication”,  +Phys. Rev. Lett. 109, 030501 (2012).  +[56] See supplementary materials.  +[57] C. Y. Hu, A. Young, J. L. O’Brien, W. J. Munro, and J. G. Rarity, “Giant optical Faraday rotation  +induced by a single‐electron spin in a quantum dot: Applications to entangling remote spins via a  +single photon”, Phys. Rev. B 78, 085307 (2008).  +[58] C. W. Gardiner, and M. J. Collett, “Input and output in damped quantum systems: Quantum  +stochastic differential equations and the master equation”, Phys. Rev. A 31, 3761 (1985).  +[59] D. F. Walls, and G. J. Milburn, Quantum Optics (Springer‐ Verlag, Berlin, 1994).  +[60] S. Daiss, S. Langenfeld, S. Welte, E. Distante, P. Thomas, L. Hartung, O. Morin, G. Rempe, “A  + +quantum‐logic gate between distant quantum‐network modules”, Science 371, 614 (2021).  +[61] S. Welte, P. Thomas, L. Hartung, S. Daiss, S. Langenfeld, O. Morin, G. Rempe and E. Distante,  +“A nondestructive Bell‐state measurement on two distant atomic qubits”, Nature Photonics 15,  +504 (2021).  +[62] L. J. Wang, Z.‐H. Li, J. P. Xu, Y. P. Yang, M. Al‐Amri, and M. S. Zubairy, “Exchange unknown  +quantum states with almost invisible photons”, Opt. Express 27, 20525 (2019).  +[63] Z. L. Wang, Z. H. Bao, Y. K. Wu, Y. Li, W. Z. Cai, W. T. Wang, Y. W. Ma, T. Q. Cai, X. Y. Han, J. H.  +Wang, Y. P. Song, L. Y. Sun, H. Y. Zhang, L. M. Duan, “A flying Schrödinger’s cat in multipartite  +entangled states”, Sci. Adv. 8, eabn1778 (2022).  +[64] Z. H. Bao, Z. L. Wang, Y. K. Wu, Y. Li, W. Z. Cai, W. T. Wang, Y. W. Ma, T. Q. Cai, X. Y. Han, J. H.  +Wang, Y. P. Song, L. Y. Sun, H. Y. Zhang, and L. M. Duan,  “Experimental preparation of generalized  +cat states for itinerant microwave photons”, arXiv:2207.04617 (2022).  +  + +Supplementary material  +A. Calculations for Equation 1  +In the main text, we have mentioned that 𝑎� and 𝑎� represent the annihilation operators of  +the  light  field  in  Zone  0  and  Zone  1,  respectively.  Based  on  this,  the  function  of 𝐵𝑆 can  be  +described as  𝑎� +� → 𝑎� +� cos 𝜃� � 𝑎� +� sin 𝜃� and 𝑎� +� → 𝑎� +� cos 𝜃� � 𝑎� +� sin 𝜃� where 𝜃� � 𝜋/2𝑀.  +Now, we consider an arbitrary initial photon state   +  +|𝐼𝑛𝑖𝑡𝑖𝑎𝑙⟩ � |𝑢, 𝑣⟩ � exp�𝑢𝑎� +� � 𝑢∗𝑎��exp�𝑣𝑎� +� � 𝑣∗𝑎��|0,0⟩,  +(A1)  +which represents that a coherent state |𝑢⟩ is in Zone 0 and a coherent state |𝑣⟩ is in Zone 1. After  +passing through the 𝐵𝑆, we have the final state  +  +|𝐹𝑖𝑛𝑎𝑙⟩ � exp�𝑢�𝑎� +� cos 𝜃� � 𝑎� +� sin 𝜃�� � 𝑢∗�𝑎� cos 𝜃� � 𝑎� sin 𝜃���,   +  + � exp�𝑣�𝑎� +� cos 𝜃� � 𝑎� +� sin 𝜃�� � 𝑣∗�𝑎� cos 𝜃� � 𝑎� sin 𝜃��� |0,0⟩  +� |𝑢 cos 𝜃� � 𝑣 sin 𝜃� , 𝑢 sin 𝜃� � 𝑣 cos 𝜃�⟩.   +(A2)  +Similarly, consider an arbitrary phase operation 𝑎� → 𝑒��𝑎�. For the initial state |𝐼𝑛𝑖𝑡𝑖𝑎𝑙⟩ � +|𝑢⟩, after the operation, the final state is  +|𝐹𝑖𝑛𝑎𝑙⟩ � exp �� 1 +2 |𝑢|�� � 𝑢� +√𝑛! +1 +√𝑛! +�𝑒��𝑎�� +�|0⟩ +∞ +��� +  +� exp �� +� +� |𝑢|�� ∑ +� +√�! �𝑢𝑒��� +�|𝑛⟩ +∞ +��� +� |𝑢𝑒��⟩                          (A3)  +Based on Eqs. (A2) and (A3), we provide the calculation of Eq. (1) in the main text.   +At the beginning of the preparation, the wave‐function of the whole system is  +  +�𝜓���� � √� +� |𝛼, 0⟩�|↑⟩ � |↓⟩�   +(A4)  +In the first cycle, after the photons pass through  𝐵𝑆 for the first time, the system state is  +  +�𝜓���� � √� +� |𝛼 cos 𝜃� , 𝛼 sin 𝜃�⟩�|↑⟩ � |↓⟩�   +(A5)  +Before  +the  +photons  +are  +reflected  +by  +𝑆𝑆𝐶 +,  +the  +system  +state  +becomes  +√� +� |�𝛼 cos 𝜃� , �𝛼 sin 𝜃�⟩�|↑⟩ � |↓⟩� due  to 𝑃𝑆.  Regarding  the  reflection,  we  emphasize  that  +only when the atom is in |↑⟩, 𝑆𝑆𝐶� does not change the phase of the reflected field. As a result,  +the  +wave‐function  +of  +the  +whole  +system  +becomes  +√� +� |𝛼 𝑐𝑜𝑠 𝜃� , �𝛼 𝑠𝑖𝑛 𝜃�⟩|↑⟩ � +√� +� |𝛼 𝑐𝑜𝑠 𝜃� , 𝛼 𝑠𝑖𝑛 𝜃�⟩|↓⟩. Subsequently, after the second time that the photons pass through  +𝐵𝑆, we have  +  +�𝜓���� � √� +� |𝛼, 0⟩|↑⟩ � √� +� |cos 2 𝜃�𝛼, sin 2 𝜃�𝛼⟩|↓⟩   +(A6)  + +This state becomes the initial state of the second cycle, and the process is repeated. It is not  +difficult to obtain that after 𝑚 cycles, the wave‐function of the whole system is  +  +�𝜓����� � √� +� |𝛼, 0⟩|↑⟩ � √� +� |𝛼 cos 2 𝑚𝜃�, 𝛼 sin 2 𝑚𝜃�⟩|↓⟩  +(A7)  +Here the superscript of �𝜓����� represents the photons pass through 𝐵𝑆 2𝑚 times.  +  +B. Calculations for Equations 2‐4  +The Hamiltonian of cavity‐atom system (𝑆𝑆𝐶�) can be described as  +𝐻 � ℏ𝜔�𝜎�� � ℏ𝜔↑𝜎↑↑ � ℏ𝜔�𝑎�𝑎 � ℏ � +� +𝜔�𝑏� +��𝜔��𝑏��𝜔��𝑑𝜔� +� +－� +���,�,� +  +�ℏ𝑔�𝜎↑�𝑎� � 𝜎�↑𝑎� � ℏ� +� +� � +�𝜎↑�𝑏� +��𝜔�� � 𝜎�↑𝑏��𝜔���𝑑𝜔� +� +－� +   +     �𝑖ℏ� +�� +� � +�𝑎𝑏� +��𝜔�� � 𝑎�𝑏��𝜔���𝑑𝜔� +� +－� +� 𝑖ℏ� +�� +� � +�𝑎𝑏� +��𝜔�� � 𝑎�𝑏��𝜔���𝑑𝜔� +� +－� +       (B1)  +where ℏ𝜔� is the energy of excited atomic state |𝑒⟩, ℏ𝜔↑ is the energy of the atomic state |↑⟩, 𝜔𝑐  +is the frequency of the cavity mode described by annihilation operator 𝑎, 𝜔𝐽 is the frequency of  +external field described by annihilation operator 𝑏�𝜔�� with �𝑏𝐽�𝜔𝐽�, 𝑏𝐽 +†�𝜔𝐽 +′�� � 𝛿�𝜔𝐽 � 𝜔𝐽 +′�, and  +the subscript 𝑅 represents the external multi‐mode field on 𝐶𝑀� side, 𝑇 represents the external  +field on 𝐶𝑀� side, 𝑆 represents the scattering field due to the atomic spontaneous emission. In  +addition, 𝑔 is coupling constant between the cavity and the atomic transition between |𝑒⟩ and |↑⟩,  +2𝛾 is the spontaneous atomic decay rate on the same transition, 𝜅𝑅 and 𝜅𝑇 are cavity field decay  +rates. We also set that 𝜎↑𝑒 � |↑⟩⟨𝑒|, 𝜎𝑒𝑒 � |𝑒⟩⟨𝑒| and 𝜎↑↑ � |↑⟩⟨↑|.        +Based on the above Hamiltonian, it is not difficult to obtain the following Heisenberg equations  +  +𝑑𝑎�𝑡� +𝑑𝑡 � �𝑖𝜔𝑐𝑎�𝑡� � 𝑖𝑔𝜎↑𝑒�𝑡� � ∑ +� +𝜅𝐽 +𝜋 +𝐽�𝑅,𝑇 +� +𝑏𝐽�𝜔𝐽, 𝑡�𝑑𝜔𝐽 +∞ +－∞ +,   +(B2)  +𝑑 +𝑑𝑡 𝜎↑𝑒�𝑡� � �𝑖�𝜔𝑒 � 𝜔↑�𝜎↑𝑒�𝑡� � 𝑖𝑔�𝜎𝑒𝑒�𝑡� � 𝜎↑↑�𝑡��𝑎�𝑡�   +�𝑖� +� +� � +�𝜎���𝑡� � 𝜎↑↑�𝑡�� +� +－� +𝑏��𝜔�, 𝑡�𝑑𝜔�   +  +� �𝑖�𝜔� � 𝜔↑�𝜎↑��𝑡� � 𝑖𝑔𝑎�𝑡� � 𝑖� +� +� � +𝑏��𝜔�, 𝑡�𝑑𝜔� +� +－� +.   +(B3)  +In the approximation, we have assumed that [1‐3]  +  +⟨�𝜎𝑒𝑒 � 𝜎↑↑�𝑎⟩ � �⟨𝑎⟩,   +(B4)  +which indicates that the atom stays in the state |↑⟩ most of the time. This can be satisfied when  +the input is weak.  + +In addition, we can also obtain Heisenberg equations for 𝑏�𝜔�. They are  +  +𝑑𝑏𝐽�𝜔𝐽,𝑡� +𝑑𝑡 +� �𝑖𝜔𝐽𝑏𝐽�𝜔𝐽, 𝑡� � � +𝜅𝑅 +𝜋 𝑎�𝑡�, 𝐽 � 𝑅, 𝑇,  +(B5)  +  +𝑑𝑏𝑆�𝜔𝑆,𝑡� +𝑑𝑡 +� �𝑖𝜔𝑆𝑏𝑆�𝜔𝑆, 𝑡� � 𝑖� +𝛾 +𝜋 𝜎↑𝑒�𝑡�.  +(B6)  +Eqs. (B5) and (B6) can be rewritten in integral form. If we assume that the atom‐light interaction  +begins at time 𝑇𝑖𝑛 � 𝑡, we have  +  +𝑏𝐽�𝜔𝐽, 𝑡� � 𝑏𝐽�𝜔𝐽, 𝑇𝑖𝑛�𝑒𝑖𝜔𝐽�𝑇𝑖𝑛�𝑡� � � +𝜅𝐽 +𝜋 � +𝑎�𝑡′�𝑒𝑖𝜔𝐽�𝑡′�𝑡�𝑑𝑡′ +𝑡 +𝑇𝑖𝑛 +,   +(B7)  +  +𝑏𝑆�𝜔𝑆, 𝑡� � 𝑏𝑆�𝜔𝑆, 𝑇𝑖𝑛�𝑒𝑖𝜔𝑆�𝑇𝑖𝑛�𝑡� � 𝑖� +𝛾 +𝜋 � +𝜎↑𝑒�𝑡′�𝑒𝑖𝜔𝑆�𝑡′�𝑡�𝑑𝑡′ +𝑡 +𝑇𝑖𝑛 +.   +(B8)  +If we assume that the atom‐light interaction ends at time 𝑇𝑜𝑢𝑡 � 𝑡, we have  +  +𝑏𝐽�𝜔𝐽, 𝑡� � 𝑏𝐽�𝜔𝐽, 𝑇𝑜𝑢𝑡�𝑒𝑖𝜔𝐽�𝑇𝑜𝑢𝑡�𝑡� � � +𝜅𝐽 +𝜋 � +𝑎�𝑡′�𝑒𝑖𝜔𝐽�𝑡′�𝑡�𝑑𝑡′ +𝑇𝑜𝑢𝑡 +𝑡 +,   +(B9)  +  +𝑏𝑆�𝜔𝑆, 𝑡� � 𝑏𝑆�𝜔𝑆, 𝑇𝑜𝑢𝑡�𝑒𝑖𝜔𝑆�𝑇𝑜𝑢𝑡�𝑡� � 𝑖� +𝛾 +𝜋 � +𝜎↑𝑒�𝑡′�𝑒𝑖𝜔𝑆�𝑡′�𝑡�𝑑𝑡′ +𝑇𝑜𝑢𝑡 +𝑡0 +.   +(B10)  +By integrating Eq. (B7) with frequency, it is not difficult to obtain that  +� +𝜅𝐽 +𝜋 � +𝑏𝐽�𝜔𝐽, 𝑡�𝑑𝜔𝐽 +∞ +－∞ +   +� � +�� +� � +𝑏��𝜔�, 𝑇���𝑒����������𝑑𝜔� +� +－� +� 2𝜅� � +𝑎�𝑡��𝑑𝑡� +� +��� +� +�� � +𝑒���������𝑑𝜔� +� +－� +   +  +� �2𝜅�𝑎�,���𝑡� � 𝜅�𝑎�𝑡�.   +(B11)  +where we have used the relation [4]  +  +� 𝑓�𝑡′�𝛿�𝑡 � 𝑡′�𝑑𝑡′ +𝑡 +𝑡0 +� � +𝑓�𝑡′�𝛿�𝑡 � 𝑡′�𝑑𝑡′ +𝑡1 +𝑡 +� 1 +2 𝑓�𝑡�, �𝑡0 � 𝑡 � 𝑡1�,   +(B12)  +and the assumptions (𝐽 � 𝑅, 𝑇)  +  +𝑎𝐽,𝑖𝑛�𝑡� � +1 +√2𝜋 � +𝑏𝐽�𝜔𝐽, 𝑇𝑖𝑛�𝑒𝑖𝜔𝐽�𝑇𝑖𝑛�𝑡�𝑑𝜔𝐽 +∞ +－∞ +,  +(B13)  +  +𝑎𝐽,𝑜𝑢𝑡�𝑡� � +1 +√2𝜋 � +𝑏𝐽�𝜔𝐽, 𝑇𝑜𝑢𝑡�𝑒𝑖𝜔𝐽�𝑇𝑜𝑢𝑡�𝑡�𝑑𝜔𝐽 +∞ +－∞ +.  +(B14)  +  +𝑎𝑆,𝑖𝑛�𝑡� � +1 +√2𝜋 � +𝑏𝑆�𝜔𝑆, 𝑇𝑖𝑛�𝑒𝑖𝜔𝑆�𝑇𝑖𝑛�𝑡�𝑑𝜔𝑆 +∞ +－∞ +,  +(B15)  +  +𝑎𝑆,𝑜𝑢𝑡�𝑡� � +1 +√2𝜋 � +𝑏𝑆�𝜔𝑆, 𝑇𝑜𝑢𝑡�𝑒𝑖𝜔𝑆�𝑇𝑜𝑢𝑡�𝑡�𝑑𝜔𝑆 +∞ +－∞ +.  +(B16)  +Similarly, from Eqs. (B8)‐(B10), we have  +  +� +𝛾 +𝜋 � +𝑏𝑆�𝜔𝑆, 𝑡�𝑑𝜔𝑆 +∞ +－∞ +� �2𝛾𝑎𝑆,𝑖𝑛�𝑡� � 𝑖𝛾𝜎↑𝑒�𝑡�,   +(B17)  + +  +� +𝜅𝐽 +𝜋 � +𝑏𝐽�𝜔𝐽, 𝑡�𝑑𝜔𝐽 +∞ +－∞ +� �2𝜅𝐽𝑎𝐽,𝑜𝑢𝑡�𝑡� � 𝜅𝐽𝑎�𝑡�,   +(B18)  +  +� +𝛾 +𝜋 � +𝑏𝑆�𝜔𝑆, 𝑡�𝑑𝜔𝑆 +∞ +－∞ +� �2𝛾𝑎𝑆,𝑜𝑢𝑡�𝑡� � 𝑖𝛾𝜎↑𝑒�𝑡�.   +(B19)  +Then, by substituting Eqs. (B11)(B18) into (B2), we can obtain the dynamic equations  +  +𝑑𝑎�𝑡� +𝑑𝑡 � �𝑖𝜔𝑐𝑎�𝑡� � 𝑖𝑔𝜎↑𝑒�𝑡� � �2𝜅𝑅𝑎𝑅,𝑖𝑛�𝑡� � 𝜅𝑅𝑎�𝑡� � �2𝜅𝑇𝑎𝑇,𝑖𝑛�𝑡� � 𝜅𝑇𝑎�𝑡�,   (B20)  +  +𝑑𝑎�𝑡� +𝑑𝑡 � �𝑖𝜔𝑐𝑎�𝑡� � 𝑖𝑔𝜎↑𝑒�𝑡� � �2𝜅𝑅𝑎𝑅,𝑜𝑢𝑡�𝑡� � 𝜅𝑅𝑎�𝑡� � �2𝜅𝑇𝑎𝑇,𝑖𝑛�𝑡� � 𝜅𝑇𝑎�𝑡�,   (B21)  +  +𝑑𝑎�𝑡� +𝑑𝑡 � �𝑖𝜔𝑐𝑎�𝑡� � 𝑖𝑔𝜎↑𝑒�𝑡� � �2𝜅𝑅𝑎𝑅,𝑖𝑛�𝑡� � 𝜅𝑅𝑎�𝑡� � �2𝜅𝑇𝑎𝑇,𝑜𝑢𝑡�𝑡� � 𝜅𝑇𝑎�𝑡�.   (B22)  +By substituting Eqs. (B17)(B19) into (B3), we have  +  +𝑑 +𝑑𝑡 𝜎↑𝑒�𝑡� � �𝑖�𝜔𝑒 � 𝜔↑�𝜎↑𝑒�𝑡� � 𝑖𝑔𝑎�𝑡� � 𝑖�2𝛾𝑎𝑆,𝑖𝑛�𝑡� � 𝛾𝜎↑𝑒�𝑡�,   +(B23)  +  +𝑑 +𝑑𝑡 𝜎↑𝑒�𝑡� � �𝑖�𝜔𝑒 � 𝜔↑�𝜎↑𝑒�𝑡� � 𝑖𝑔𝑎�𝑡� � 𝑖�2𝛾𝑎𝑆,𝑜𝑢𝑡�𝑡� � 𝛾𝜎↑𝑒�𝑡�.   +(B24)  +In  the  following,  we  assume  that  only  the  input  on  𝐶𝑀�  side  is  none‐zero,  i.e.,  𝑎𝑇,𝑖𝑛�𝑡� � +𝑎𝑆,𝑖𝑛�𝑡� � 0. Then, by subtracting (B20) and (B21), we can get the relation between the input  +𝑎𝑅,𝑖𝑛�𝑡� and output 𝑎𝑅,𝑜𝑢𝑡�𝑡�,  +  +𝑎𝑅,𝑖𝑛�𝑡� � �2𝜅𝑅𝑎�𝑡� � 𝑎𝑅,𝑜𝑢𝑡�𝑡�.   +(B25)  +By subtracting (B20) and (B22), we have  +  +𝑎𝑇,𝑜𝑢𝑡�𝑡� � �2𝜅𝑇𝑎�𝑡�.  +(B26)  +By subtracting (B23) and (B24), we have  +  +𝑎𝑆,𝑜𝑢𝑡�𝑡� � �𝑖�2𝛾𝜎↑𝑒�𝑡�.   +(B27)  +In addition to the above relations, we next calculate the steady‐state solution of the dynamic  +equations (B20)(B21)(B23) by assuming that the cavity‐atom system is driven by the input light  +field with frequency 𝜔. We suppose that   +𝑎𝑅,𝑖𝑛�𝑡� � 𝛼𝑖,1↑𝑒�𝑖𝜔𝑡, +𝑎�𝑡� � 𝛼𝑒����, +𝜎↑𝑒�𝑡� � 𝜎�𝑒�𝑖𝜔𝑡, +𝑎𝑅,𝑜𝑢𝑡�𝑡� � 𝛼𝑅,1↑𝑒�𝑖𝜔𝑡, +𝑎𝑇,𝑜𝑢𝑡�𝑡� � 𝛼𝑇,1↑𝑒�𝑖𝜔𝑡,  +                                                                   𝑎𝑆,𝑜𝑢𝑡�𝑡� � 𝛼𝑆,1↑𝑒�𝑖𝜔𝑡.                                 +(B28)  +Then, we can obtain that  +  +��𝑖�𝜔𝑐 � 𝜔� � 𝜅𝑅 � 𝜅𝑇�𝛼 � 𝑖𝑔𝜎� � �2𝜅𝑅𝛼𝑖,1↑ � 0,   +(B29)  +  +��𝑖�𝜔𝑐 � 𝜔� � 𝜅𝑅 � 𝜅𝑇�𝛼 � 𝑖𝑔𝜎� � �2𝜅𝑅𝛼𝑅,1↑ � 0,   +(B30)  + +  +��𝑖�𝜔𝑒 � 𝜔↑ � 𝜔� � 𝛾�𝜎� � 𝑖𝑔𝛼 � 0.   +(B31)  +It is not difficult to get that  +  +𝛼𝑖,1↑ � � +�𝑖�𝜔𝑐�𝜔��𝜅𝑅�𝜅𝑇��𝑖�𝜔𝑒�𝜔↑�𝜔��𝛾��𝑔2 +�2𝜅𝑅�𝑖�𝜔𝑒�𝜔↑�𝜔��𝛾� +𝛼,   +(B32)  +  +𝛼𝑅,1↑ � � +�𝑖�𝜔𝑐�𝜔��𝜅𝑅�𝜅𝑇��𝑖�𝜔𝑒�𝜔↑�𝜔��𝛾��𝑔2 +�2𝜅𝑅�𝑖�𝜔𝑒�𝜔↑�𝜔��𝛾� +𝛼.   +(B33)  +Eq. (B32) shows that when the input increases, the intensity of the cavity field also increase,  +resulting in the condition (B4) not being satisfied.  +With Eqs. (B32) and (B33), we can calculate Eq. (2) in the main text. Suppose that the cavity and  +the external field are resonant, i.e., 𝜔 � 𝜔�, and 𝛥 � 𝜔� � 𝜔↑ � 𝜔�, we obtain   +  +𝛼𝑅,1↑ +𝛼𝑖,1↑ � 1 � +2𝜅𝑅�𝑖𝛥�𝛾� +�𝜅𝑅�𝜅𝑇��𝑖𝛥�𝛾��𝑔2.   +(B34)  +By using Eqs. (B26) and (B27), we can also have Eqs. (3) and (4) in the main text, which are  +  +𝛼𝑇,1↑ +𝛼𝑖,1↑ � � +2√𝜅𝑅𝜅𝑇�𝑖𝛥�𝛾� +�𝜅𝑅�𝜅𝑇��𝑖𝛥�𝛾��𝑔2,   +(B35)  +  +𝛼𝑆,1↑ +𝛼𝑖,1↑ � +2�𝜅𝑅𝛾𝑔 +�𝜅𝑅�𝜅𝑇��𝑖𝛥�𝛾��𝑔2.   +(B36)  +  +  +  +[1] C. Y. Hu, A. Young, J. L. O’Brien, W. J. Munro, and J. G. Rarity, “Giant optical Faraday rotation  +induced by a single‐electron spin in a quantum dot: Applications to entangling remote spins via a  +single photon”, Phys. Rev. B 78, 085307 (2008).  +[2] A. Reiserer, and G. Rempe, “Cavity‐ based quantum networks with single atoms and optical  +photons”, Rev. Mod. Phys. 87, 1379 (2015).  +[3] B. Hacker, S. Welte, S. Daiss, A. Shaukat, S. Ritter, L. Li, and G. Rempe, “Deterministic creation  +of entangled atom–light Schrödinger‐cat states”, Nature Photonics 13, 110 (2019).   +[4] D. F. Walls, and G. J. Milburn, Quantum Optics (Springer‐ Verlag, Berlin, 1994).  + diff --git a/3tE1T4oBgHgl3EQfAgIm/content/tmp_files/load_file.txt b/3tE1T4oBgHgl3EQfAgIm/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e7e4c786aa43393869c28f9ba6ce6bddd0161529 --- /dev/null +++ b/3tE1T4oBgHgl3EQfAgIm/content/tmp_files/load_file.txt @@ -0,0 +1,1101 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf,len=1100 +page_content='Method to deterministically generate large‐amplitude Optical Schrödinger‐cat states Zheng-Hong Li,1,2,* Zhen-Ya Li,1 Fei Yu,1 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Al-Amri,3,4,5 and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Suhail Zubairy3 1 Department of Physics, Shanghai University, Shanghai 200444, China 2 Shanghai Key Laboratory of High Temperature Superconductors, Shanghai University, Shanghai 200444, China 3 Institute for Quantum Science and Engineering (IQSE) and Department of Physics and Astronomy, Texas A&M University, College Station, Texas 77843‐4242, USA 4 NCQOQI, KACST, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='Box 6086, Riyadh 11442, Saudi Arabia 5 The National Center for Quantum Optics and Quantum Informatics, KACST, Riyadh 11442, Saudi Arabi A deterministic preparation method for large‐amplitude optical Schrödinger‐cat state is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' The key ingredient is to entangle an atom buried in a single‐side cavity with a large‐amplitude coherent light pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' To achieve this purpose, a multiple reflection Michelson interferometer is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' The light pulse can go back and forth inside the interferometer and interact with the atom many times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' However, in every interaction, the average photon number of the light field that manipulated by the atom is much less than 1, which ensures that the atom‐cavity system can properly control the phase of the reflected field, and thus achieve the entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Not only that, but we also further demonstrate that due to quantum Zeno effect, our scheme is insensitive to both atomic spontaneous emission and detuning between the atom and the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Therefore, the fidelity of the cat state can be increased by improving the linear optical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Introduction Schrödinger’s gedanken experiment involving a cat in a superposition of dead and alive states played a crucial role in elucidating certain counterintuitive aspects of quantum mechanics [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' In modern physics, this Schrödinger cat state (CS) is usually represented by the superposition of two distinct coherent states |�𝛼⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' With the increase of amplitude |𝛼| , CS gets closer to the macroscopic superposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' It is not only attractive from a fundamental point of view [2,3], but also valuable for applications including quantum teleportation [4‐7], quantum computing [8‐13], quantum error correction [14‐17] and quantum metrology [18‐24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' It is well‐known that a quasi‐ideal CS requires |𝛼| to be large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' In response to the above demands, after decades of efforts, CS has been generated on various platforms [3,25‐28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' However, in the optical domain, in the best experimental results so far, |𝛼| remains less than 2 [29‐37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Needless to say, optical field is an excellent medium for information transmission [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' It is necessary and valuable to create optical CS of large amplitudes with propagation properties that are on demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Although there have been some probabilistic methods, for example, the photon subtraction method [29‐33,38,39], they have low probability of success for generating large‐amplitude CS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' As for synthesis method proposed in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [36,40,41], it is limited by the amplitude of the pre‐ prepared CS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' In addition, the light‐matter interaction to generate CS has become an important research direction recently [37,42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' In the experiment of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [37], CS is deterministically generated by the interaction of an incident coherent pulse with a single‐side cavity containing a single atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' According to different atomic states, the reflected light field evolves in different ways, and eventually produces π phase difference leading to entanglement between the atom and the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Applying the measurement on the atom collapses the wave function into the optical CS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' It is worth noting that in such scheme [37,42], only one reflection happens between the light and the atom‐cavity system (Hereinafter we call it the single reflection scheme).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' This means preparing a large‐amplitude CS requires the atom to control a strong coherent light field through a single interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' It is obviously unrealistic, and in the experiment [37], the amplitude of the output CS is only |𝛼| � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' In light of this discussion, it is clear that a deterministic generation of large‐amplitude CS in the optical regime remains elusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' In this article, we propose a deterministic method to generate flying optical CS whose amplitude can be arbitrarily large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Our starting point is Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [37,42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' However, our approach differs distinctly by employing a multiple reflection model to achieve multiple phase operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' This allows, on one hand, for more interactions between the light field and the atom, but, on the other hand, only a small fraction of light is reflected by the atom‐cavity system during each interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Through repeated interactions, we demonstrate that just one atom is possible to control a macroscopic light field and become entangled with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' In addition, it is worth emphasizing that the atom‐cavity system presented in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [37,42] is not the keystone for our multiple reflection scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' It can be replaced by any other quantum systems say Rydberg blockade [43‐45], photon blockade [46], nondemolition measurement of an optical photon [47,48] and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' The physics behind our scheme is similar to the interaction free measurement along with quantum Zeno effect [49‐51], which explain another important result of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' When the atom‐cavity system is used, the simulation shows that our scheme becomes insensitive to both atomic spontaneous emission and detuning between the atom and the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' The insensitivity increases as the number of interactions increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Consequently, our scheme can achieve better performance by just enhancing the quality of the linear optical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' RESULTS Multiple reflection scheme As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='1, the scheme consists of a Michelson interferometer and a single‐side cavity‐ atom system [37,42,52‐54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' The interference occurs between the light fields in Zones 0 and 1, which are located on the left and right sides of the beam splitter �𝐵𝑆�, respectively, separated by a dotted line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Assume that 𝑎� � (𝑧 � 0,1) represents the creation operator of the light field in Zone 𝑧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' The function of 𝐵𝑆 can be described by 𝑎� � → 𝑎� � cos 𝜃� � 𝑎� � sin 𝜃� and 𝑎� � → 𝑎� � cos 𝜃� � 𝑎� � sin 𝜃� [51], where cos� 𝜃� represents the reflectivity of 𝐵𝑆 and 𝜃� � 𝜋 2𝑀 ⁄ ( 𝑀 is an integer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' In addition, 𝑆 stands for light source, 𝐶 stands for optical circulator, 𝑆𝑀 stands for switchable mirror (In the experiment, it can be realized by fiber switch and mirrors [55]), which is transparent when it is turned off, and 𝑃𝑆 stands for phase shifter, which adds a π phase shift to the light field only as it propagates from 𝐵𝑆 to single‐side cavity 𝑆𝑆𝐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' When 𝑃𝑆 works, its function can be described as 𝑎� � → �𝑎� �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' As for 𝑆𝑆𝐶����, it is constituted by two facing mirrors 𝐶𝑀����� and 𝐶𝑀�����.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Ideally, 𝐶𝑀� is assumed to have perfect reflection, but 𝐶𝑀� is allowed for in‐ and outcoupling of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' 𝑆𝑆𝐶� is an empty cavity, while 𝑆𝑆𝐶� traps a three‐level atom whose level configuration is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Only the transition between levels |↑⟩ and |𝑒⟩ is strongly coupled by the cavity mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' According to Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [37,42], when the atom is in |↑⟩, due to normal‐mode splitting [53], an incident weak coherent light pulse |𝛼⟩, which is resonant with the empty cavity, does not enter the cavity, but is reflected directly with no phase change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' The corresponding description of the reflection due to 𝑆𝑆𝐶� is 𝑎� � → 𝑎� �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' As for the transition between |↓⟩ and |↑⟩, it is decoupled from the cavity mode due to large detuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Therefore, when the atom is in |↓⟩, the cavity can be treated as empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' The incident pulse enters the cavity and is reflected back but with a 𝜋 phase [37,42], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=', 𝑎� � → �𝑎� �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Last but not least, the feature of our scheme is that 𝑆𝑀 can be turned on so that a coherent light pulse travels back and forth inside the interferometer and hence interacts with the atom 𝑀 cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' One cycle is defined as a wave packet starting at 𝑆𝑀, going through 𝐵𝑆 twice, and returning to 𝑆𝑀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' 1 Multiple reflection scheme based on a Michelson interferometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' When the switchable mirrors (𝑆𝑀) are turned on, the coherent light pulse is bounced inside the interferometer and interact with the single‐ side cavity (𝑆𝑆𝐶) for 𝑀 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Inside 𝑆𝑆𝐶� there is an atom whose level structure is shown on the up‐left side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' At the beginning of the preparation, 𝑆𝑀 is transparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' The light source emits a coherent pulse into the interferometer, while the light field in Zone 1 is in a vacuum state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' The corresponding initial state of the light field is |𝛼, 0⟩ � exp�𝛼𝑎� � � 𝛼∗𝑎��|0,0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' When the pulse passes, 𝑆𝑀 turns on to start 𝑀 cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Supposing that the atom is prepared in a superposition state �|↑⟩ � |↓⟩�/√2 initially, after 𝑚 cycles, the wave‐function of the whole system becomes [56] �𝜓����� � 1 √2 �|𝛼, 0⟩|↑⟩ � |𝛼 cos 2 𝑚𝜃�, 𝛼 sin 2 𝑚𝜃�⟩|↓⟩�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' �1� When 𝑚 � 𝑀, we have the light‐atom entangled state �|𝛼, 0⟩|↑⟩ � |�𝛼, 0⟩|↓⟩� √2 ⁄ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Apparently, no photons appear at 𝑆𝑀� side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' After measuring the atom with basis �|↑⟩ � |↓⟩� √2 ⁄ , the corresponding even/odd optical CS, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=', �|𝛼⟩ � |�𝛼⟩� √2 ⁄ , is output from 𝑆𝑀� side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' So far, we have only focused on the ideal case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' In the following, nonetheless, we analyze the performance of the multiple reflection scheme for non‐ideal situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' We show that our scheme highly durable when it comes to parameter variations such as atomic spontaneous emission decay and atom‐cavity detuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Practical parameter analysis Regarding the practical atom‐cavity system (𝑆𝑆𝐶�), the incident light field is not only reflected, but also transmitted and scattered [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' To evaluate these effects, we set that 2𝛾 and 𝜔� as the Atomiclevelstructure CMTO ISSCO e) Cavity Empty CMROD 11> PS (/+<)PS Input ■ SM BS c Atom S CMR1 CMT1 Output SM, Zone 0 Zone 1spontaneous emission decay rate and transition frequency of the atomic transition between |𝑒⟩ and |↑⟩, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' The coupling constant between the cavity mode with frequency 𝜔� and the atomic transition is 𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' The atom‐cavity detuning is Δ � 𝜔� � 𝜔�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Moreover, we set 𝜅���� as the cavity field decay rate into the external light field on the 𝐶𝑀���� side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Considering that the atom is hardly excited in our scheme, as long as the condition of slowly varying light intensities is satisfied [37,54,57], 𝑆𝑆𝐶� can be well described by the input‐output theory [58,59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Suppose that |𝛼�,�↑⟩ is the incident coherent light field from 𝐶𝑀�� side when the atom is in |↑⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' The cavity reflection |𝛼�,�↑⟩ satisfies [56] 𝛼�,�↑ � �1 � 2𝜅��𝑖𝛥 � 𝛾� 𝜅�𝑖𝛥 � 𝛾� � 𝑔�� 𝛼�,�↑ � �𝜂�,�↑�𝑒���,�↑𝛼�,�↑, �2� where 𝜅 � 𝜅� � 𝜅�, �𝜂�,�↑� � is the reflectivity and 𝛽�,�↑ describes the phase of the reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Similarly, for the transmission of the cavity �𝛼�,�↑�, we have 𝛼�,�↑ � � 2�𝑖𝛥 � 𝛾�√𝜅�𝜅� 𝜅�𝑖𝛥 � 𝛾� � 𝑔� 𝛼�,�↑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' �3� Regarding the scattering field �𝛼�,�↑� due to the atomic spontaneous emission, we have 𝛼�,�↑ � 2𝑔√𝜅�𝛾 𝜅�𝑖𝛥 � 𝛾� � 𝑔� 𝑎�,�↑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' �4� As for the situation that the atom is in |↓⟩, we still assume that the atom is completely unaffected by the cavity mode due to the large detuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Therefore, 𝑆𝑆𝐶� in such case can be treated the same as the empty cavity 𝑆𝑆𝐶�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' By setting 𝑔 � 0 in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' (2)‐(4), we can immediately obtain the corresponding reflection and transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' As for the scattering light field, it is obviously 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Based on the above mathematical description of 𝑆𝑆𝐶� and 𝑆𝑆𝐶�, we can numerically simulate the dynamic evolution process of the input coherent pulse |𝛼⟩ and the fidelity of the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' We suppose that the target state is |𝜓�⟩ � �|𝛼⟩|↑⟩ � |�𝛼⟩|↓⟩� √2 ⁄ , and the final state of the whole system after 𝑀 cycles is �𝜓�� � �|𝐶�↑⟩|𝑙𝑜𝑠𝑠↑⟩|↑⟩ � |𝐶�↓⟩|𝑙𝑜𝑠𝑠↓⟩|↓⟩� √2 ⁄ with �𝑙𝑜𝑠𝑠↑�↓�� � �𝐶�↑�↓�� ⊗ ∏ �𝛼�,�↑�↓� ��� � �𝛼�,�↑�↓� ��� � �𝛼�,�↑�↓� ��� � �𝛼�,�↑�↓� ��� � � ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Here state |𝐶�↑�↓�⟩ (𝑧 � 0,1) denotes the outputs appearing at 𝑆𝑀� side when the atom is in state |↑ �↓�⟩, and �𝛼�,�↑�↓� ��� � ��𝛼�,�↑�↓� ��� �� denotes the transmission (scattering) field generated by 𝑆𝑆𝐶� in 𝑚‐th cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Therefore, �𝑙𝑜𝑠𝑠↑�↓�� includes all optical losses, while the fidelity is obtained by tracing �𝑙𝑜𝑠𝑠↑�↓�� , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=', 𝐹 � 𝑇𝑟�����⟨𝜓�|𝜓�〉�𝜓��𝜓�〉� � �|⟨𝛼|𝐶�↑⟩|� � |⟨�𝛼|𝐶�↓⟩|� � 2Re�⟨𝛼|𝐶�↑〉⟨𝐶�↓|�𝛼⟩⟨𝑙𝑜𝑠𝑠↓|𝑙𝑜𝑠𝑠↑⟩��/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' As a comparison, we also consider the single reflection model in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='[37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' More specifically, the input |𝛼⟩ is directly reflected by 𝑆𝑆𝐶� , and the corresponding output state is ��𝛼�,�↑�|𝑙𝑜𝑠𝑠↑⟩|↑⟩ � �𝛼�,�↓�|𝑙𝑜𝑠𝑠↓⟩|↓⟩� √2 ⁄ with �𝑙𝑜𝑠𝑠↑�↓�� � �𝛼�,�↑�↓���𝛼�,�↑�↓��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' In this model, the constraints on the atomic parameters 𝛾 and Δ can be directly obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' For the empty cavity case (atom is in |↓⟩), as long as 𝜅� � 0, the ideal reflection 𝛼� � �𝛼� can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' As for the case where the atom is in |↑⟩, the condition for ideal reflection 𝛼� � 𝛼� is Δ � 𝛾 � 𝜅� � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' If only 𝛾 is non‐zero, we can see that the ideal reflection can be approximately achieved when 𝛾 ≪ 𝑔�/𝜅� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' As 𝛾 increases, the cavity reflectivity �𝜂�,�↑� � decreases monotonically until it drops to 0 when 𝛾 � 𝑔�/𝜅�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' If we focus on Δ, however, it only affects 𝛽�,�↑ when 𝛾 � 𝜅� � 0, since �𝜂�,�↑� � � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' As Δ varies from �∞ to ∞, 𝛽�,↑ decreases monotonically from 𝜋 to �𝜋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' In order to ensure that 𝛽�,�↑ is close to 0, the constraint Δ ≪ 𝑔�/𝜅� is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='0 \uf067=2\uf070x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='0MHz \uf06emax Color: |\uf061|2=4 |\uf061|2=10 |\uf061|2=16 Style: M=5 M=20 M=100 \uf06emax Multiple reflection model Fidelity \uf067 |\uf061|2=4 Single reflection model Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Fidelity 𝐹 versus dimensionless 𝛾� � 𝜅�𝛾/𝑔�with 𝑔 � 2𝜋 � 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='8𝑀𝐻𝑧, 𝜅� � 𝜅 � 2𝜋 � 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='3𝑀𝐻𝑧 and 𝛥 � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' The dashed double doted pink curve is for the single reflection case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Other curves are for the multiple reflection case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Different colors represent different |𝛼|�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Different styles represent different 𝑀, except that the dotted curves are plotted for 𝑣��� with 𝑀 � 20, which is the maximum value of the average photon number reaching 𝑆𝑆𝐶� in each cycle when the atom is in |↑⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' In our multiple reflection scheme, however, the above constraints are relaxed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' In the following, we show that our scheme can be insensitive to atomic parameters 𝛾 and Δ, thus the fidelity of the CS depends only on the quality of the linear optical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' In order to analyze the effect of 𝛾 , we plot the fidelity against 𝛾� � 𝜅�𝛾/𝑔� with 𝑔 � 2𝜋 � 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='8𝑀𝐻𝑧 , 𝜅� � 𝜅 � 2𝜋 � 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='3𝑀𝐻𝑧 and Δ � 0 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' The pink dot‐dot‐dash curve is plotted for the single reflection model with |𝛼|� � 4, which has almost reached the upper limit of such model [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Other curves are plotted for the multiple reflection model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' The color black/red/blue represents |𝛼|� � 4 10 16 ⁄ ⁄ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' The curve style dash/solid/dot‐dash denotes 𝑀 � 5 20 100 ⁄ ⁄ , while the dotted curves are drawn for 𝑣��� with 𝑀 � 20 instead of fidelity, where 𝑣��� � max ��𝛼�,�↑ ��� � � , �𝛼�,�↑ ��� � � , … , �𝛼�,�↑ ���� � … � is the maximum value of the average photon number reaching 𝑆𝑆𝐶� in each cycle when the atom is in |↑⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' As shown in the figure, 𝑣��� is always less than 1 (For other 𝑀, the situation is similar), which validates the low atomic excitation probability condition, hence Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' (2)‐(4) are valid for simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' By comparison, we can see that the multiple reflection scheme outperforms the single reflection scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' In our scheme, it is evident that fidelity increases as 𝑀 increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Whereas for larger |𝛼|�, larger 𝑀 is required to achieve the same fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' More importantly, for 𝛾 much larger than 2𝜋 � 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='0𝑀𝐻𝑧 (This value is taken from the experiment in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' It corresponds to 𝛾� � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='11 and has been marked in the figure), our scheme can still provide large 𝐹.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' To better explain the result, we consider the extreme case when 𝛾� � 1, which means all photons reaching 𝑆𝑆𝐶� in a single cycle are lost when the atom is in |↑⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Under such conditions, the interference between Zone 0 and Zone 1 is continuously interrupted, resulting in the output light field state in Zone 0 becomes |𝛼 cos� 𝜃� cos��� 2 𝜃�⟩ [50,51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Since cos� 𝜃� cos��� 2 𝜃� � 1 � 𝜋� 2𝑀 ⁄ tends to 1 as 𝑀 tends to infinity, this implies that the light field is frozen in its initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Such result is exactly what we look for, and the mechanism is called the quantum Zeno effect [49,50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' In practice, 𝑀 is finite, hence, the quantum Zeno effect is inevitably accompanied by photon loss, which is proportional to |𝛼|�, but tends to 0 as 𝑀 increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Subsequently, this can explain that in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='2, the larger 𝑀 and the smaller |𝛼|�, the better the fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' So far, our discussion is about 𝛾� � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' As for the case of 0 � 𝛾� � 1, the situation is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' There is a mixture of two physical mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' The first is to maintain the initial state by phase modulation, which does not bring any photon loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' The second is the quantum Zeno effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' It is worth mentioning that the fidelity in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='2 decreases monotonically as 𝛾� increases, which implies that the upper limit of the total photon loss of our scheme is determined by the quantum Zeno effect, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=', 𝑀 and |𝛼| only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Together, the two mechanisms ensure that our scheme has higher fidelity and higher tolerance to 𝛾 than the single reflection scheme as 𝑀 increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' In addition, since the condition 𝛾 ≪ 𝑔�/𝜅� is relaxed, it implies that our scheme does not require strong coupling between atom and cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Following the analysis of 𝛾, we discuss the impact of Δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' We have shown that by interrupting the interference, the transmission of the light field from Zone 0 to Zone 1 can be suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Note that the phase mismatch between the two Zones also interrupts the interference, we expect that our scheme can have high tolerance of Δ as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' 3, we plot the fidelity against Δ� � 𝜅�Δ/𝑔�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Solid curves are for 𝛾 � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Dotted dashed curves are for 𝛾 � 2𝜋 � 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='0𝑀𝐻𝑧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' The values of 𝑔, 𝜅� and 𝜅� are the same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' In addition, the pink curves are plotted for the single reflection model with |𝛼|� � 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' As for the multiple reflection model, the black curves are for |𝛼|� � 4, 𝑀 � 5, the red curves are for |𝛼|� � 10, 𝑀 � 20 and the blue curves are for |𝛼|� � 16, 𝑀 � 100, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='3, even for large |𝛼|, as long as 𝑀 is large, our scheme can be insensitive to Δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='0 Style: \uf067=0 \uf067=2\uf070x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='0MHz Color: Single reflection model |\uf061|2=4, M=5 |\uf061|2=16, M=100 |\uf061|2=10, M=20 Fidelity \uf044 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Fidelity 𝐹 versus dimensionless Δ� � 𝜅�Δ/𝑔� with 𝑔 � 2𝜋 � 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='8𝑀𝐻𝑧 and 𝜅� � 𝜅 � 2𝜋 � 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='3𝑀𝐻𝑧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' The solid curves are for 𝛾 � 0, and the dotted dashed curves are for 𝛾 � 2𝜋 � 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='0𝑀𝐻𝑧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' The pink curves are for the single reflection model with |𝛼|� � 4, and other curves are for the multiple reflection cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' In the above analyses, we ignore the influence of the cavity parameter 𝜅�, which will be discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' (2), the reflectivity of an empty cavity is |�𝜅� � 𝜅�� �𝜅� � 𝜅� ⁄ �|� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [37], 𝜅� � 2𝜋 � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='2 𝑀𝐻𝑧 and 𝜅� � 2𝜋 � 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='3 𝑀𝐻𝑧 , which results in a reflectivity of only about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='7 for single reflection, while after a few reflections, almost all photons are lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Therefore, the cavity employed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [37] is unfortunately not suitable for our scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' To increase reflectivity, one needs either decrease 𝜅�, or increase 𝜅�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' The latter is simpler in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' However, although increasing 𝜅� can reduce the photon loss during the interference of two empty cavities (The atom is in state |↓⟩), it also increases the photon loss in the presence of atom‐cavity coupling (The atom is in state |↑⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' To verify this, we plot effective fidelity 𝐹�� � 𝑇𝑟������𝜓���𝜓�⟩ � against 𝜅� with |𝛼|� � 8, 𝑀 � 10, 𝛾 � 2𝜋 � 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='0𝑀𝐻𝑧 and Δ � 0 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Here, the target state is set as �𝜓��� � ��𝛼���|↑⟩ � ��𝛼���|↓⟩� √2 ⁄ with 𝛼�� � �𝐶�↓ � 𝛼 ��𝜅� � 𝜅�� �𝜅� � 𝜅�� ⁄ �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Note that |𝐶�↓⟩ is the output when the atom is in |↓⟩ , where interference occurs between the two empty cavities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' If the optical parameters of these two cavities are the same, only intensity of the output is affected and reduced from |𝛼|� to �𝛼��� �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' In the figure, the solid (dashed) curves are plotted for 𝜅� � 2𝜋 � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='02 �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='002�𝑀𝐻𝑧 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' The black/red/blue curves are plotted for 𝑔 � 2𝜋 � 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='8 15 30 ⁄ ⁄ 𝑀𝐻𝑧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' In addition, the pink curves are plotted for �𝛼��� �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' We can see that 𝐹�� can be significantly improved as 𝜅� decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' As for 𝜅�, when it increases at the beginning, �𝛼��� � rapidly rises to its maximum value 8, which causes 𝐹�� to increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Subsequently, photon loss due to atom‐cavity coupling plays a major role, resulting in the decrease of 𝐹��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Particularly, we note that for the black curve, when 𝐹�� starts to decrease, its corresponding �𝛼��� � is not close to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' The reason is that 𝜅� is approaching to the limit 𝑔� 𝛾 ⁄ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Under such limit, the photon loss of a single reflection on 𝑆𝑆𝐶� when the atom is in |↑⟩ is almost 100%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Therefore, we plot for larger 𝑔 in order to increase the limit so that �𝛼��� � can get closer to the maximum value 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' We can see that the maximum value of 𝐹�� increases as 𝑔 increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Moreover, 𝜅� maintains wide range of high fidelity (see blue curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' This is because the constraint 𝑔� ≫ 𝜅�𝛾 in our scheme is relaxed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' However, we must emphasize that the larger 𝑔 is not necessary for high fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' By decreasing 𝜅�, we can achieve the same purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' In fact, the motivation of this work is to reduce the influence of the atom, and to show that the performance of our protocol can be improved by just upgrading the linear optical system, such as the parameters 𝑀 and 𝜅�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Besides the atomic parameters (𝛾, Δ) and linear optical system parameters (𝜅�, 𝑀), next we provide a discussion about the influence of the decoherence between the atomic states |↓⟩ and |↑⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Obviously, our scheme requires the atom to remain in superposition at least until the end of 𝑀 cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Nevertheless, we need to mention that the multiple reflection processes hardly affect the atomic decoherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' When the atom is in |↑⟩, the low atomic excitation probability can be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' As for the atom in |↓⟩, it is not coupled to the light field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' While the atomic superposition state has been reported to last about 400𝜇𝑠 [60,61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' The full‐width at half‐maximum of the light pulse that is employed in the experiment of single reflection model is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='3𝜇𝑠 [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Therefore, it is possible for our scheme to be completed before the decoherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' 10 20 30 40 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='0 10 20 30 40 50 0 2 4 6 8 Effective Fidelity, Fef Color : Fef , g=2\uf070x7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='8MHz Fef , g=2\uf070x15MHz Fef , g=2\uf070x30MHz Style : \uf06b\uf054=2\uf070x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='02MHz \uf06b\uf054=2\uf070x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='002MHz Multiple reflection model |\uf061|2=8 M=10 |\uf061ef|2 \uf06bR(x2\uf070MHz) |\uf061ef|2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Effective fidelity 𝐹�� and the output intensity �𝛼��� � versus 𝜅� with different 𝑔 and 𝜅� for the multiple reflection model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' In addition, |𝛼|� � 8, 𝑀 � 10, 𝛾 � 2𝜋 � 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='0𝑀𝐻𝑧 and 𝛥 � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' The initial state of the system is �|𝛼⟩|↑⟩ � |�𝛼⟩| ↓⟩� √2 ⁄ and the target state is ��𝛼���|↑⟩ � ��𝛼���� ↓�� √2 ⁄ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' DISCUSSION Advantages of multiple reflection scheme Compared with the single reflection model, our multiple reflection scheme has two main advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' First, our scheme provides the single atom with the means to manipulate a strong coherent light field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' When the atom is in |↓ �↑�⟩, the light field |𝛼⟩ evolves to | � 𝛼⟩�|𝛼⟩�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' We emphasizes that in the single reflection model [37], the above phase manipulation can only be realized when |𝛼|� is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' As |𝛼|� increases, the single atom can no longer prevent the light field from entering the cavity (In this case, regardless of the state of the atom, the reflected light field carries a 𝜋 phase shift just like the empty cavity case), causing the atom to be excited from state |↑⟩ to |𝑒⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' As a result, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' (2) is no longer valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' In our scheme, however, when the atom is in |↑⟩, only a small fraction of light touches 𝑆𝑆𝐶� in each cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Its average photon number is |𝛼 sin 𝜃�|� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' By adjusting the transmittance of 𝐵𝑆, this value can be far less than 1, thus preventing the atom from being excited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Consequently, even after a large number of cycles, the phase of a strong coherent light field still can be manipulated by the single atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' This result illustrates that our multiple reflection scheme provides a single qubit with the ability to control large amplitude light field, even at macroscopic level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Second, our scheme does not require a high‐quality atom‐cavity coupling system, and it has a high tolerance for atomic parameters (𝛾 and Δ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' In the single reflection model, the phase manipulation depends on the interaction between the atom and the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Hence, the constraint 𝑔� ≫ 𝜅�𝛾 is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' In our multiple reflection model, however, the phase manipulation depends on the interference of light between Zones 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' If the interference continues uninterrupted, the light field eventually carries a 𝜋 phase shift ( |𝛼⟩ → | � 𝛼⟩ ), whereas if interrupted, the phase remains unchanged (|𝛼⟩ → |𝛼⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' It is worth noting that in the process of generating 𝜋 phase, the interference occurs only between two empty cavities and the atom is not involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Unlike the interference case, the interruption of the interference is more likely to occur, bearing in mind that atom‐cavity system from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [37] is not the only way to realize the interruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' For example, if we replace 𝑆𝑆𝐶�(𝑆𝑆𝐶�) by a photon‐absorbing object (mirror), the scheme in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='1 becomes a typical interaction‐free measurement scheme based on quantum Zeno effect [49] (the difference from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [49] is that here we use a Michelson interferometer instead of a chain of Mach‐Zehnder interferometers, and a coherent light source instead of a single photon source).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Since the photons entering Zone 1 are absorbed in each cycle, the light field is suppressed in Zone 0, maintaining its initial state |𝛼⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' In fact, some studies have further shown that even if the object causes only a partial loss of light, it still can interrupt the interference process and prevent the evolution of the light field [62], which is consistent with our numerical analysis results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Note that in our CS preparation scheme, the main role of atom‐cavity system is just to interrupt the interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Therefore, our scheme does not require a high‐quality atom‐ cavity coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Even if the atom‐cavity system has imperfections such as photon scattering by the atom, CS can be still prepared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Scalability of multiple reflection scheme From the above analysis, we can see that the atom‐cavity system from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [37] is not necessary to accomplish our CS preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' It can certainly be replaced by any quantum object that is in a superposition of passing/absorbing photons such as Rydberg blockade [43‐45] and photon blockade [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Moreover, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' 3 implies that if the object adds an additional phase to those photons passing through it instead of absorbing them, it also leads to freezing the evolution of the initial state |𝛼⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' This suggests that the three‐level atomic model, used in nondemolition measurement [47,48], can also be used to replace the atom‐cavity system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Therefore, in our CS preparation method, the multiple reflection model is more indispensable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' In addition, our scheme can be used beyond the preparation of CS, and realize the entangled coherent state required in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [20,21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' To do so, we just need to turn off 𝑆𝑀 when 𝑚 � 𝑀/2 instead of 𝑚 � 𝑀, so that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' (2) becomes �|𝛼, 0⟩|↑⟩ � |0, 𝛼⟩|↓⟩� √2 ⁄ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Last but not least, we focus on the optical platform so far, nevertheless, our method also works for other platforms such as superconducting microwave resonator [63,64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' In summary, we have proposed a deterministic method to entangle an atom to a large‐ amplitude coherent pulse, thus realizing the preparation of a large‐amplitude optical CS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' A multiple reflection scheme is used, which brings two advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' First, in each reflection, the actual number of photons manipulated by the atom is very small, which ensures that the single atom can properly control the phase of the reflected field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Second, due to quantum Zeno effect, our scheme becomes insensitive to atomic parameters 𝛾 and Δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' The sensitivity continues to decrease as the number of reflections 𝑀 increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' This allows our scheme to improve the fidelity of the output CS only by improving the linear optical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Data availability: Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Code availability: The code generated to analyze the protocol is available from the corresponding author upon reasonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Acknowledgements: This work is supported by a grant from the King Abdulaziz City for Science and Technology (KACST), and Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' NPRP 13S‐0205‐200258 of the Qatar National Research Fund (QNRF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Author contributions: The theory was conceived by Z‐H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Numerical calculations were performed by Z‐Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' under the supervision of Z‐H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' The project was supervised by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' All the authors participated in the manuscript preparation, discussions, and checks of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Competing interests: The authors declare no competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Additional information: Correspondence and requests for materials should be addressed to Z‐H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Reference: [1] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Schrödinger, “Die gegenwärtige Situation in der Quantenmechanik”, Naturwissenschaften 23, 807 (1935).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [2] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Haroche, “Nobel Lecture: Controlling photons in a box and exploring the quantum to classical boundary”, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' 85, 1083 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [3] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Wineland, “Nobel Lecture: Superposition, entanglement, and raising Schrödinger’s cat”, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' 85, 1103 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [4] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Li, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='‐L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Zou, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Albert, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Muralidharan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Girvin, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Jiang, “Cat codes with optimal decoherence suppression for a lossy bosonic channel”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' 119, 030502 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [5] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='‐W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Lee, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Jeong, “Near‐deterministic quantum teleportation and resource efficient quantum computation using linear optics and hybrid qubits”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' A 87, 022326 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [6] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Sangouard, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Simon, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Gisin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Laurat, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Tualle‐Brouri, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Grangier, “Quantum repeaters with entangled coherent states”, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' B 27, A137 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [7] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Brask, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Rigas, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Polzik, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Andersen, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Sørensen, “Hybrid long distance entanglement distribution protocol”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' 105, 160501 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [8] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Jeong and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Kim, “Efficient quantum computation using coherent states”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' A 65, 042305 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [9] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Ralph, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Gilchrist, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Milburn, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Munro, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Glancy, “Quantum computation with optical coherent states”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' A 68, 042319 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [10] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Lund, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Ralph, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Haselgrove, “ Fault‐tolerant linear optical quantum computing with small‐amplitude coherent states”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' 100, 030503 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [11] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Mirrahimi, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Leghtas, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Albert, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Touzard, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Schoelkopf, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Jiang, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Devoret, “Dynamically protected cat‐qubits: a new paradigm for universal quantum computation”, New Journal of Physics 16, 045014 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [12] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Su, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Dhand, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Ralph, “Universal quantum computation with optical four‐ component cat qubits”, arXiv:2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='12278 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [13] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Hastrup and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Andersen, “All‐optical catcode quantum error correction”, arXiv:2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='12225 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [14] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Cochrane, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Milburn, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Munro, “Macroscopically distinct quantum‐ superposition states as a bosonic code for amplitude damping”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' A 59, 2631 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [15] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Leghtas, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Kirchmair, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Vlastakis, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Schoelkopf, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Devoret, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Mirrahimi, “Hardware‐efficient autonomous quantum memory protection”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' 111, 120501 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [16] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Bergmann and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' van Loock, “Quantum error correction against photon loss using multicomponent cat states”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' A 94, 042332 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [17] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Ofek, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Petrenko, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Heeres, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Reinhold, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Leghtas, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Vlastakis, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Liu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Frunzio, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Girvin, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Jiang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Mirrahimi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Devoret, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Schoelkopf, Extending the lifetime of a quantum bit with error correction in superconducting circuits, Nature 536, 441 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [18] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Munro, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Nemoto, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Milburn, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Braunstein, “Weak‐force detection with superposed coherent states”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' A 66, 023819 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [19] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Gilchrist, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Nemoto, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Munro, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Ralph, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Glancy, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Braunstein, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Milburn, “Schrödinger cats and their power for quantum information processing”, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' B 6, S828 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [20] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Joo, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Munro, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Spiller, “Quantum metrology with entangled coherent states”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' 107, 083601 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [21] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Li, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Yang, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Jin, “Quantum Fisher information of entangled coherent states in the presence of photon loss”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' A 88, 043832 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [22] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Ghosh, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Sharma, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Roy, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Panigrahi, “Mesoscopic quantum superposition of the generalized cat state: A diffraction limit”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' A 92, 053819 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [23] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Facon, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='‐K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Dietsche, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Grosso, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Haroche, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='‐M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Raimond, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Brune, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Gleyzes, “A sensitive electrometer based on a Rydberg atom in a Schrödinger‐cat state”, Nature 535, 262– 265 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [24] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Polino, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Valeri, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Spagnolo, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Sciarrino, “Photonic quantum metrology”, AVS Quantum Science 2, 024703 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [25] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Monroe, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Meekhof, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' King, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Wineland, “A “Schrödinger cat” superposition state of an atom”, Science 272, 1131 (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [26] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Kienzler, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Flühmann, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Negnevitsky, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='‐Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Lo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Marinelli, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Nadlinger, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Home, “Observation of quantum interference between separated mechanical oscillator wave packets”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' 116, 140402 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [27] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Deleglise, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Dotsenko, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Sayrin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Bernu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Brune, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='‐M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Raimond, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Haroche, “Reconstruction of non‐classical cavity field states with snapshots of their decoherence”, Nature 455, 510 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [28] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Vlastakis, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Kirchmair, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Leghtas, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Nigg, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Frunzio, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Girvin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Mirrahimi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Devoret, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Schoelkopf, “Deterministically encoding quantum information using 100‐ photon Schrödinger cat states”, Science 342, 607 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [29] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Neergaard‐Nielsen, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Nielsen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Hettich, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Mølmer, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Polzik, “Generation of a superposition of odd photon number states for quantum information networks”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' 97, 083604 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [30] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Ourjoumtsev, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Tualle‐Brouri, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Laurat, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Grangier, “Generating optical Schrödinger kittens for quantum information processing”, Science 312, 83 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [31] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Gerrits, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Glancy, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Clement, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Calkins, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Lita, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Miller, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Migdall, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Nam, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Mirin, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Knill, “Generation of optical coherent‐state superpositions by number‐resolved photon subtraction from the squeezed vacuum”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' A 82, 031802(R) (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [32] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Takahashi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Wakui, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Suzuki, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Takeoka, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Hayasaka, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Furusawa, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Sasaki, “Generation of Large‐Amplitude Coherent‐State Superposition Via Ancilla‐Assisted Photon Subtraction”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' 101, 233605 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [33] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Huang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Le Jeannic, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Ruaudel, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Verma, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Shaw, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Marsili, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Nam, E Wu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Zeng, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='‐C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Jeong, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Filip, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Morin, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Laurat, “Optical Synthesis of Large‐Amplitude Squeezed Coherent‐State Superpositions with Minimal Resources”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' 115, 023602 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [34] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Ourjoumtsev, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Jeong, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Tualle‐Brouri, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Grangier, “Generation of optical ‘Schrödinger cats’ from photon number states”, Nature (London) 448, 784 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [35] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Ulanov, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Fedorov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Sychev, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Grangier, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Lvovsky, “Loss‐tolerant state engineering for quantum enhanced metrology via the reverse Hong‐Ou‐Mandel effect”, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' 7, 11925 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [36] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Sychev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Ulanov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Pushkina, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Richards, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Fedorov, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=" Lvovsky, “Enlargement of optical Schrödinger's cat states”, Nature Photonics 11, 379 (2017)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [37] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Hacker, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Welte, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Daiss, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Shaukat, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Ritter, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Li, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Rempe, “Deterministic creation of entangled atom–light Schrödinger‐cat states”, Nature Photonics 13, 110 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [38] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Dakna, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Anhut, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Opatrný, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Knöll, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='‐G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Welsch, “Generating Schrödinger‐cat‐like states by means of conditional measurements on a beam splitter”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' A 55, 3184 (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [39] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Takase, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='‐i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Yoshikawa, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Asavanant, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Endo, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Furusawa, “Generation of optical Schrödinger cat states by generalized photon subtraction”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' A 103, 013710 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [40] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Lund, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Jeong, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Ralph, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Kim, “Conditional production of superpositions of coherent states with inefficient photon detection”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' A 70, 020101(R) (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [41] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Laghaout, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Neergaard‐Nielsen, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Rigas, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Kragh, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Tipsmark, and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Andersen, “Amplification of realistic Schrödinger‐cat‐state‐like states by homodyne heralding”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' A 87, 043826 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [42] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Wang and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='‐M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Duan, “Engineering superpositions of coherent states in coherent optical pulses through cavity‐assisted interaction”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' A 72, 022320 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [43] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Urban, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Johnson, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Henage, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Isenhower, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Yavuz, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Walker, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Saffman, “Observation of Rydberg blockade between two atoms”, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' 5, 110 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [44] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Johnson, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Urban, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Henage, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Isenhower, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Yavuz, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Walker, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Saffman, “Rabi Oscillations between Ground and Rydberg States with Dipole‐Dipole Atomic Interactions”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' 100, 113003 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [45] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Guo, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Cheng, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Chen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Wang, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Zhang, “Counterfactual quantum‐information transfer without transmitting any physical particles”, Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' 5, 8416 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [46] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Birnbaum, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Boca, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Miller, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Boozer, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Northup, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Kimble, “Photon blockade in an optical cavity with one trapped atom”, Nature (London) 436, 87 (2005) [47] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Nogues, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Rauschenbeutel, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Osnaghi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Brune, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Raimond, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Haroche, “Seeing a single photon without destroying it”, Nature (London) 400, 239 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [48] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Scully and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Zubairy, Quantum Optics (Cambridge University Press, Cambridge, 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [49] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Kwiat, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Weinfurter, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Herzog, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Zeilinger, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Kasevich, “Interaction‐free measurement”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' 74, 4763 (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [50] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Salih, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='‐H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Li, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Al‐Amri, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Zubairy, “Protocol for direct counterfactual quantum communication”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' 110, 170502 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [51] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='‐H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Li, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='‐Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Feng, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Al‐Amri, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Zubairy, “Direct counterfactual quantum communication protocol beyond single photon source”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' A 106, 032610 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [52] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='‐M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Duan, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Kimble, “Scalable Photonic Quantum Computation through Cavity‐ Assisted Interactions”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' 92, 127902 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [53] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Reiserer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Ritter, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Rempe, “Nondestructive Detection of an Optical Photon”, Science 342, 1349 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [54] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Reiserer, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Rempe, “Cavity‐ based quantum networks with single atoms and optical photons”, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' 87, 1379 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [55] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Liu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Ju, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='‐L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Liang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='‐B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Tang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='‐L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Shen Tu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Zhou, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='‐Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Peng, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Chen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='‐Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Chen, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='‐B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Chen, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='‐W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Pan, “Experimental Demonstration of Counterfactual Quantum Communication”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' 109, 030501 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [56] See supplementary materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [57] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Hu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Young, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' O’Brien, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Munro, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Rarity, “Giant optical Faraday rotation induced by a single‐electron spin in a quantum dot: Applications to entangling remote spins via a single photon”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' B 78, 085307 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [58] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Gardiner, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Collett, “Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' A 31, 3761 (1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [59] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Walls, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Milburn, Quantum Optics (Springer‐ Verlag, Berlin, 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [60] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Daiss, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Langenfeld, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Welte, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Distante, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Thomas, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Hartung, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Morin, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Rempe, “A quantum‐logic gate between distant quantum‐network modules”, Science 371, 614 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [61] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Welte, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Thomas, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Hartung, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Daiss, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Langenfeld, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Morin, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Rempe and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Distante, “A nondestructive Bell‐state measurement on two distant atomic qubits”, Nature Photonics 15, 504 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [62] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Wang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='‐H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Xu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Yang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Al‐Amri, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Zubairy, “Exchange unknown quantum states with almost invisible photons”, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Express 27, 20525 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [63] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Wang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Bao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Wu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Li, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Cai, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Ma, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Cai, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Han, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Song, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Sun, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Zhang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Duan, “A flying Schrödinger’s cat in multipartite entangled states”, Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' 8, eabn1778 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [64] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Bao, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Wu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Li, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Cai, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Ma, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Cai, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Han, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Song, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Sun, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Zhang, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Duan, “Experimental preparation of generalized cat states for itinerant microwave photons”, arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='04617 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Supplementary material A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Calculations for Equation 1 In the main text, we have mentioned that 𝑎� and 𝑎� represent the annihilation operators of the light field in Zone 0 and Zone 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Based on this, the function of 𝐵𝑆 can be described as 𝑎� � → 𝑎� � cos 𝜃� � 𝑎� � sin 𝜃� and 𝑎� � → 𝑎� � cos 𝜃� � 𝑎� � sin 𝜃� where 𝜃� � 𝜋/2𝑀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Now, we consider an arbitrary initial photon state |𝐼𝑛𝑖𝑡𝑖𝑎𝑙⟩ � |𝑢, 𝑣⟩ � exp�𝑢𝑎� � � 𝑢∗𝑎��exp�𝑣𝑎� � � 𝑣∗𝑎��|0,0⟩, (A1) which represents that a coherent state |𝑢⟩ is in Zone 0 and a coherent state |𝑣⟩ is in Zone 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' After passing through the 𝐵𝑆, we have the final state |𝐹𝑖𝑛𝑎𝑙⟩ � exp�𝑢�𝑎� � cos 𝜃� � 𝑎� � sin 𝜃�� � 𝑢∗�𝑎� cos 𝜃� � 𝑎� sin 𝜃���, � exp�𝑣�𝑎� � cos 𝜃� � 𝑎� � sin 𝜃�� � 𝑣∗�𝑎� cos 𝜃� � 𝑎� sin 𝜃��� |0,0⟩ � |𝑢 cos 𝜃� � 𝑣 sin 𝜃� , 𝑢 sin 𝜃� � 𝑣 cos 𝜃�⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' (A2) Similarly, consider an arbitrary phase operation 𝑎� → 𝑒��𝑎�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' For the initial state |𝐼𝑛𝑖𝑡𝑖𝑎𝑙⟩ � |𝑢⟩, after the operation, the final state is |𝐹𝑖𝑛𝑎𝑙⟩ � exp �� 1 2 |𝑢|�� � 𝑢� √𝑛!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' 1 √𝑛!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' �𝑒��𝑎�� �|0⟩ ∞ ��� � exp �� � � |𝑢|�� ∑ � √�!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' �𝑢𝑒��� �|𝑛⟩ ∞ ��� � |𝑢𝑒��⟩ (A3) Based on Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' (A2) and (A3), we provide the calculation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' (1) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' At the beginning of the preparation, the wave‐function of the whole system is �𝜓���� � √� � |𝛼, 0⟩�|↑⟩ � |↓⟩� (A4) In the first cycle, after the photons pass through 𝐵𝑆 for the first time, the system state is �𝜓���� � √� � |𝛼 cos 𝜃� , 𝛼 sin 𝜃�⟩�|↑⟩ � |↓⟩� (A5) Before the photons are reflected by 𝑆𝑆𝐶 , the system state becomes √� � |�𝛼 cos 𝜃� , �𝛼 sin 𝜃�⟩�|↑⟩ � |↓⟩� due to 𝑃𝑆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Regarding the reflection, we emphasize that only when the atom is in |↑⟩, 𝑆𝑆𝐶� does not change the phase of the reflected field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' As a result, the wave‐function of the whole system becomes √� � |𝛼 𝑐𝑜𝑠 𝜃� , �𝛼 𝑠𝑖𝑛 𝜃�⟩|↑⟩ � √� � |𝛼 𝑐𝑜𝑠 𝜃� , 𝛼 𝑠𝑖𝑛 𝜃�⟩|↓⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Subsequently, after the second time that the photons pass through 𝐵𝑆, we have �𝜓���� � √� � |𝛼, 0⟩|↑⟩ � √� � |cos 2 𝜃�𝛼, sin 2 𝜃�𝛼⟩|↓⟩ (A6) This state becomes the initial state of the second cycle, and the process is repeated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' It is not difficult to obtain that after 𝑚 cycles, the wave‐function of the whole system is �𝜓����� � √� � |𝛼, 0⟩|↑⟩ � √� � |𝛼 cos 2 𝑚𝜃�, 𝛼 sin 2 𝑚𝜃�⟩|↓⟩ (A7) Here the superscript of �𝜓����� represents the photons pass through 𝐵𝑆 2𝑚 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Calculations for Equations 2‐4 The Hamiltonian of cavity‐atom system (𝑆𝑆𝐶�) can be described as 𝐻 � ℏ𝜔�𝜎�� � ℏ𝜔↑𝜎↑↑ � ℏ𝜔�𝑎�𝑎 � ℏ � � 𝜔�𝑏� ��𝜔��𝑏��𝜔��𝑑𝜔� � －� ���,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='� �ℏ𝑔�𝜎↑�𝑎� � 𝜎�↑𝑎� � ℏ� � � � �𝜎↑�𝑏� ��𝜔�� � 𝜎�↑𝑏��𝜔���𝑑𝜔� � －� �𝑖ℏ� �� � � �𝑎𝑏� ��𝜔�� � 𝑎�𝑏��𝜔���𝑑𝜔� � －� � 𝑖ℏ� �� � � �𝑎𝑏� ��𝜔�� � 𝑎�𝑏��𝜔���𝑑𝜔� � －� (B1) where ℏ𝜔� is the energy of excited atomic state |𝑒⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' ℏ𝜔↑ is the energy of the atomic state |↑⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' 𝜔𝑐 is the frequency of the cavity mode described by annihilation operator 𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' 𝜔𝐽 is the frequency of external field described by annihilation operator 𝑏�𝜔�� with �𝑏𝐽�𝜔𝐽�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' 𝑏𝐽 †�𝜔𝐽 ′�� � 𝛿�𝜔𝐽 � 𝜔𝐽 ′�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' and the subscript 𝑅 represents the external multi‐mode field on 𝐶𝑀� side,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' 𝑇 represents the external field on 𝐶𝑀� side,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' 𝑆 represents the scattering field due to the atomic spontaneous emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' In addition, 𝑔 is coupling constant between the cavity and the atomic transition between |𝑒⟩ and |↑⟩, 2𝛾 is the spontaneous atomic decay rate on the same transition, 𝜅𝑅 and 𝜅𝑇 are cavity field decay rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' We also set that 𝜎↑𝑒 � |↑⟩⟨𝑒|, 𝜎𝑒𝑒 � |𝑒⟩⟨𝑒| and 𝜎↑↑ � |↑⟩⟨↑|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Based on the above Hamiltonian, it is not difficult to obtain the following Heisenberg equations 𝑑𝑎�𝑡� 𝑑𝑡 � �𝑖𝜔𝑐𝑎�𝑡� � 𝑖𝑔𝜎↑𝑒�𝑡� � ∑ � 𝜅𝐽 𝜋 𝐽�𝑅,𝑇 � 𝑏𝐽�𝜔𝐽, 𝑡�𝑑𝜔𝐽 ∞ －∞ , (B2) 𝑑 𝑑𝑡 𝜎↑𝑒�𝑡� � �𝑖�𝜔𝑒 � 𝜔↑�𝜎↑𝑒�𝑡� � 𝑖𝑔�𝜎𝑒𝑒�𝑡� � 𝜎↑↑�𝑡��𝑎�𝑡� �𝑖� � � � �𝜎���𝑡� � 𝜎↑↑�𝑡�� � －� 𝑏��𝜔�, 𝑡�𝑑𝜔� � �𝑖�𝜔� � 𝜔↑�𝜎↑��𝑡� � 𝑖𝑔𝑎�𝑡� � 𝑖� � � � 𝑏��𝜔�, 𝑡�𝑑𝜔� � －� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' (B3) In the approximation, we have assumed that [1‐3] ⟨�𝜎𝑒𝑒 � 𝜎↑↑�𝑎⟩ � �⟨𝑎⟩, (B4) which indicates that the atom stays in the state |↑⟩ most of the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' This can be satisfied when the input is weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' In addition, we can also obtain Heisenberg equations for 𝑏�𝜔�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' They are 𝑑𝑏𝐽�𝜔𝐽,𝑡� 𝑑𝑡 � �𝑖𝜔𝐽𝑏𝐽�𝜔𝐽, 𝑡� � � 𝜅𝑅 𝜋 𝑎�𝑡�, 𝐽 � 𝑅, 𝑇, (B5) 𝑑𝑏𝑆�𝜔𝑆,𝑡� 𝑑𝑡 � �𝑖𝜔𝑆𝑏𝑆�𝜔𝑆, 𝑡� � 𝑖� 𝛾 𝜋 𝜎↑𝑒�𝑡�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' (B6) Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' (B5) and (B6) can be rewritten in integral form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' If we assume that the atom‐light interaction begins at time 𝑇𝑖𝑛 � 𝑡, we have 𝑏𝐽�𝜔𝐽, 𝑡� � 𝑏𝐽�𝜔𝐽, 𝑇𝑖𝑛�𝑒𝑖𝜔𝐽�𝑇𝑖𝑛�𝑡� � � 𝜅𝐽 𝜋 � 𝑎�𝑡′�𝑒𝑖𝜔𝐽�𝑡′�𝑡�𝑑𝑡′ 𝑡 𝑇𝑖𝑛 , (B7) 𝑏𝑆�𝜔𝑆, 𝑡� � 𝑏𝑆�𝜔𝑆, 𝑇𝑖𝑛�𝑒𝑖𝜔𝑆�𝑇𝑖𝑛�𝑡� � 𝑖� 𝛾 𝜋 � 𝜎↑𝑒�𝑡′�𝑒𝑖𝜔𝑆�𝑡′�𝑡�𝑑𝑡′ 𝑡 𝑇𝑖𝑛 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' (B8) If we assume that the atom‐light interaction ends at time 𝑇𝑜𝑢𝑡 � 𝑡, we have 𝑏𝐽�𝜔𝐽, 𝑡� � 𝑏𝐽�𝜔𝐽, 𝑇𝑜𝑢𝑡�𝑒𝑖𝜔𝐽�𝑇𝑜𝑢𝑡�𝑡� � � 𝜅𝐽 𝜋 � 𝑎�𝑡′�𝑒𝑖𝜔𝐽�𝑡′�𝑡�𝑑𝑡′ 𝑇𝑜𝑢𝑡 𝑡 , (B9) 𝑏𝑆�𝜔𝑆, 𝑡� � 𝑏𝑆�𝜔𝑆, 𝑇𝑜𝑢𝑡�𝑒𝑖𝜔𝑆�𝑇𝑜𝑢𝑡�𝑡� � 𝑖� 𝛾 𝜋 � 𝜎↑𝑒�𝑡′�𝑒𝑖𝜔𝑆�𝑡′�𝑡�𝑑𝑡′ 𝑇𝑜𝑢𝑡 𝑡0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' (B10) By integrating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' (B7) with frequency, it is not difficult to obtain that � 𝜅𝐽 𝜋 � 𝑏𝐽�𝜔𝐽, 𝑡�𝑑𝜔𝐽 ∞ －∞ � � �� � � 𝑏��𝜔�, 𝑇���𝑒����������𝑑𝜔� � －� � 2𝜅� � 𝑎�𝑡��𝑑𝑡� � ��� � �� � 𝑒���������𝑑𝜔� � －� � �2𝜅�𝑎�,���𝑡� � 𝜅�𝑎�𝑡�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' (B11) where we have used the relation [4] � 𝑓�𝑡′�𝛿�𝑡 � 𝑡′�𝑑𝑡′ 𝑡 𝑡0 � � 𝑓�𝑡′�𝛿�𝑡 � 𝑡′�𝑑𝑡′ 𝑡1 𝑡 � 1 2 𝑓�𝑡�, �𝑡0 � 𝑡 � 𝑡1�, (B12) and the assumptions (𝐽 � 𝑅, 𝑇) 𝑎𝐽,𝑖𝑛�𝑡� � 1 √2𝜋 � 𝑏𝐽�𝜔𝐽, 𝑇𝑖𝑛�𝑒𝑖𝜔𝐽�𝑇𝑖𝑛�𝑡�𝑑𝜔𝐽 ∞ －∞ , (B13) 𝑎𝐽,𝑜𝑢𝑡�𝑡� � 1 √2𝜋 � 𝑏𝐽�𝜔𝐽, 𝑇𝑜𝑢𝑡�𝑒𝑖𝜔𝐽�𝑇𝑜𝑢𝑡�𝑡�𝑑𝜔𝐽 ∞ －∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' (B14) 𝑎𝑆,𝑖𝑛�𝑡� � 1 √2𝜋 � 𝑏𝑆�𝜔𝑆, 𝑇𝑖𝑛�𝑒𝑖𝜔𝑆�𝑇𝑖𝑛�𝑡�𝑑𝜔𝑆 ∞ －∞ , (B15) 𝑎𝑆,𝑜𝑢𝑡�𝑡� � 1 √2𝜋 � 𝑏𝑆�𝜔𝑆, 𝑇𝑜𝑢𝑡�𝑒𝑖𝜔𝑆�𝑇𝑜𝑢𝑡�𝑡�𝑑𝜔𝑆 ∞ －∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' (B16) Similarly, from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' (B8)‐(B10), we have � 𝛾 𝜋 � 𝑏𝑆�𝜔𝑆, 𝑡�𝑑𝜔𝑆 ∞ －∞ � �2𝛾𝑎𝑆,𝑖𝑛�𝑡� � 𝑖𝛾𝜎↑𝑒�𝑡�, (B17) � 𝜅𝐽 𝜋 � 𝑏𝐽�𝜔𝐽, 𝑡�𝑑𝜔𝐽 ∞ －∞ � �2𝜅𝐽𝑎𝐽,𝑜𝑢𝑡�𝑡� � 𝜅𝐽𝑎�𝑡�, (B18) � 𝛾 𝜋 � 𝑏𝑆�𝜔𝑆, 𝑡�𝑑𝜔𝑆 ∞ －∞ � �2𝛾𝑎𝑆,𝑜𝑢𝑡�𝑡� � 𝑖𝛾𝜎↑𝑒�𝑡�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' (B19) Then, by substituting Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' (B11)(B18) into (B2), we can obtain the dynamic equations 𝑑𝑎�𝑡� 𝑑𝑡 � �𝑖𝜔𝑐𝑎�𝑡� � 𝑖𝑔𝜎↑𝑒�𝑡� � �2𝜅𝑅𝑎𝑅,𝑖𝑛�𝑡� � 𝜅𝑅𝑎�𝑡� � �2𝜅𝑇𝑎𝑇,𝑖𝑛�𝑡� � 𝜅𝑇𝑎�𝑡�, (B20) 𝑑𝑎�𝑡� 𝑑𝑡 � �𝑖𝜔𝑐𝑎�𝑡� � 𝑖𝑔𝜎↑𝑒�𝑡� � �2𝜅𝑅𝑎𝑅,𝑜𝑢𝑡�𝑡� � 𝜅𝑅𝑎�𝑡� � �2𝜅𝑇𝑎𝑇,𝑖𝑛�𝑡� � 𝜅𝑇𝑎�𝑡�, (B21) 𝑑𝑎�𝑡� 𝑑𝑡 � �𝑖𝜔𝑐𝑎�𝑡� � 𝑖𝑔𝜎↑𝑒�𝑡� � �2𝜅𝑅𝑎𝑅,𝑖𝑛�𝑡� � 𝜅𝑅𝑎�𝑡� � �2𝜅𝑇𝑎𝑇,𝑜𝑢𝑡�𝑡� � 𝜅𝑇𝑎�𝑡�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' (B22) By substituting Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' (B17)(B19) into (B3), we have 𝑑 𝑑𝑡 𝜎↑𝑒�𝑡� � �𝑖�𝜔𝑒 � 𝜔↑�𝜎↑𝑒�𝑡� � 𝑖𝑔𝑎�𝑡� � 𝑖�2𝛾𝑎𝑆,𝑖𝑛�𝑡� � 𝛾𝜎↑𝑒�𝑡�, (B23) 𝑑 𝑑𝑡 𝜎↑𝑒�𝑡� � �𝑖�𝜔𝑒 � 𝜔↑�𝜎↑𝑒�𝑡� � 𝑖𝑔𝑎�𝑡� � 𝑖�2𝛾𝑎𝑆,𝑜𝑢𝑡�𝑡� � 𝛾𝜎↑𝑒�𝑡�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' (B24) In the following, we assume that only the input on 𝐶𝑀� side is none‐zero, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=', 𝑎𝑇,𝑖𝑛�𝑡� � 𝑎𝑆,𝑖𝑛�𝑡� � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Then, by subtracting (B20) and (B21), we can get the relation between the input 𝑎𝑅,𝑖𝑛�𝑡� and output 𝑎𝑅,𝑜𝑢𝑡�𝑡�, 𝑎𝑅,𝑖𝑛�𝑡� � �2𝜅𝑅𝑎�𝑡� � 𝑎𝑅,𝑜𝑢𝑡�𝑡�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' (B25) By subtracting (B20) and (B22), we have 𝑎𝑇,𝑜𝑢𝑡�𝑡� � �2𝜅𝑇𝑎�𝑡�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' (B26) By subtracting (B23) and (B24), we have 𝑎𝑆,𝑜𝑢𝑡�𝑡� � �𝑖�2𝛾𝜎↑𝑒�𝑡�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' (B27) In addition to the above relations, we next calculate the steady‐state solution of the dynamic equations (B20)(B21)(B23) by assuming that the cavity‐atom system is driven by the input light field with frequency 𝜔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' We suppose that 𝑎𝑅,𝑖𝑛�𝑡� � 𝛼𝑖,1↑𝑒�𝑖𝜔𝑡, 𝑎�𝑡� � 𝛼𝑒����, 𝜎↑𝑒�𝑡� � 𝜎�𝑒�𝑖𝜔𝑡, 𝑎𝑅,𝑜𝑢𝑡�𝑡� � 𝛼𝑅,1↑𝑒�𝑖𝜔𝑡, 𝑎𝑇,𝑜𝑢𝑡�𝑡� � 𝛼𝑇,1↑𝑒�𝑖𝜔𝑡, 𝑎𝑆,𝑜𝑢𝑡�𝑡� � 𝛼𝑆,1↑𝑒�𝑖𝜔𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' (B28) Then, we can obtain that ��𝑖�𝜔𝑐 � 𝜔� � 𝜅𝑅 � 𝜅𝑇�𝛼 � 𝑖𝑔𝜎� � �2𝜅𝑅𝛼𝑖,1↑ � 0, (B29) ��𝑖�𝜔𝑐 � 𝜔� � 𝜅𝑅 � 𝜅𝑇�𝛼 � 𝑖𝑔𝜎� � �2𝜅𝑅𝛼𝑅,1↑ � 0, (B30) ��𝑖�𝜔𝑒 � 𝜔↑ � 𝜔� � 𝛾�𝜎� � 𝑖𝑔𝛼 � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' (B31) It is not difficult to get that 𝛼𝑖,1↑ � � �𝑖�𝜔𝑐�𝜔��𝜅𝑅�𝜅𝑇��𝑖�𝜔𝑒�𝜔↑�𝜔��𝛾��𝑔2 �2𝜅𝑅�𝑖�𝜔𝑒�𝜔↑�𝜔��𝛾� 𝛼, (B32) 𝛼𝑅,1↑ � � �𝑖�𝜔𝑐�𝜔��𝜅𝑅�𝜅𝑇��𝑖�𝜔𝑒�𝜔↑�𝜔��𝛾��𝑔2 �2𝜅𝑅�𝑖�𝜔𝑒�𝜔↑�𝜔��𝛾� 𝛼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' (B33) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' (B32) shows that when the input increases, the intensity of the cavity field also increase, resulting in the condition (B4) not being satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' With Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' (B32) and (B33), we can calculate Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' (2) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Suppose that the cavity and the external field are resonant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=', 𝜔 � 𝜔�, and 𝛥 � 𝜔� � 𝜔↑ � 𝜔�, we obtain 𝛼𝑅,1↑ 𝛼𝑖,1↑ � 1 � 2𝜅𝑅�𝑖𝛥�𝛾� �𝜅𝑅�𝜅𝑇��𝑖𝛥�𝛾��𝑔2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' (B34) By using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' (B26) and (B27), we can also have Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' (3) and (4) in the main text, which are 𝛼𝑇,1↑ 𝛼𝑖,1↑ � � 2√𝜅𝑅𝜅𝑇�𝑖𝛥�𝛾� �𝜅𝑅�𝜅𝑇��𝑖𝛥�𝛾��𝑔2, (B35) 𝛼𝑆,1↑ 𝛼𝑖,1↑ � 2�𝜅𝑅𝛾𝑔 �𝜅𝑅�𝜅𝑇��𝑖𝛥�𝛾��𝑔2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' (B36) [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Hu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Young, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' O’Brien, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Munro, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Rarity, “Giant optical Faraday rotation induced by a single‐electron spin in a quantum dot: Applications to entangling remote spins via a single photon”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' B 78, 085307 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Reiserer, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Rempe, “Cavity‐ based quantum networks with single atoms and optical photons”, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' 87, 1379 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [3] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Hacker, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Welte, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Daiss, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Shaukat, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Ritter, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Li, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Rempe, “Deterministic creation of entangled atom–light Schrödinger‐cat states”, Nature Photonics 13, 110 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' [4] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Walls, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} +page_content=' Milburn, Quantum Optics (Springer‐ Verlag, Berlin, 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfAgIm/content/2301.02839v1.pdf'} diff --git a/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf b/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..730b7578e798201bb33c97b4ea419beed4586135 --- /dev/null +++ b/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:126873c8147edd46a8e68fc2ba8da86e66275b0c56b755a3c84a4cb8fb4d8ad7 +size 1769868 diff --git a/49A0T4oBgHgl3EQfNv86/vector_store/index.pkl b/49A0T4oBgHgl3EQfNv86/vector_store/index.pkl new file mode 100644 index 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McCalluma, b, G. Milluzzoc, a, M. Borghesia, A. Subielb, F. Romanod +a Centre for Plasma Physics, Queen’s University Belfast, +BT7 1NN, United Kingdom +b Medical Radiation Science, National Physical Laboratory, +Teddington, TW11 0LW, United Kingdom +c Istituto Nazionale di Fisica Nucleare, Laboratori Nazionali del Sud, +Via S. Sofia 62, 95123 Catania, Italy +d Istituto Nazionale di Fisica Nucleare, Sezione di Catania, +Via S. Sofia 64, 95123 Catania, Italy + +E-mail: smccallum05@qub.ac.uk +ABSTRACT: Laser-driven ion beams are defined by a number of unique features, including a large +spread in energy. A stack configuration of radiochromic film (RCF) can be utilized to characterize +such beams through measurements of their energy spectra. A spectroscopic procedure is reported +that allows the proton energy density within each active layer of a radiochromic film (RCF) stack +to be retrieved. This is based upon on a deconvolution algorithm developed through Geant4 Monte +Carlo simulations to correct the contributions of energy depositions within a given film layer. +Through Monte Carlo calculations, the spectrum retrieved from a simulated film stack can be +retrieved and compared with a known energy spectrum, providing an examination of the efficacy +of this tool. Application of the developed deconvolution procedure thus offers the potential to +correctly reconstruct the incident energy spectrum of a laser-driven proton and ion beam from a +stack of irradiated RCF. +KEYWORDS: Detector modelling and simulations I, dE/dx detectors, Plasma diagnostics - +charged-particle spectroscopy, Simulation methods and programs. + + + + + +– 1 – +Contents +1. Introduction +1 +2. Methodology +2 +3. Monte Carlo Analysis + +4. Conclusions + +5. References +2 + + +1. Introduction +Whilst laser-driven proton and light ion acceleration has attracted significant interest for over 20 +years [1, 2], conducting accurate measurements of these beams has proven to be technically +challenging [3-5]. In particular, the ultra-high dose rates and wide spectral distributions make +conventional measurement techniques impracticable [6-8]. For applications, including clinical +and radiobiological ones requiring a precise energy selection, characterisation of such beams +through accurate measurement of their energy spectra is necessary. Spectroscopic methods reliant +on stacked configurations of radiochromic films (RCF) are well-established for measurements of +accelerated proton beams, with several approaches of radiochromic film imaging spectroscopy +(RIS) reported in the literature [9-14]. A stacked configuration of films placed perpendicularly to +the beam orientation can be used to perform an energy resolved measurement of an impinging ion +beam. Differential energy loss results in each particle depositing a fraction of its initial kinetic +energy on every film it passes before coming to arrest. For polyenergetic sources such as laser- +driven beams, a superposition of kinetic energy contributions is amassed across the films, +requiring a calculation for correction of higher energies. This is achieved through a deconvolution +or unfolding of the energy transferred to each film in the stack, so that only the particles stopping +within a given film remain. The aim of the work reported here was to investigate and assess a +developed algorithm for spectroscopy of laser-driven proton and ion beams through Monte Carlo +simulations, studying the possible limitations. This procedure requires knowledge of the RCF +energy sensitivity values, and an algorithm to unfold the proton energy spectrum from the RCF +response, both of which have been evaluated using the Geant4 toolkit [15-17]. Further, the same +Monte Carlo methods were utilised to conduct analysis of the performance and limitations of the +developed technique in acquiring the energy spectrum. Once validated, the spectroscopic +procedure reported offers the potential to reliably extract the laser-driven proton spectra from a +stack of irradiated RCF. + +2. Methodology +Energy resolved measurements of impinging proton and ion beams can be performed using +multiple RCF arranged into a stack configuration. The differing stopping positions for protons of +a given energy within an RCF stack, means each layer can be defined by a unique energy + + + +– 2 – +sensitivity. This is chosen to correspond to the energy required to generate a Bragg peak at that +given depth, defining the energy of protons that will be referred to as peak region protons. Low +energy components stop in the first few layers of the stack, whilst higher energies penetrate +further downstream, giving a total energy composition of stopping protons, in addition to the +fractional contributions of those exceeding the energy sensitivity of a given film layer. Unfolding +the peak energy from the total energy deposited within any RCF can be achieved through the +development of a deconvolution procedure for proton spectroscopy. This relies on an algorithm +utilising weight factors to describe the fractional contributions of each energy component within +every film. This process is detailed in figure 1. + + + +The developed algorithm performs a backwards weighted subtraction of contributions, starting +from the final layer, as a singular energy is contained on this film. Careful subtraction of weighted +components discriminates the energy of stopping protons within each film from passing energies. +This remaining peak or stopping energy is then converted into a measurement of the stopping +particle fluence through the corresponding stopping power of every given layer. + +𝑁!"#$#%& = +'!" +!#' +$%& +'()*+,(! +' !" +!#../' +0(12,3 + (𝐸𝑞. 1), + +The numerator of Eq. 1 represents the remaining peak stopping energy within every active layer +after the deconvolution algorithm has been applied to the total deposited energy within each. The +denominator denotes the energy transfer as a function of the thickness of film material crossed, +found through Monte Carlo simulation. A processing script was written using the MATLAB +software [18], that compiles all of the required input parameters and procedures of this +spectroscopic method into a single program. This provides the possibility to directly input scanned +Figure 1. Visual representation of the calculation of weight factors. The water equivalent depths of the +active layers, in addition to the energy required to produce a Bragg peak at the depth of each, are both +well-known. Extrapolating the peak contributions allowed weighting factors to be calculated through +normalization of the deposited energy contribution to that of the respective peak value. For example, to +calculate the weighting factor provided by peak B to peak A, the ratio of the energy deposited by peak B +at the position of peak A, EdepB(x), to the maximum ionization of B itself, EdepB(peak), is found. This +process is performed for each energy component, at each active layer depth, and a matrix of weight factors +is then constructed. + +0.0007 +0.0006 +Edepa(peak) +0.0005 +Edep(peak) +Dose [a.u.] +0.0004 +Edepc(peak) +Peak (A) +0.0003 +Peak (B) +Peak (C) +0.0002 +EdepB(x) +0.0001 +Edepc(x) +0 +0 +0.5 +1 +1.5 +2 +2.5 +3 +Depthin water[mm] + +– 3 – +RCF images, and through simple modification, data from simulation, for a direct reconstruction +of the proton energy spectrum. A typical reconstructed spectrum is highlighted in figure 2, with +data obtained at a laser-driven proton facility. + + +The resultant energy spectrum displays an exponentially decreasing behaviour typical of laser- +driven beams produced through the target normal sheath acceleration process [1, 2]. The fact +that this is observed in the data in figure 2 gives some confidence that the procedure can +reproduce the expected spectral profile. + +3. Monte Carlo Analysis +To further assess the effectiveness of the developed deconvolution procedure, a Monte Carlo +analysis was conducted through Geant4. A replicated RCF stack configuration of the model +GafChromic EBT3, with symmetrical structure of a 28 𝜇𝑚 active layer, sandwiched between two +polyester dead layers, was constructed as outlined within the manufacturer’s specifications [19]. +Detailed simulation of this film stack is a vital first step in the spectroscopic procedure +development, allowing the energy sensitivity and corresponding stopping power values to be +evaluated for each film layer, in addition to the weighting factors required in the deconvolution +algorithm. + +The reliability of the developed deconvolution procedure as a tool for spectroscopy was assessed +through examination of the retrieved deconvolution spectrum, with one that is known. Through +Geant4 simulation, a proton source with tailored energy could be sent into the constructed RCF +stack, and the deposited energy converted to a measurement of the particle number (energy +fluence) at each active layer node using the deconvolution algorithm developed. This arrangement +was used for input proton sources with both exponential and flat energy spectra. The latter of +these source spectra proved more useful in highlighting potential discrepancies between the actual +and expected spectra. From analysis, it was noticed that particularly for laser-driven energy +spectra, with particle numbers extending orders of magnitude, the differences between the +Figure 2. Proton energy spectrum found from an irradiated stack of RCF of the model GafChromic +HDV2. This data was taken from a laser-plasma experiment at the LULI facility (Laboratoire pour +l'Utilisation des Lasers Intenses, École Polytechnique, France). The stack was placed immediately after +the target, from which protons were generated with the typical TNSA exponential behaviour. + +14 +1010 +12 +10 +Number of Protons +8 +9 +4 +2 +0 +0 +5 +10 +15 +20 +25 +30 +Energy [MeV] + +– 4 – +retrieved spectral particle numbers can be quite large, whilst still maintaining an apparently good +degree of agreement. During cross-comparison, the potential to disguise discrepancies between +spectra was reduced with the use of a flat spectrum. Once the proton energy spectrum had been +recovered from the energy deconvolution data, it was cross-compared with the original energy +spectrum. A measurement of the spectrum that originates at the source can be obtained from the +simulation through examination of the particle flux at a thin region coinciding with the front face +of the film stack. This eliminates interaction of the impinging proton beam with the RCF material, +and potential errors induced through conversion of the measured deposited energy to particle flux. +Analysis of the retrieved spectrum through application of the deconvolution algorithm in Geant4 +is shown in figure 3. + + + +A reasonable agreement between the deconvolution and entrance spectra is observed from fig. 4, +outlining the accuracy of the developed procedure in obtaining the correct particle flux at each +measurement node. This systematic Monte Carlo investigation thus gives an insight into the +working order of the algorithm for deconvolution, providing an indication of its reliability in +correctly reconstructing the energy spectrum. Within previous works concerning RCF +spectroscopy, the final spectrum is often assumed to be correct, with no such systematic check +performed. Analysis has shown that this cannot be taken for granted, and so by carrying out this +procedure some confidence is gained concerning the reliability of this spectroscopic tool. + +Conclusions +A spectroscopic procedure for the measurement of laser-driven proton energy spectra based on +the use of a stacked configuration of radiochromic films has been developed and reported here. A +deconvolution algorithm that operates through an iterative backwards weighted subtraction of +energy components from successive films has been developed to unfold the stopping proton +energy from the total energy deposited in each film layer. Initial tests demonstrated reconstruction +of a typical exponential-like spectrum with large energy spread for films irradiated using a laser- +driven proton beam. Further analysis of the developed spectroscopic procedure was conducted +through Monte Carlo methods utilising the Geant4 particle simulation toolkit. Comparison of the +Figure 3. Cross-comparison of the proton energy spectrum obtained through a deconvolution of the total +energy deposited in each film layer, with the proton fluence spectrum originating at the source as measured +at the stack entrance. + +250000 +200000 +150000 +100000 +Energydeconvolution +Spectrum originatingatsource +50000 +0 +0 +10 +20 +30 +40 +50 +60 +Kineticenergy[MeV] + +– 5 – +spectrum retrieved through deconvolution of the energy transferred to each film, to that +originating at the source for a flat energy spectrum showed a good agreement, indicating the +applicability of this tool in the spectral reconstruction of a laser-driven proton source. Although +the analysis reported is promising, a thorough examination of experimental data should be carried +out to validate the developed procedure. A reasonable result would outline the potential of this +tool in deriving a fast measurement of the energy spectrum from an irradiated stack of +radiochromic films. Nonetheless, this systematic investigation based on analysis of spectral +deconvolution through detailed Monte Carlo simulations represents one that has not been tried +before. Through cross-comparison within simulation, this has allowed an effective evaluation of +the performance of such a spectroscopic tool required for accurate measurement of the proton +energy spectrum generated through laser-driven beams. + +References +[1] Macchi, A., et al, (2013), Rev. Mod. Phys. 85, 751 +[2] Gibbon, P., (2005), Imperial College Press +[3] Badziak, J., et al, (2010), Appl. Phys. Lett. 96, 251502 +[4] Bolton, P., et al, (2018), CRC Press +[5] Borghesi, M., (2014), NIMA, 740;6-9 +[6] Bolton, P., et al, (2014), Physica Medica, 30, 3;255-270 +[7] Schreiber, J., et al, (2016), Review of Scientific Instruments, 87, 7 +[8] Margarone, D., et al, (2018), Quantum Beam Sci. 2(2), 8 +[9] Breschi, E., et al., (2004), Laser Part. Beams, 22, 393. +[10] Schollmeier, M., et al., (2008), Phys. Rev. Lett. 101, 055004 +[11] Cowan, T. E., et al., (2004), Phys. Rev. Lett. 92, 204801 +[12] Hey, D. S., et al, (2008), Rev. Sci. Instrum. 79, 053501 +[13] Nuernberg, F., et al., (2009), Rev. Sci. Instrum. 80, 033301 +[14] Kirby, D., et al., (2011), Laser and Particle Beams 29(02) +[15] Agostinelli, S., et al. Nuclear Methods and Instruments in Physics Research, 506(3), (2003) 250 +[16] Allison, J., et al. Nucl. Instrum. Meth. A 835 (2016) 186-225 +[17] http://geant4.web.cern.ch +[18] MATLAB, R2019b, (www.mathworks.com) +[19] Ashland ISP Advanced Materials, NJ, USA, (www.gafchromic.com) + diff --git a/6dFAT4oBgHgl3EQfnh1q/content/tmp_files/load_file.txt b/6dFAT4oBgHgl3EQfnh1q/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..dfbc2f54abc975bf3e6d2353f347b9d6dbc36abb --- /dev/null +++ b/6dFAT4oBgHgl3EQfnh1q/content/tmp_files/load_file.txt @@ -0,0 +1,173 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf,len=172 +page_content='Assessment of the reliability of Deconvolution Procedures for RCF Spectroscopy of Laser-Driven Ion Beams S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' McCalluma, b, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' Milluzzoc, a, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' Borghesia, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' Subielb, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' Romanod a Centre for Plasma Physics, Queen’s University Belfast, BT7 1NN, United Kingdom b Medical Radiation Science, National Physical Laboratory, Teddington, TW11 0LW, United Kingdom c Istituto Nazionale di Fisica Nucleare, Laboratori Nazionali del Sud, Via S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' Sofia 62, 95123 Catania, Italy d Istituto Nazionale di Fisica Nucleare, Sezione di Catania, Via S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' Sofia 64, 95123 Catania, Italy E-mail: smccallum05@qub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content='uk ABSTRACT: Laser-driven ion beams are defined by a number of unique features, including a large spread in energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' A stack configuration of radiochromic film (RCF) can be utilized to characterize such beams through measurements of their energy spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' A spectroscopic procedure is reported that allows the proton energy density within each active layer of a radiochromic film (RCF) stack to be retrieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' This is based upon on a deconvolution algorithm developed through Geant4 Monte Carlo simulations to correct the contributions of energy depositions within a given film layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' Through Monte Carlo calculations, the spectrum retrieved from a simulated film stack can be retrieved and compared with a known energy spectrum, providing an examination of the efficacy of this tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' Application of the developed deconvolution procedure thus offers the potential to correctly reconstruct the incident energy spectrum of a laser-driven proton and ion beam from a stack of irradiated RCF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' KEYWORDS: Detector modelling and simulations I, dE/dx detectors, Plasma diagnostics - charged-particle spectroscopy, Simulation methods and programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' – 1 – Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' Introduction 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' Methodology 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' Monte Carlo Analysis 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' Conclusions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' References 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' Introduction Whilst laser-driven proton and light ion acceleration has attracted significant interest for over 20 years [1, 2], conducting accurate measurements of these beams has proven to be technically challenging [3-5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' In particular, the ultra-high dose rates and wide spectral distributions make conventional measurement techniques impracticable [6-8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' For applications, including clinical and radiobiological ones requiring a precise energy selection, characterisation of such beams through accurate measurement of their energy spectra is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' Spectroscopic methods reliant on stacked configurations of radiochromic films (RCF) are well-established for measurements of accelerated proton beams, with several approaches of radiochromic film imaging spectroscopy (RIS) reported in the literature [9-14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' A stacked configuration of films placed perpendicularly to the beam orientation can be used to perform an energy resolved measurement of an impinging ion beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' Differential energy loss results in each particle depositing a fraction of its initial kinetic energy on every film it passes before coming to arrest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' For polyenergetic sources such as laser- driven beams, a superposition of kinetic energy contributions is amassed across the films, requiring a calculation for correction of higher energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' This is achieved through a deconvolution or unfolding of the energy transferred to each film in the stack, so that only the particles stopping within a given film remain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' The aim of the work reported here was to investigate and assess a developed algorithm for spectroscopy of laser-driven proton and ion beams through Monte Carlo simulations, studying the possible limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' This procedure requires knowledge of the RCF energy sensitivity values, and an algorithm to unfold the proton energy spectrum from the RCF response, both of which have been evaluated using the Geant4 toolkit [15-17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' Further, the same Monte Carlo methods were utilised to conduct analysis of the performance and limitations of the developed technique in acquiring the energy spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' Once validated, the spectroscopic procedure reported offers the potential to reliably extract the laser-driven proton spectra from a stack of irradiated RCF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' Methodology Energy resolved measurements of impinging proton and ion beams can be performed using multiple RCF arranged into a stack configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' The differing stopping positions for protons of a given energy within an RCF stack, means each layer can be defined by a unique energy – 2 – sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' This is chosen to correspond to the energy required to generate a Bragg peak at that given depth, defining the energy of protons that will be referred to as peak region protons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' Low energy components stop in the first few layers of the stack, whilst higher energies penetrate further downstream, giving a total energy composition of stopping protons, in addition to the fractional contributions of those exceeding the energy sensitivity of a given film layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' Unfolding the peak energy from the total energy deposited within any RCF can be achieved through the development of a deconvolution procedure for proton spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' This relies on an algorithm utilising weight factors to describe the fractional contributions of each energy component within every film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' This process is detailed in figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' The developed algorithm performs a backwards weighted subtraction of contributions, starting from the final layer, as a singular energy is contained on this film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' Careful subtraction of weighted components discriminates the energy of stopping protons within each film from passing energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' This remaining peak or stopping energy is then converted into a measurement of the stopping particle fluence through the corresponding stopping power of every given layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' 𝑁!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' "#$#%& = \'!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content='" !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content="#' $%& '() +,(!" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=" ' !" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content='" !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content='#.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content="./' 0(12,3 (𝐸𝑞." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' 1), The numerator of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' 1 represents the remaining peak stopping energy within every active layer after the deconvolution algorithm has been applied to the total deposited energy within each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' The denominator denotes the energy transfer as a function of the thickness of film material crossed, found through Monte Carlo simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' A processing script was written using the MATLAB software [18], that compiles all of the required input parameters and procedures of this spectroscopic method into a single program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' This provides the possibility to directly input scanned Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' Visual representation of the calculation of weight factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' The water equivalent depths of the active layers, in addition to the energy required to produce a Bragg peak at the depth of each, are both well-known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' Extrapolating the peak contributions allowed weighting factors to be calculated through normalization of the deposited energy contribution to that of the respective peak value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' For example, to calculate the weighting factor provided by peak B to peak A, the ratio of the energy deposited by peak B at the position of peak A, EdepB(x), to the maximum ionization of B itself, EdepB(peak), is found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' This process is performed for each energy component, at each active layer depth, and a matrix of weight factors is then constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content='0007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content='0006 Edepa(peak) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content='0005 Edep(peak) Dose [a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content='] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content='0004 Edepc(peak) Peak (A) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content='0003 Peak (B) Peak (C) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content='0002 EdepB(x) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content='0001 Edepc(x) 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content='5 3 Depthin water[mm] – 3 – RCF images, and through simple modification, data from simulation, for a direct reconstruction of the proton energy spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' A typical reconstructed spectrum is highlighted in figure 2, with data obtained at a laser-driven proton facility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' The resultant energy spectrum displays an exponentially decreasing behaviour typical of laser- driven beams produced through the target normal sheath acceleration process [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' The fact that this is observed in the data in figure 2 gives some confidence that the procedure can reproduce the expected spectral profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' Monte Carlo Analysis To further assess the effectiveness of the developed deconvolution procedure, a Monte Carlo analysis was conducted through Geant4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' A replicated RCF stack configuration of the model GafChromic EBT3, with symmetrical structure of a 28 𝜇𝑚 active layer, sandwiched between two polyester dead layers, was constructed as outlined within the manufacturer’s specifications [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' Detailed simulation of this film stack is a vital first step in the spectroscopic procedure development, allowing the energy sensitivity and corresponding stopping power values to be evaluated for each film layer, in addition to the weighting factors required in the deconvolution algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' The reliability of the developed deconvolution procedure as a tool for spectroscopy was assessed through examination of the retrieved deconvolution spectrum, with one that is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' Through Geant4 simulation, a proton source with tailored energy could be sent into the constructed RCF stack, and the deposited energy converted to a measurement of the particle number (energy fluence) at each active layer node using the deconvolution algorithm developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' This arrangement was used for input proton sources with both exponential and flat energy spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' The latter of these source spectra proved more useful in highlighting potential discrepancies between the actual and expected spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' From analysis, it was noticed that particularly for laser-driven energy spectra, with particle numbers extending orders of magnitude, the differences between the Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' Proton energy spectrum found from an irradiated stack of RCF of the model GafChromic HDV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=" This data was taken from a laser-plasma experiment at the LULI facility (Laboratoire pour l'Utilisation des Lasers Intenses, École Polytechnique, France)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' The stack was placed immediately after the target, from which protons were generated with the typical TNSA exponential behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' 14 1010 12 10 Number of Protons 8 9 4 2 0 0 5 10 15 20 25 30 Energy [MeV] – 4 – retrieved spectral particle numbers can be quite large, whilst still maintaining an apparently good degree of agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' During cross-comparison, the potential to disguise discrepancies between spectra was reduced with the use of a flat spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' Once the proton energy spectrum had been recovered from the energy deconvolution data, it was cross-compared with the original energy spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' A measurement of the spectrum that originates at the source can be obtained from the simulation through examination of the particle flux at a thin region coinciding with the front face of the film stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' This eliminates interaction of the impinging proton beam with the RCF material, and potential errors induced through conversion of the measured deposited energy to particle flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' Analysis of the retrieved spectrum through application of the deconvolution algorithm in Geant4 is shown in figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' A reasonable agreement between the deconvolution and entrance spectra is observed from fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' 4, outlining the accuracy of the developed procedure in obtaining the correct particle flux at each measurement node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' This systematic Monte Carlo investigation thus gives an insight into the working order of the algorithm for deconvolution, providing an indication of its reliability in correctly reconstructing the energy spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' Within previous works concerning RCF spectroscopy, the final spectrum is often assumed to be correct, with no such systematic check performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' Analysis has shown that this cannot be taken for granted, and so by carrying out this procedure some confidence is gained concerning the reliability of this spectroscopic tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' Conclusions A spectroscopic procedure for the measurement of laser-driven proton energy spectra based on the use of a stacked configuration of radiochromic films has been developed and reported here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' A deconvolution algorithm that operates through an iterative backwards weighted subtraction of energy components from successive films has been developed to unfold the stopping proton energy from the total energy deposited in each film layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' Initial tests demonstrated reconstruction of a typical exponential-like spectrum with large energy spread for films irradiated using a laser- driven proton beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' Further analysis of the developed spectroscopic procedure was conducted through Monte Carlo methods utilising the Geant4 particle simulation toolkit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' Comparison of the Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' Cross-comparison of the proton energy spectrum obtained through a deconvolution of the total energy deposited in each film layer, with the proton fluence spectrum originating at the source as measured at the stack entrance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' 250000 200000 150000 100000 Energydeconvolution Spectrum originatingatsource 50000 0 0 10 20 30 40 50 60 Kineticenergy[MeV] – 5 – spectrum retrieved through deconvolution of the energy transferred to each film, to that originating at the source for a flat energy spectrum showed a good agreement, indicating the applicability of this tool in the spectral reconstruction of a laser-driven proton source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' Although the analysis reported is promising, a thorough examination of experimental data should be carried out to validate the developed procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' A reasonable result would outline the potential of this tool in deriving a fast measurement of the energy spectrum from an irradiated stack of radiochromic films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' Nonetheless, this systematic investigation based on analysis of spectral deconvolution through detailed Monte Carlo simulations represents one that has not been tried before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' Through cross-comparison within simulation, this has allowed an effective evaluation of the performance of such a spectroscopic tool required for accurate measurement of the proton energy spectrum generated through laser-driven beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' References [1] Macchi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=', et al, (2013), Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' 85, 751 [2] Gibbon, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=', (2005), Imperial College Press [3] Badziak, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=', et al, (2010), Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' 96, 251502 [4] Bolton, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=', et al, (2018), CRC Press [5] Borghesi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=', (2014), NIMA, 740;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content='6-9 [6] Bolton, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=', et al, (2014), Physica Medica, 30, 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content='255-270 [7] Schreiber, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=', et al, (2016), Review of Scientific Instruments, 87, 7 [8] Margarone, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=', et al, (2018), Quantum Beam Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' 2(2), 8 [9] Breschi, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=', (2004), Laser Part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' Beams, 22, 393.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' [10] Schollmeier, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=', (2008), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' 101, 055004 [11] Cowan, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=', (2004), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' 92, 204801 [12] Hey, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=', et al, (2008), Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' Instrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' 79, 053501 [13] Nuernberg, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=', (2009), Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' Instrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' 80, 033301 [14] Kirby, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=', (2011), Laser and Particle Beams 29(02) [15] Agostinelli, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' Nuclear Methods and Instruments in Physics Research, 506(3), (2003) 250 [16] Allison, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' Instrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' Meth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content=' A 835 (2016) 186-225 [17] http://geant4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content='web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content='cern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content='ch [18] MATLAB, R2019b, (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content='mathworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content='com) [19] Ashland ISP Advanced Materials, NJ, USA, (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content='gafchromic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} +page_content='com)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'} diff --git a/79AyT4oBgHgl3EQfp_gy/content/tmp_files/2301.00534v1.pdf.txt b/79AyT4oBgHgl3EQfp_gy/content/tmp_files/2301.00534v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1cea293eb59d09c56545e3429cbe3d412aa100f5 --- /dev/null +++ b/79AyT4oBgHgl3EQfp_gy/content/tmp_files/2301.00534v1.pdf.txt @@ -0,0 +1,2438 @@ +FROM MORPHISM CATEGORIES TO FUNCTOR CATEGORIES +HOSSEIN ESHRAGHI AND RASOOL HAFEZI +Abstract. For a nice-enough category C, we construct both the morphism category H(C) of +C and the category mod-C of all finitely presented contravariant additive functors over C with +values in Abelian groups. The main theme of this paper, is to translate some representation- +theoretic attributes back and forth from one category to the other. This process is done by +using an appropriate functor between these two categories, an approach which seems quite +promising in particular when we show that many of almost split sequences are preserved by +this functor. We apply our results to the case of wide subcategories of module categories to +obtain certain auto-equivalences over them. Another part of the paper deals with Auslander +algebras arising from algebras of finite representation type. In fact, we apply our results to +study the Auslander-Reiten translates of simple modules over such algebras. In the last parts, +we try to recognize particular components in the stable Auslander-Reiten quiver of Auslander +algebras arising from self-injective algebras of finite representation type. +1. Introduction +As a popular belief, it is said that the introduction of the language of functor categories to +the study of categories of modules over rings dates back to Auslander and his colleagues’ works. +These works trace back mainly to the papers [A65, A71, A76, AR74, AR78]. +In particular +Auslander’s Formulae [A65] that suggests to recover the category mod-Λ of finitely generated +modules over an Artin algebra Λ as the quotient +mod-Λ ≃ mod-(mod-Λ) +{F : F(Λ) = 0} +deserves attention; here and throughout, mod-(mod-Λ) denotes the category of additive con- +travariant coherent functors on mod-Λ with values in Ab, the category of Abelian groups. While +talking about the exchange between two categories consisting objects that are apparently of +different types, one expects to encounter with functors transferring from one category to the +other. Concerning the morphism categories and the functor categories, such a study has initi- +ated probably in [A71]. Roughly, the general theme of the current paper is to figure out how some +representation-theoretic attributes transfer between functor and morphism categories. However, +to be more precise, we prefer to provide a layout of the paper section by section. Prior to this, +we want to point out that the morphism category of Λ has on its own right been systematically +studied from various aspects: deriving its Auslander-Reiten theory in the language of AR-theory +of Λ [RS, XZZ, E, HE], establishing its links to Gorenstein homological algebra [Z, LZ, ZX], and +looking at a particular subcategory of it, namely the monomorphism category, in order to study +the so-called Auslander algebras [AR76, HM]. +2020 Mathematics Subject Classification. 18A25, 16G70, 16G10. +Key words and phrases. Functor Category, Morphism Category, Auslander-Reiten Components. +1 +arXiv:2301.00534v1 [math.RT] 2 Jan 2023 + +2 +HOSSEIN ESHRAGHI AND RASOOL HAFEZI +First of all, to keep the results as general as possible, we try to deal with the morphism category +H(C) of a nice-enough category C (definitions are recalled later on). Namely, if we assume that +C is an idempotent-complete additive category that admits pseudokernels then, in Section 3, we +endow H(C) with an exact structure defined by degree-wise split exact sequences in C, denoted +Hcw(C). Even though such constructions have been considered in some particular cases, e.g. +in [Ba] where the category of morphisms between projective modules over an Artin algebra +have come to play, we do it in a most general possible circumstance as declared above. The +motivation behind such considerations comes from two origins. Firstly, we look for a reasonable +structure on H(C) with respect to which one may define almost split sequences. Note, secondly, +that if one imposes tougher conditions on C, for instance taking C to be an extension-closed +subcategory of mod-Λ, then H(C) inherits an exact structure as an extension-closed subcategory +of the morphism category of Λ. So now a natural question arises: What are intrinsic similarities +between these two exact structures on H(C)? +To get more involved with the aforementioned question, we need to take a glance at the +contents of Section 4. For, we recall form [A71] that there exists a functor Θ : H(C) −→ mod-C, +where mod-C is the category of contravariant additive coherent functors on C. The objective in +Section 4 is to study Θ form the point of view of Auslander-Reiten theory. We show that Θ +induces an equivalence H(C)/ +� +(M → 0), (M +1→ M) +� +≃ mod-C where M runs through the objects +of C. Using this, we show that Hcw(C) admits almost split sequences whenever C is assumed +to be a dualizing variety. Furthermore, to conquer the question posed above, it is shown that +if C is an extension-closed dualizing subvariety of mod-Λ then, in many cases, the almost split +sequences in Hcw(C) and H(C) coincide. Not going off-topic, one more thing will be proved: Θ +respects almost split sequences. +In Section 5, we turn to apply some of the results to the case of wide subcategories. To +illuminate the role and importance of wide subcategories of mod-Λ, we must point out that such +subcategories arise naturally in the study of τ-tiling theory of Λ [AIR] and in connection with +determination of certain torsion classes in mod-Λ [MS]. These also play significant role in the +study of certain classes of universal localizations over Λ [MS, HMV1, HMV2]. Such classes of +modules also appear in classification problems for the so-called τ-tilting finite algebras. Among +other things, for a given functorially finite wide subcategory X of mod-Λ we construct, based on +our previous results, an auto-equivalence σX : X → X which fulfills the exact sequence +0 → (−, σX τX (X)) → D(P, −) → D(Q, −) → D(X, −) +in mod-X for every indecomposable module X which is not projective in X; here τX denotes the +Auslander-Reiten translation of X and P → Q → X → 0 is the minimal projective presentation +of X with respect to X. In this regard, recall that for a non-projective Λ-module M with minimal +projective presentation P → Q → M → 0, there exists an exact sequence 0 → τ(M) → ν(P) → +ν(Q) → ν(M) → 0 where τ and ν stand respectively for the Auslander-Reiten translation and the +Nakayama functor over mod-Λ. Hence the aforementioned exact sequence of functors resembles, +and generalizes, the latter one. This is more clarified by showing that when X is the whole +category mod-Λ, then σX is nothing but the identity functor. We believe that this observation +is convincing-enough to say that the rich treasury behind functorially finite wide subcategories +of mod-Λ might be discovered by applying some instruments from functor categories. +In Section 6, we switch to algebras Λ of finite representation type. The main impetus for such +a study comes from the fact that in this case, one may construct the Auslander algebra A of Λ +which is, by definition, the endomorphism algebra of a representation-generator M of Λ. Then + +FROM MORPHISM CATEGORIES TO FUNCTOR CATEGORIES +3 +there is a nice interpretation of the category mod-A in terms of the functor category; namely, +there is a categorical equivalence mod-A ≃ mod-(mod-Λ). In the meanwhile, it is known [A76] +that simple functors over mod-Λ correspond bijectively to indecomposable Λ-modules. Hence +this categorical equivalence provides a framework in which one tries to understand in more details +the simple modules over A and its projectively stable version A. The results presented in this +section come up by analyzing certain almost split sequences mainly provided in [HE] and also in +[HZ]. The main results discover a relation between the (inverse) Auslander-Reiten translation of +simple A-(resp. A-) modules and the cosyzygies (resp. syzygies) of simple A-(resp. A-) modules. +The last section is devoted to study certain components in the (stable) Auslander-Reiten +quiver ΓA of the Auslander algebra A whenever Λ is self-injective of finite representation type. +Note that recognition of such components have already been the subject of some earlier researches +[IPTZ]. +To this end, we firstly deal with τH-periodic objects by invoking some almost split +sequences already obtained in [HE]. In this direction, it turns out that the auto-equivalence +A = ντ 3 of the stable category mod-Λ, as defined in [HE], plays a significant role. In fact, +we show that the existence of certain A -periodic Λ- modules makes ΓA into a finite oriented +cycle, and in particular, makes A into an algebra of finite representation type. Another result +asserts that for Λ self-injective of finite representation type, any component Ξ of the stable +Auslander-Reiten quiver of A that contains a certain simple module is either infinite or is of the +form Z∆/G for a Dynkin quiver ∆ and an automorphism group G of Z∆; this is based on a +structural theorem due to Liu [L]. +2. preliminaries and notation +In this section, we collect very briefly some necessary background material of the paper. When +required, explicit references are provided. +2.1. Functor Categories. Let k be a commutative Artinian ring and let C be a k-linear Krull- +Schmidt category. A C-module is a contravariant additive functor from C to the category Ab +of Abelian groups. +We denote by Mod-C the category of all C-modules, and by mod-C the +full subcategory of Mod-C consisting of finitely presented modules. Recall from [A65] that a +C-module M is called finitely presented if there exists an exact sequence +HomC(−, A) → HomC(−, B) → M → 0 +in Mod-C, for some objects A, B of C. Moreover, proj-C and inj-C denote the full subcategories of +mod-C consisting of projective and injective objects in mod-C, respectively. The category mod-C +is an abelian category if and only if C admits pseudokernels; see page 315 of [AR74]. We shall +sometimes write (−, X) instead of the representable functor HomC(−, X). +2.2. Dualizing k-varieties. Let r be the radical of k and E(k/r) be the injective envelope of +the k-module k/r. A Hom-finite k-linear Krull-Schmidt category C is called a dualizing k-variety +[AR74] if the k-dual functors D : Mod-C → Mod-(Cop) and D : Mod-(Cop) → Mod-C given by +D(F)(C) = Homk(F(C), E(k/r)) for every object C of C and F ∈ Mod-(C) or Mod-(Cop) induce +dualities +D : mod-C → mod-(Cop) and D : mod-(Cop) → mod-C. +In this case, it turns out that mod-C is an abelian subcategory of Mod-C that admits enough pro- +jective and enough injective objects [AR74, Theorem 2.4]. As an example, proj-Λ, the category +of finitely generated projective modules over an Artin k-algebra Λ, is a dualizing k-variety. We + +4 +HOSSEIN ESHRAGHI AND RASOOL HAFEZI +note from [AR74, Proposition 2.6] that if C is a dualizing k-variety then so is mod-C. Further- +more, any functorially finite subcategory of a dualizing k-variety is itself a dualizing k-variety +by [AS81, Theorem 2.3]. +2.3. Morphism Categories. Let C be a category. +The morphism category H(C) of C is a +category whose objects are morphisms f : X → Y in C, and whose morphisms are given by +commutative diagrams. If we regard the morphism f : X → Y as an object in H(C), we will +usually present it as (X +f→ Y ). However, due to typographical considerations, we have to use +also the vertical notation ( X +Y )f. A morphism between the objects (X +f→ Y ) and (X′ f ′ +→ Y ′) is +presented as (σ1, σ2) : (X +f→ Y ) → (X′ f ′ +→ Y ′) or, ( σ1 +σ2 ) : ( X +Y )f → +� X′ +Y ′ +� +f ′, where σ1 : X → X′ +and σ2 : Y → Y ′ are morphisms in C with σ2f = f ′σ1. +Adapting the notation, the morphism category raised from C = mod-Λ, the category of finitely +generated right modules over an Artin k-algebra Λ, will be denoted simply by H; this will cause +no ambiguity. The same rule also applies to the monomorphism category S of Λ whose objects +are just monic Λ-maps. +2.4. Auslander-Reiten-Serre Duality. Let (C, E) be an exact category in the sense of Quillen +[Q, K] (see next section for an introduction). +Recall that a morphism v: E → Y in C is +called right almost split if it is not a retraction and each f : Z → Y which is not a retraction +factors through v. Dually, a morphism u: X → E in C is called left almost split if it is not a +section and each f : X → Z which is not a section factors through u. An admissible sequence +δ: 0 → X +u−→ E +v−→ Y → 0 in E is an almost split sequence if u is left almost split and v is right +almost split. Since δ determines X and Z in a unique way, we call X the Auslander-Reiten +translation X = τC(Y ) of Y in C. +A non-zero object X ∈ C is said to be endo-local if its +endomorphism ring EndC(X) is local. Following [INY, Definition 3.1], we say that C has almost +split sequences if endo-local non projective objects of C and endo-local non-injective objects of +C are respectively the terminal and the initial terms of some almost split sequence in E. +Assume now that C is further a k-linear category and let D be the k-dual functor. Put C and +C denote respectively the projectively and the injectively stable categories of C. An Auslander- +Reiten-Serre duality (ARS duality, in brief) is a pair (τC, η) consisting of an equivalence functor +τC : C → C together with a bi-natural isomorphism +ηX,Y : HomC(X, Y ) ≃ DExt1 +C(Y, τC(X)) +for any X, Y ∈ C. +The following lemma, taken from [INY, Theorem 3.6] (see also [J]), provides a close connection +between the existence of almost split sequences in C and the existence of an ARS-duality. Let us +recall that under the above hypothesis, C is Ext-finite if the k-modules Ext1 +C(X, Y ) are finitely +generated. +Lemma 2.1. Let C be a k-linear Ext-finite Krull-Schmidt exact category. Then the following +conditions are equivalent. +(1) C has almost split sequences. +(2) C has an Auslander-Reiten-Serre duality. +(3) The stable category C is a dualizing k-variety. +(4) The stable category C is a dualizing k-variety. +Throughout the paper, Λ will stand for a fixed Artin k-algebra and modules are, by default, +finitely generated right modules. +The Auslander-Reiten translation, the Nakayama functor, + +FROM MORPHISM CATEGORIES TO FUNCTOR CATEGORIES +5 +the syzygy and the cosyzygy functor of Λ are respectively denoted by τ, ν, Ω, and Ω−1. If +we deal with an algebra other than Λ or with a category, these functors will be accompanied +with necessary subscripts. The symbols Ker, Coker, and Im, used freely in all contexts, stand +respectively for the kernel, cokernel, and the image of morphisms. +3. Exact structures on the morphism category +An exact category (C, E) is formed by an additive category C, and a class E of composable +pairs of morphisms in C (also called kernel-cokernel pairs) satisfying certain axioms that we +refrain to exhibit here and refer the reader e.g. to [K]. The composable pair (i, p) in E is usually +denoted by 0 → A′ +i→ A +p→ A′′ → 0, where i : A′ → A and p : A → A′′ are respectively called +an E-admissible monic and an E-admissible epic. +Composable pairs, admissible monics and +admissible epics are sometimes referred to respectively as conflations, inflations and deflations. +The notion of an exact category was first introduced by Quillen in [Q] and then Keller [K] proved +the redundancy of some axioms. +Let C be an additive category. In this section, we shall put an exact structure on the morphism +category H(C) of C [Ba]. For let Ecw be the class of all pairs of composable morphisms +δ : +� X1 +X2 +� +f +� φ1 +φ2 +� +�� Z1 +Z2 +� +h +� ψ1 +ψ2 +� +�� Y1 +Y2 +� +g +such that the induced composable morphisms Xi +φi +→ Zi +ψi +→ Yi split in C for i = 1, 2. It can be +easily seen that any pair of composable morphisms in Ecw is isomorphic to a pair of composable +morphisms of the form +δ′ : +� X1 +X2 +� +f +�[ 1 +0] +[ 1 +0] +� +�� X1⊕Y1 +X2⊕Y2 +� +h +� [0 1] +[0 1] +� +�� Y1 +Y2 +� +g +where h = +� +f q +0 g +� +and q : Y1 → X2 is a possibly non-zero morphism in C. Regarding this easy +observation, without loss of generality, we usually take all kernel-cokernel pairs in H(C) to be of +this form; this is justified by the following lemma. +Lemma 3.1. Any object in Ecw is a kernel-cokernel pair in H(C). +Proof. Take the element δ′ of Ecw and assume that the composite of the morphisms (σ1, σ2) : +(X1 ⊕ Y1 +h→ X2 ⊕ Y2) → (V +s→ W) and ([ 1 +0 ], [ 1 +0 ]) vanishes. This means that the restriction of σi +on Xi, for i = 1, 2, is the zero map. This enables us to define the morphisms σ1|Y1 and σ2|Y2 and +it readily follows that (σ1, σ2) factors uniquely over ([0 1], [0 1]) via the morphism (σ1|Y1, σ2|Y2). +The remaining axioms are verified similarly. +□ +Recall that an additive category D is called idempotent-complete if every idempotent endo- +morphism in D admits a kernel. +Proposition 3.2. Assume C is idempotent-complete and admits pseudokernels. Then Ecw de- +fines an exact structure on the additive category H(C). +Proof. Since C is idempotent-complete, it is known that the Yoneda functor gives an equivalence +C ≃ proj-C. This equivalence is naturally extended to an equivalence between corresponding +morphism categories; i.e., H(C) ≃ H(proj-C). One observes that, under this equivalence, the + +6 +HOSSEIN ESHRAGHI AND RASOOL HAFEZI +kernel-cokernel pairs in Ecw provided by Lemma 3.1 correspond bijectively to the short exact se- +quences in the abelian category H(mod-C) whose terms lie inside H(proj-C). But the subcategory +H(proj-C) is closed under extensions and inherits an exact structure from H(mod-C). +□ +Remark 3.3. To make our arguments work, we had to impose some restrictions on the additive +category C to get a suitable exact structure out of H(C). However, it may be the case that +the aforementioned set of requirements is not minimal in the sense that the above family of +kernel-cokernel pairs may equip H(C) with an exact structure even if some of the hypothesis in +Proposition 3.2 are dropped. +From now on we assume that C is idempotent-complete and admits pseudokernels, and the +symbol Hcw(C) stands for the exact category (H(C), Ecw), sometimes also called the cw-exact +category. The following proposition is recorded for future use. +Proposition 3.4. Suppose (X1 +f→ X2) is an object in Hcw(C). +(1) f defines an indecomposable projective object in Hcw(C) if and only if it is isomorphic +either to (X +1→ X) or (0 → X) for some indecomposable object X in C. +(2) f defines an indecomposable injective object in Hcw(C) if and only if it is isomorphic +either to (X +1→ X) or (X → 0) for some indecomposable object X in C. +Furthermore, Hcw(C) has enough projectives and enough injectives. +Proof. This should be compared to [Ba, Corollary 3.2]. +We just remark that the last claim +follows from the short exact sequences +0 +�� 0 +X1 +� +0 +� +0 +� f +−1 +� +� +�� 0 +X2 +� +0 ⊕ +� X1 +X1 +� +1 +� +1 +[ 1 f ] +� +�� X1 +X2 +� +f +�0 +and +0 +�� X1 +X2 +� +f +� � f +1 +� +1 +� +�� X2 +X2 +� +1 ⊕ +� X1 +0 +� +0 +� [ −1 f ] +0 +� +�� X2 +0 +� +�0 +in Ecw. +□ +Now assume C is an extension-closed subcategory of mod-Λ for an Artin algebra Λ. We may +consider C as an exact category through the structure induced by the abelian category mod-Λ. +Then also H(C), as an extension-closed subcategory of the abelian category H is endowed with +the canonical exact structure inherited from H, still denoted by H(C). We also keep the cw- +exact structure Hcw(C) defined by degree-wise split sequences. It will be indicated in the next +section that if C is a k-dualizing variety, then Hcw(C) admits almost split sequences. Further, +the canonical exact category H(C) admits almost split sequences provided C is a k-dualizing +subvariety of mod-Λ. +It also becomes clear how the canonical exact category H(C) inherits +almost split sequence from Hcw(C) in the latter case. However, for technical reasons, we have to +defer the proofs until next section. +Remark 3.5. Suppose for a moment that C is further functorially finite in mod-Λ. In this case, +another approach one may take to show that H(C) has almost split sequences is to explore when +H(C) is functorially finite in H. This seems natural in view of the fact that, by [AS81, Theorem +2.4], any functorially finite extension-closed subcategory of H admits almost split sequences. + +FROM MORPHISM CATEGORIES TO FUNCTOR CATEGORIES +7 +Restricting to the case where C = mod-Λ, in the last part of this section, we put a third exact +structure on H that will turn out in Section 7 to be in connection with the stable Auslander- +Reiten quiver of Auslander algebras; see Remark 7.8. An indecomposable object in H is said +to be of type (a) (resp. (b), or (c)) provided it is isomorphic to (0 → M) (resp. (M +1→ M), +or (M → 0)) for some Λ-module M. Further, an indecomposable object is said to be of type +(d) if it is isomorphic to (P +f→ Q) where P, Q are projective Λ-modules. Let X be the smallest +subcategory of H containing all objects of types (a), (b), (c) and (d). Let also EX be the class of +all short exact sequences 0 → X → Y → Z → 0 in H such that the induced sequence +0 → HomH(V, X) → HomH(V, Y) → HomH(V, Z) → 0 +is exact for every V ∈ X. We know from [AS93] and [Bu] that EX induces an exact structure on +H denoted by HX = (H, EX ). One infers from [AS93, Theorem 1.12] that the exact category HX +has enough projectives and enough injectives. Denote by P(HX ) (resp. I(HX )) the subcategory +of projective (resp. injective) objects in HX . In view of [AS93, Corollary 1.6 and Proposition +1.10], we have P(HX ) = X ∪ proj-H and I(HX ) = τH(X) ∪ inj-H, where proj-H and inj-H +stand respectively for the subcategories of projective and injective objects in H and τH is the +Auslander-Reiten translation of H. We exploit [AS93, Proposition 1.9] to examine the almost +split sequences in HX ; it turns out that an almost split sequence 0 → X → Y → Z → 0 in H is +an almost split sequence in HX if and only if neither X ∈ I(HX ) nor Z ∈ P(HX ). +4. Interplay between morphism and functor categories +Until further notice, we assume throughout the section that C is a dualizing k-variety. In +this section, we will be involved with a functor going from morphism category to the functor +category, originally defined and studied in [A71] and then reconsidered in [HM]. This functor +is our main tool to exchange between these two categories. The construction is based on the +Yoneda functor. +Construction 4.1. Let (X1 +f→ X2) be an object of H(C). Define +(X1 +f→ X2) +Θ +�→ Coker(C(−, X1) +C(−,f) +−→ C(−, X2)). +If h = +� h1 +h2 +� +: X = +� X1 +X2 +� +f → +� +X′ +1 +X′ +2 +� +f ′ = X′ is a morphism in H(Λ), then we let Θ(h) be the +unique morphism σ that makes the following diagram commute. +HomC(−, X1) +HomC(−,f) � +HomC(−,h1) +� +HomC(−, X2) +� +HomC(−,h2) +� +Θ(X) +� +σ +� +0 +HomC(−, X′ +1) +HomC(−,f ′) � HomC(−, X′ +2) +� Θ(X′) +� 0. +It is routine to verify that this rules introduce a well-defined functor Θ : H(C) → mod-C. The +purpose of this section is to study this functor from the perspective of almost split sequences. +It turns out that Θ behaves well over such sequences. Firstly, we need to recall some facts on +objective functors; more details are provided in the Appendix of [RZ]. Let F : C −→ D be an +additive functor between additive categories. F is called an objective functor if any morphism +f in C with F(f) = 0 factors through an object K of C with F(K) = 0; such a K is then called +a kernel object of F. We say that the kernel of a functor F is generated by a subcategory X of +C if add-X, the additive closure of X in C, is the class of all kernel objects of F. + +8 +HOSSEIN ESHRAGHI AND RASOOL HAFEZI +Let F : C −→ D be a full, dense and objective functor and let the kernel of F be generated +by X. Then F induces an equivalence F : C/X −→ D where the additive quotient category C/X +of C with respect to X has the same objects as C and the morphisms are defined via the rule +C/X(X, Y ) := C(X, Y )/{φ | φ factors through an object in add-X} +for any pair of objects X, Y of C. +Theorem 4.2. The functor Θ : H(C) −→ mod-C is full, dense and objective. Thus, there exists +an equivalence Θ of categories that makes the following diagram commute. +H(C) +Θ � +π +� +mod-C +H(C) +V +Θ +� +Here, π is the natural quotient map and V is the full subcategory of H(C) generated by all finite +direct sums of objects of type (b) or (c), that is to say, objects of the form (M +1 +−→ M) and +(M −→ 0), where M runs through the objects of C. +Proof. Θ is dense; for take F ∈ mod-C with a projective presentation (−, X) +(−,g) +→ +(−, Y ) → +F → 0. It is plain that Θ(X +g→ Y ) = F. To see the fullness of Θ, take two objects (X +g→ Y ) +and (X′ +g′ +→ Y ′) of H(C). As the representable functors (−, Y ) and (−, Y ′) are projective, it +follows that any morphism σ : F = Θ(X +g→ Y ) → Θ(X′ +g′ +→ Y ′) = F ′ in mod-C might be +lifted to a map from the augmented projective presentation (−, X) +(−,g) +→ +(−, Y ) → F → 0 +to (−, X′) +(−,g′) +→ +(−, Y ′) → F ′ → 0. +Then using Yoneda’s Lemma and the aforementioned +construction, one obtains a morphism h : (X +g→ Y ) → (X′ g′ +→ Y ′) in H(C) with σ = Θ(h). +Now assume Θ(X +g→ Y ) = 0, for some object (X +g→ Y ). Then by the construction, we have +the exact sequence 0 → (−, X) +(−,g) +−→ (−, Y ) → 0 in mod-C. One then observes that the identity +map 1 : Y → Y factors over g via, say, h : Y → X. Therefore, X = Im(h) ⊕ Ker(g). This leads +to the decomposition (X +g→ Y ) = (Ker(g) → 0) ⊕ (Im(h) +g| +→ Y ) where g| is the restricted map +which must be an isomorphism since gh = 1Y . This settles that the kernel of Θ is generated by +V. +Finally, suppose Θ(h) = 0, for h = (h, h′) : (X +g→ Y ) → (X′ +g′ +→ Y ′) in H(C). +Setting +F = Θ(X +g→ Y ) and F ′ = Θ(X′ g′ +→ Y ′), this induces a chain map between complexes of functors +· · · +� (−, Z0) +(−,α0) +� +� (−, X) +(−,h) +� +(−,g) � (−, Y ) +(−,h′) +� +� F +0 +� +� 0 +· · · +� (−, Z′ +0) +� (−, X′) +(−,g′)� (−, Y ′) +� F ′ +� 0 +raised by taking projective presentations of F and F ′. +Evidently, this chain map is null- +homotopic and, according to [G, Corollary 3.5], factors through a contractible complex. +As + +FROM MORPHISM CATEGORIES TO FUNCTOR CATEGORIES +9 +any contractible complex of functors might be imagined to be a direct sum of complexes of the +form +· · · → 0 → (−, W) +1→ (−, W) → 0 → · · · +for various objects W of C, this induces a commutative diagram +· · · +� (−, Z0) +(−,α0) +� +� +� (−, X) +(−,h) +� +� +(−,g) +� (−, Y ) +(−,h′) +� +� +� 0 +· · · +� (−, Z0 ⊕ X′) +� +� +(−, X′ ⊕ Y ′) +� +� +(−, Y ′) +� +� 0 +· · · +� (−, Z′ +0) +� (−, X′) +(−,g′) � (−, Y ′) +� 0. +Therefore, there exists a factorization of h through the object (X′ → 0) ⊕ (Y ′ +1→ Y ′), which is +a kernel object according to the above paragraph. This shows that Θ is an objective functor. +Now the existence of the equivalence Θ comes up from observations prior to the theorem. +□ +Let us record here that applying a dual construction to the opposite category Cop results in +a contravariant functor +Θ′ : H(C) → mod-Cop, +(X +f→ Y ) �→ Coker(C(Y, −) +C(f,−) +−→ C(X, −)) +which is seen to induce a duality Θ′ that makes the diagram +H(C) +Θ′ � +π′ +� +mod-Cop +H(C) +V′ +Θ′ +� +commute. Here, V′ is the full subcategory of H(C) generated by all finite direct sums of objects +of type (a) or (b). +Consider the morphism category H(C), endowed with the exact structure given by Ecw. Ac- +cording to Proposition 3.4, V (resp. V′) is nothing but the subcategory of injective (resp. pro- +jective) objects of the exact category Hcw(C). Consequently, the factor categories H(C)/V′ and +H(C)/V are equivalent respectively to the projectively and injectively stable categories Hcw(C) +and Hcw(C). Hence the following proposition emerges to settle that Hcw(C) admits almost split +sequences. +Proposition 4.3. For a dualizing k-variety C, the following statements hold. +(1) Hcw(C) admits almost split sequences. +(2) Hcw(C) has an Auslander-Reiten-Serre duality. +Proof. The above observations along with Theorem 4.2 provide the equivalences H(C)/V ≃ +mod-C ≃ Hcw(C). Note that by [AR74, Proposition 2.6], mod-C is a dualizing k-variety as well. +Hence Lemma 2.1 completes the proof. +□ +We are now in a position to prove the assertion in previous section concerning the existence +of almost split sequences in H(C) where C is an extension-closed dualizing subvariety of mod-Λ +for an Artin algebra Λ. The following lemma which is taken from [MO, Proposition 3.1] will be + +10 +HOSSEIN ESHRAGHI AND RASOOL HAFEZI +fruitful. While the special case C = mod-Λ has been dealt with in [MO], the same proof still +works for general C as we consider here. +Lemma 4.4. Assume δ : 0 → A +f→ B +g→ C → 0 is an almost split sequence in C. Then +(1) The almost split sequence in H(C) ending at (0 → C) is of the form +0 +�( A +A )1 +� 1 +f +� +�( A +B )f +� 0 +g +� +�( 0 +C )0 +�0. +(2) The almost split sequence in H(C) ending at (C +1→ C) is of the form +0 +�( A +0 )0 +� f +0 +� +�( B +C )g +( g +1) +�( C +C )1 +�0. +Proposition 4.5. Let C be an extension-closed k-dualizing subvariety of mod-Λ. +Then the +canonical exact category H(C) admits almost split sequences. +Proof. Let Z be an indecomposable non-projective object in H(C). Assume first that Z is of either +types (0 → X) or (X +1→ X), for an object X ∈ C. Then since C admits almost split sequences +by [AS81, Theorem 1.1], from Lemma 4.4 we infer that Z is the end term of an almost split +sequence in H(C). Otherwise, Z is not projective in the exact category Hcw(C) by Proposition +3.4. Hence, by Proposition 4.3, there exists an almost split sequence ending at Z in the exact +category Hcw(C). However, following the definitions, it is easy to verify that this is an almost +split sequence in H(C) as well. +□ +The following corollary is an immediate consequence of the arguments above. +Corollary 4.6. Let C be an extension-closed k-dualizing subvariety of mod-Λ and let +0 +�� X1 +X2 +� +f +� φ1 +φ2 +� +�� Z1 +Z2 +� +h +� ψ1 +ψ2 +� +�� Y1 +Y2 +� +g +�0 +be an almost split sequence in H(C). Then the sequences 0 → Xi +φi +→ Zi +Ψi +→ Yi → 0, i = 1, 2, split +provided that either of the following situations occur. +(1) The terminal term (Y1 +g→ Y2) is not of type (a) or (b). +(2) The initial term (X1 +f→ X2) is not of type (b) or (c). +We now turn to show that Θ respects almost split sequences; so we return to the setting that +C is a dualizing k-variety. Let Y = (Y1 +g→ Y2) be an indecomposable non-projective object in +H(C). Take +δ : +� X1 +X2 +� +f +� φ1 +φ2 +� +�� Z1 +Z2 +� +h +� ψ1 +ψ2 +� +�� Y1 +Y2 +� +g +to be the almost split sequence in Hcw(C) ending at Y. For simplicity, set Z = (Z1 +h→ Z2), +X = (X1 +f→ X2), φ = (φ1, φ2) and ψ = (ψ1, ψ2). Note that δ induces degree-wise split sequences + +FROM MORPHISM CATEGORIES TO FUNCTOR CATEGORIES +11 +and, applying Θ, one gets the commutative diagram with exact rows +0 +� +0 +� +0 +� +0 +� K1 +� +� K2 +i +� +η +� K3 +λ +� +0 +� (−, X1) +(−,f) +� +� (−, Z1) +(−,h) +� +(−,ψ1)� (−, Y1) +(−,g) +� +� 0 +0 +� (−, X2) +� +� (−, Z2) +� +(−,ψ2)� (−, Y2) +� +� 0 +Θ(X) +Θ(φ) +� +� +Θ(Z) +Θ(ψ) � +� +Θ(Y) +� +� +0 +0 +0 +0 +in mod-C whose bottom row is indeed the image Θ(δ) of δ under the functor Θ. +Lemma 4.7. The map η in the above diagram is an epimorphism. As an upshot, Θ(δ) is a +short exact sequence in mod-C. +Proof. Let (−, P) +σ→ K3 → 0 be an epimorphism in mod-C and let d : P → Y1 be a morphism +in C which represents the composite λσ : (−, P) → K3 → (−, Y1). Since gd = 0, Yoneda’s +lemma induces a morphism (d, 0) : (P → 0) → Y which is plainly not a retraction. Hence it +must factor over the right almost split map (ψ1, ψ2) via, say, (a, 0) for some a : P → Z1 in +C. Consequently, the map (−, a) in mod-C satisfies (−, h)(−, a) = 0. Adding that (−, P) is a +projective functor, this gives a map γ : (−, P) → K2 in mod-C with (−, a) = iγ. Note that +ληγ = (−, ψ1)iγ = (−, ψ1)(−, a) = λσ. But λ is a monomorphism; thus ηγ = σ whence the +surjectivity of η. The second claim comes up immediately from the Snake Lemma. +□ +The following theorem is another main result of the section. +Theorem 4.8. Under the above notation, Θ(δ) is an almost split sequence in mod-C. +Proof. The indecomposability of X and Y imply that Θ(X) and Θ(Y) are indecomposable. By +previous lemma, Θ(δ) is an exact sequence that, moreover, does not split. Indeed, if it did, then +(−, f) ⊕ (−, g) would be a minimal projective presentation for Θ(Z) which should comply with +the one provided by the middle column of the above diagram. In view of the form of kernel +elements of the functor Θ declared by Theorem 4.2, an application of Yoneda’s lemma gives +that, for some objects A, B of C, there should exist an isomorphism +(Z1 +h→ Z2) = (X1 +f→ X2) ⊕ (Y1 +g→ Y2) ⊕ (A +1→ A) ⊕ (B → 0) +of objects in H(C). As stated before, we may assume Zi ≃ Xi ⊕ Yi, i = 1, 2 since δ belongs to +Ecw. However, C being a Krull-Schmidt category implies A = B = 0 which makes δ split. This +contradiction shows that Θ(δ) does not split. + +12 +HOSSEIN ESHRAGHI AND RASOOL HAFEZI +Now, as [AR74, Theorem 2.4] guarantees that mod-C is abelian in this case, invoking [AR77, +Theorem 2.14], it suffices to show that Θ(ψ) is right almost split. For let q : F → Θ(Y) be a +non-retraction in mod-C and take a projective presentation (−, W1) +(−,d) +→ (−, W2) → F → 0 of +F. Note that by definition, Θ(W1 +d→ W2) = F. The morphism q lifts to a morphism between the +projective presentations (−, W1) +(−,d) +→ (−, W2) → F → 0 and (−, Y1) +(−,g) +→ (−, Y2) → Θ(Y ) → 0. +The lifted morphism induces, again by the Yoneda lemma, a map +( σ1 +σ2 ) : +� W1 +W2 +� +d → +� Y1 +Y2 +� +g +in H(C) such that Θ(σ1, σ2) = q. +Then (σ1, σ2) is not a retraction since otherwise q would +be so. +Now, δ being an almost split sequence in Hcw(C), (σ1, σ2) factors over ψ via some +(η1, η2) : (W1 +d→ W2) → (Z1 +h→ Z2). Then applying Θ, we see that the morphism q factors over +Θ(ψ) via Θ(η1, η2). +□ +5. The case of wide subcategories +Our objective in this section is to study the morphism categories raised by functorially finite +wide subcategories of mod-Λ. +Some results from previous section will come to play. +After- +wards, we shall switch to functor categories and obtain some results in this direction that extend +others from the module category. So let firstly X be a functorially finite idempotent-complete +subcategory of mod-Λ. By [AS81, Theorem 2.3], X itself is a dualizing variety. +Following Proposition 4.3, for a dualizing k-variety C, there is an equivalence τH(C) : Hcw(C) → +Hcw(C) that, based on what we said in previous section, might be considered as an equivalence +from H(C)/V′ to H(C)/V. Pictorially, there exists a composition of equivalences +H(C)/V +τ −1 +H(C) � H(C)/V′ +Θ′ +� +mod-C +(Θ)−1 +� +� mod-Cop +D +� mod-C +denoted throughout by ∆C, or simply by ∆. Applied to the functorially finite subcategory X +of mod-Λ, this yields an equivalence ∆X : mod-X → mod-X which is restricted to the category +proj-X of projective functors. Since X is idempotent-complete, the Yoneda functor induces an +equivalence proj-X ≃ X. Summing up, one obtains an equivalence σX : X → X which agrees +with the restricted equivalence ∆X via the latter identification. We notice that, going through +the definitions, one figures out that for an object X of X, there are A, B ∈ X and an exact +sequence +(B, −) +(f,−) +→ (A, −) → D(−, σX (X)) → 0 +in mod-X such that τ −1 +H(X)(0 → X) = (A +f→ B). +Definition 5.1. A minimal projective presentation of an object C ∈ X with respect to X is +an exact sequence P1 +f→ P0 +h→ C with P1, P0 ∈ P(X), the class of projective objects of X, +and is computed by taking minimal right P(X)-approximations consecutively. Minimal injective +presentations with respect to X are defined dually via I(X), the class of injective objects of X. +Recall that a subcategory M of mod-Λ is said to be closed under kernels (resp. cokernels, +images) if for every morphism X +f→ Y in M also Ker(f) (resp. Coker(f), Im(f)) belongs to + +FROM MORPHISM CATEGORIES TO FUNCTOR CATEGORIES +13 +M. Further, M is called a wide subcategory of mod-Λ if it is closed under extensions, kernels +and cokernels. It is clear that a wide subcategory is closed under images and is automatically +idempotent-complete. +The following couple of propositions are quite useful. +Proposition 5.2. Assume X is a functorially finite wide subcategory of mod-Λ and δ : 0 → +A +f→ B +g→ C → 0 is an almost split sequence in X. Let also A +d→ I0 +q→ I1 be a minimal injective +presentation with respect to X, where b : Coker(d) → I1 is a minimal left I(X)-approximation, +a : I0 → Coker(d) is the canonical quotient map and q = ba. Then the exact sequence +0 +�� I0 +I1 +� +q +( u +1 ) +�� W +I1 +� +br +( v +0 ) +�( C +0 )0 +�0 +in H(X) raised by forming the push out diagram +A +d +� +f +� B +h +� +g +� C +I0 +a +� +u +� W +r +� +v +� C +Coker(d) +Coker(d) +in the exact category X, is almost split. +Proof. Using that A +d→ I0 +s→ I1 is a minimal injective presentation and A is indecomposable, +we deduce that (I0 +s→ I1) is indecomposable. Hence it suffices to show that any non-retraction +(φ, 0) : (M +p→ N) → (C → 0) in X factors over (v, 0). If φ is a non-retraction, then, since δ is +an almost split sequence, it factors in X over g via, say, w : M → B. Then it is easy that the +morphism (hw, 0) : (M +p→ N) → (W +br +→ I1) factors the morphism (φ, 0) over (v, 0). So now take +φ to be a retraction. Without loss of generality, we may assume M = C and φ = 1C. Two cases +might be distinguished: +Case 1: p is a monomorphism. Since v is a retraction in X, there exists s : C → W such that +vs = 1. As p : C → N is a monomorphism in X, there exists an extension of brs : C → I1 to a +map z : N → I1; that is to say, zp = brs. It follows then that (s, z) : (C +p→ N) → (W +br +→ I1) +produces the desired factorization. +Case 2: Assume Ker(p) ̸= 0. Note that since X is a wide subcategory, Ker(p) lies in X. The +fact that (φ, 0) is a non-retraction implies that Ker(p) is a proper submodule of C and thus the +canonical inclusion i : Ker(p) → C is a non-retraction in X. According to the hypothesis, we +infer the existence of a map y : Ker(p) → B such that gy = i. Note further that, since v is +retraction, one may write W = Im(s)⊕Ker(v) and, consequently, present h as h = [l1, l2]t, where +l1 : B → Im(s) and l2 : B → Ker(v). Using the injectivity of Ker(v) in X yields an extension of +l2y : Ker(p) → Ker(v) to C; that is, there exists y′ : C → Ker(v) such that y′i = l2y. Putting +together, we get a diagram + +14 +HOSSEIN ESHRAGHI AND RASOOL HAFEZI +Ker(p) +y +� +i +� C +[s y′]t +� +� Im(p) +� 0 +B +h +� W +r � Coker(h) +� 0 +with commutative left part. This induces a map y′′ : Im(p) → Cok(h) completing the diagram. +Again, as X is wide, the monomorphism Im(p) +i′ +→ N lies inside X and, hence, the injectivity +of I1 in X gives a map z′ : N → I1 with z′i′ = by′′. Finally, one verifies that the morphism +([s y′]t, z′) : (C +p→ N) → (W +br +→ I1) gives the required factorization. +□ +As a dual statement, we record the following proposition. +Proposition 5.3. Assume X is a functorially finite wide subcategory of mod-Λ and δ : 0 → A +f→ +B +g→ C → 0 is an almost split sequence in X. Let also P1 +ℓ→ P0 +h→ C be a minimal projective +presentation with respect to X, where k : P1 → Ker(h) is a minimal right P(X)-approximation, +i : Ker(h) → P0 is the canonical inclusion and ℓ = ik. Then the exact sequence +0 +�( 0 +A ) +( 0 +u) +�� P1 +Z +� +wk +( 1 +v ) +�� P1 +P0 +� +ℓ +�0 +in H(X) raised by forming the pull back diagram +Ker(h) +w +� +Ker(h) +i +� +A +u +� Z +r +� +v +� P0 +h +� +A +f +� B +g +� C +in the exact category X, is almost split. +Any functorially finite wide subcategory X of mod-Λ admits almost split sequences by [AS81, +Theorem 2.4]. Hence, following Lemma 2.1, we let τX denote the Auslander-Reiten translation +over X. +Corollary 5.4. Let X be a functorially finite wide subcategory of mod-Λ and consider the auto- +equivalence σX : X → X introduced earlier. Assume that X ∈ X is an indecomposable module +not belonging to P(X), and that P +f→ Q → X is a minimal projective presentation with respect +to X. Then there is an exact sequence +0 → (−, σX τX (X)) → D(P, −) → D(Q, −) → D(X, −) +in mod-X +Proof. Proposition 5.3 yields that the inverse Auslander-Reiten translation τ −1 +H(X)(0 → τX (X)) +of (0 → τX (X)) in H(X) coincides with (P +f→ Q). Taking into account our previous observations +on the functor σX gives the result. +□ + +FROM MORPHISM CATEGORIES TO FUNCTOR CATEGORIES +15 +Remark 5.5. For every non-projective indecomposable Λ-module M, we know that there exists +an exact sequence +0 → τ(M) → ν(P) → ν(Q) → ν(M) → 0 +where P → Q → M → 0 is the minimal projective presentation of M. In some sense, the +sequence presented by Corollary 5.4 goes parallel to, and generalizes this observation. This is +more justified as we show in the sequel that for the case X = mod-Λ, σX is just the identity +functor. However, it would be interesting to explore σX further by considering other functorially +finite wide subcategories X. +Denote by σ := σX : mod-Λ → mod-Λ the auto-equivalence obtained above in the case +where X = mod-Λ. We refer e.g. to [HE] for further explanation on how the Auslander-Reiten +translation τH and its inverse τ −1 +H work in this case and suffice to recall that the standard duality +functor DH might be computed in a local manner in terms of the standard duality D of Λ. That +is to say, DH(X +f→ Y ) = (D(Y ) +D(f) +→ D(X)). +Note that since σ is an equivalence, it clearly restricts to an equivalence σ′ : inj-Λ → inj-Λ on +the subcategory of injective modules. In the sequel, it will be shown that σ′, and consequently +σ, are nothing but the identity functor on the corresponding categories. +Lemma 5.6. The restricted equivalence σ′ is isomorphic to the identity functor on inj-Λ. +Proof. Let I be an injective Λ-module. There exists a minimal injective resolution in H +0 → ( 0 +I )0 → ( I +I )1 → ( I +0 )0 → 0 +of the object (0 → I). Applying the duality DH leads to the projective presentation in Hop +0 → ( 0 +DI )0 → ( DI +DI )1 → ( DI +0 )0 → 0 +of the object DH(0 → I). Then, we compute the transpose and deduce that τ −1 +H (0 → I) ≃ +(ν−1(I) → 0). As we pointed out earlier in this section, this results in an equivalence (ν−1(I), −) ≃ +D(−, σ(I)) in mod-(mod-Λ)op. Hence, evaluating on the regular module Λ, yields a natural iso- +morphism σ(I) ≃ νν−1(I) ≃ I. +□ +Theorem 5.7. The equivalence σ is isomorphic to the identity functor on mod-Λ. +Proof. According to Lemma 5.6, the restricted equivalence σ′ is naturally isomorphic to the +identity functor on the subcategory of injective modules. Using injective resolutions, it is then +straightforward to see that the same holds for σ itself. +□ +In the rest of this section, we will provide some applications of the aforementioned theorem. +Corollary 5.8. Let F be a functor in mod-(mod-Λ) with a minimal projective presentation +(−, X) +(−,f) +→ (−, Y ) → F → 0. Then there is an exact sequence +(Y ′, −) +(g,−) +→ (X′, −) → DF → 0 +in mod-(mod-Λ)op where (X′ +g→ Y ′) is the inverse Auslander-Reiten translation of (X +f→ Y ) in +H. +Proof. Again we specify our constructions to the dualizing variety C = mod-Λ. By virtue of +Theorem 5.7, the functor ∆ := ∆mod-Λ acts identically on projective functors. Using projective +presentations, it follows that ∆ is isomorphic to the identity functor on the whole mod-(mod-Λ). +Definition of ∆ then implies that the duality functor D : mod-(mod-Λ) → mod-(mod-Λ)op is + +16 +HOSSEIN ESHRAGHI AND RASOOL HAFEZI +isomorphic to Θ′ ◦τ −1 +H ◦(Θ)−1. This proves the claim by following the definitions of the functors +involved. +□ +Corollary 5.9. Let M be an indecomposable Λ-module with a minimal projective presentation +P +f→ Q → M → 0. Then there is an exact sequence +0 → (−, τ(M)) → D(P, −) → D(Q, −) → D(M, −) → 0 +in mod-(mod-Λ). +Proof. If M is projective, then such a sequence exists trivially. Otherwise, applying Corollary +5.4 for X = mod-Λ, there exists an exact sequence +0 → (−, σX τX (M)) → D(P, −) → D(Q, −) → D(M, −) → 0 +in mod-(mod-Λ). Now Theorem 5.7 settles the statement. +□ +Let us exploit Corollary 5.9 to observe a connection between the inverse Auslander-Reiten +translation of an indecomposable non-projective Λ-module M with the second syzygies of injec- +tive functors. For, replace M in Corollary 5.9 by τ −1M to get the exact sequence +0 → (−, M) → D(P, −) → D(Q, −) → D(τ −1(M), −) → 0 +in mod-(mod-Λ)op in which P → Q → τ −1(M) → 0 is a minimal projective presentation. +Applying the duality D : mod-(mod-Λ)op → mod-(mod-Λ) gives the exact sequence +0 → (τ −1(M), −) → (Q, −) → (P, −) → D(−, M) → 0. +This shows that the functor (τ −1(M), −) might be interpreted as a second syzygy of the injective +functor D(−, M). +6. simple modules over (stable) Auslander algebra +Assume Λ is of finite representation type and let M be a basic representation generator of +mod-Λ; that is, M is the direct sum of all pairwise non-isomorphic indecomposable finitely +generated Λ-modules. +The endomorphism algebra A(Λ) = EndΛ(M), simply denoted by A +throughout the section, is called the Auslander algebra of Λ. Moreover, the stable Auslander +algebra of Λ is by definition A = EndΛ(M)/P, where P is the ideal in EndΛ(M) consisting of +those endomorphisms factoring through a projective module. In this case, we can identify mod-A +with mod-(mod-Λ) via the equivalence induced by the evaluation functor eM : mod-(mod-Λ) → +mod-A, F �→ F(M). It is also easy to see that eM induces an equivalence between mod-(mod-Λ) +and mod-A. +It is known [A76] that indecomposable modules in mod-Λ correspond bijectively to sim- +ple functors in mod-(mod-Λ) by sending an indecomposable module M to the simple functor +SM := (−, M)/rad(−, M). Further, for any indecomposable non-projective module M, there is +a minimal projective resolution +0 → (−, N) +(−,f) +→ (−, K) +(−,g) +→ (−, M) → SM → 0 +of SM such that 0 → N +f→ K +g→ M → 0 is an almost split sequence in mod-Λ ([A76, §2]). +Combined to the above observations on the Auslander algebra A, one may identify simple A- +modules (resp. simple A-modules) and indecomposable (resp. indecomposable non-projective) +Λ-modules. + +FROM MORPHISM CATEGORIES TO FUNCTOR CATEGORIES +17 +Specializing [H, Construction 3.1] to the module category mod-Λ gives a functor Ψ : S → +mod-(mod-Λ), S being the monomorphism category of Λ. This is defined by sending (X +f→ Y ) +in S to the functor F ∈ mod-(mod-Λ) lying in the exact sequence +0 → (−, X) +(−,f) +→ (−, Y ) → (−, Coker(f)) → F → 0 +in mod-(mod-Λ). +As the following result says, Ψ behaves well with respect to almost split +sequences. +Lemma 6.1. ([H, Proposition 5.7]) Let 0 → U → V → W → 0 be an almost split sequence +in S. Assume W is neither of types (a) or (b), nor of the form (Ω(X) → P), where X is a +non-projective indecomposable Λ-module with projective cover P. Then +0 → Ψ(U) → Ψ(V) → Ψ(W) → 0 +is an almost split sequence in mod-(mod-Λ). +The following theorem is one of the main results in this section. +Theorem 6.2. Assume Λ is of finite representation type and A is its stable Auslander algebra. +Let S be a simple non-projective A-module. Then, exactly one of the followings hold: +(1) the Auslander-Reiten translate τA(S) is projective. +(2) there exists a simple A-module S′ such that τA(S) ≃ Ω−1 +A (S′). In this case, Ext2 +A(S, S′) ≃ +DHomA(S, S). +Proof. According to aforementioned remarks, the simple non-projective module S corresponds +to a simple functor (−, C)/rad(−, C) lying in the exact sequence +0 → (−, A) +(−,f) +→ (−, B) +(−,g) +→ (−, C) → (−, C)/rad(−, C) → 0 +in mod-(mod-Λ) in such a way that λ : 0 → A +f→ B +g→ C → 0 is an almost split sequence +in mod-Λ. Note that the middle term B may not be projective since otherwise there exists an +isomorphism (−, C)/rad(−, C) ≃ (−, C) which is against non-projectivity of S. +We distinguish two cases: Assume first that A is projective. So by Proposition 3.3 of [HE], +there exists an almost split sequence +0 +�� rad(A) +A +� +i +�( A +A )1 ⊕ +� rad(A) +B +� +fi +�( A +B )f +�0 +in S(Λ). Hence, in view of Lemma 6.1, we get the almost split sequence +0 → Ψ +� rad(A) +A +� +i → Ψ +� rad(A) +B +� +fi → Ψ( A +B )f → 0 +in mod-(mod-Λ). +Since A is projective, the definition of Ψ shows that Ψ(rad(A) +i→ A) ≃ +(−, A/rad(A)). Likewise, as λ does not split, we have Ψ(A +f→ B) ≃ (−, C)/rad(−, C). Hence +τA((−, C)/rad(−, C)) ≃ (−, A/rad(A)) that proves the claim in this case. +Assume next that A is not projective. Then there exists an almost split sequence +ϵ : 0 → A′ → B′ → A → 0 +in mod-Λ. Applying [H, Lemma 6.3] on λ and ϵ, one infers the almost split sequences +0 +�( A +A )1 +�( A +B )f +�( 0 +C )0 +�0 and + +18 +HOSSEIN ESHRAGHI AND RASOOL HAFEZI +0 +�� A′ +I +� +e +�� B′ +I⊕A +� +h +�( A +A )1 +�0 +in S where the second one is obtained from the push-out diagram +A′ +e +� +� B′ +h +� +� A +(†) +I +e +� +� I ⊕ A +d +� +� A +Ω−1 +Λ (A) +Ω−1 +Λ (A) +in which e : A′ → I is the injective envelope. From [HZ, Lemma 3.3], we can write (B′ +h→ +I ⊕ A) ≃ X ⊕ (J +1→ J), where X is an indecomposable non-projective object and J is either zero +or isomorphic to I. It follows then that τS(A +f→ B) ≃ X. Accordingly, by taking into account +that (−, C)/rad(−, C) ≃ Ψ(A +f→ B) by the exact sequence mentioned at the beginning of the +proof, another application of Lemma 6.1 shows that +(6.1) +F := τmod-(mod-Λ)((−, C)/rad(−, C)) ≃ Ψ(X). +However, the definition of Ψ yields F = Ψ(B′ +h→ I ⊕ A). Therefore, abusing the notation, we +may write τA(S) ≃ F. +Regarding the definition of Ψ, the middle column of (†) gives the long exact sequence +0 +� (−, B′) +� (−, I ⊕ A) +� (−, Ω−1 +Λ (A)) +� +� +Ext1 +Λ(−, B′) +� Ext1 +Λ(−, I ⊕ A) +F +� +in mod-(mod-Λ) that implies F = Ker(Ext1 +Λ(−, B′) → Ext1 +Λ(−, A)) because I is injective. On +the other hand, since ϵ is an almost split sequence, our previous considerations show that there +exists an exact sequence +0 +� (−, A′) +� (−, B′) +� (−, A) +� +� +Ext1 +Λ(−, A′) +� +� +Ext1 +Λ(−, B′) +� Ext1 +Λ(−, A) +(−, A)/rad(−, A) +� +F +� +of functors. Invoking [AR74, Proposition 7.4], we see that Ext1 +Λ(−, A′) is an injective functor in +mod-(mod-Λ) and so the induced short exact sequence 0 → (−, A)/rad(−, A) → Ext1 +Λ(−, A′) → +F → 0 gives F = Ω−1 +A ((−, A)/rad(−, A)). Now it suffices to set S′ = (−, A)/rad(−, A). Notice +that the last assertion in the theorem is an upshot of the Auslander-Reiten formula. +□ +Based on previous theorem, in the following result we establish a bijection between certain +simple modules over A and Λ. This provides an interesting application concerning the stable +equivalences of Artin algebras. +Corollary 6.3. Let Λ be of finite representation type and A be its stable Auslander algebra. +There exists a bijection between + +FROM MORPHISM CATEGORIES TO FUNCTOR CATEGORIES +19 +(1) the set of isomorphism classes of non-projective simple modules S ∈ mod-A whose +Auslander-Reiten translate τA(S) is projective; and +(2) the set of isomorphism classes of indecomposable non-injective projective modules P ∈ +mod-Λ such that the middle term of the almost split sequence starting from P is not +projective; and +(3) the set of isomorphism classes of simple modules S ∈ mod-Λ whose projective cover P(S) +is non-injective, and the middle term of the almost split sequence starting from P(S) is +not projective. +Proof. The bijection between (2) and (3) might be shown by restricting the well-known bijection +between simple and indecomposable projective modules. The map from (2) to (1) is given by +sending P to (−, τ −1(P))/rad(−, τ −1(P)), which is well-defined due to the argument given in +Theorem 6.2. Let now S be a simple non-projective module in mod-A with τA(S) projective. +We already know that there is an indecomposable non-projective Λ-module C such that S ≃ +(−, C)/rad(−, C). We claim that τ(C) is projective; otherwise, as in Theorem 6.2, there exists +a simple A-module S′ such that τA(S) ≃ Ω−1 +A (S′). Hence the short exact sequence 0 → S′ → +I → Ω−1 +A (S′) → 0, in which I′ is the injective envelop of S′, splits. This means that Ω−1 +A (S′) = 0 +and so τA(S) = 0 which is against non-projectivity of S. Thus τ(C) is projective and setting +P := τ(C) completes the proof. +□ +Recall that two Artin algebras Λ and Λ′ are said to be stably equivalent if there is an equiv- +alence of categories mod-Λ ≃ mod-Λ′. +Denote by n(Λ) the number of iso classes of simple +Λ-modules satisfying the third condition of the above corollary. As a byproduct, we show that +n(Λ) is an invariant of the stable equivalences. +Proposition 6.4. Let Λ and Λ′ be of finite representation type and stably equivalent. Then +n(Λ) = n(Λ′). +Proof. Since Λ and Λ′ are stably equivalent, it follows that the corresponding stable Auslander +algebras A and A′ are Morita equivalent. By Corollary 6.3, we see that simple modules in mod-Λ +(resp. mod-Λ′) that satisfy condition (3) correspond bijectively to non-projective simple modules +over the stable Auslander algebra A (resp. A′) with projective Auslander-Reiten translates. We +are done since the modules of latter type are preserved under Morita equivalences. +□ +The following lemma is taken from [HE, Proposition 3.2]. +Lemma 6.5. Let δ : 0 → A +f→ B +g→ C → 0 and δ′ : 0 → A′ f ′ +→ B′ g′ +→ A → 0 be almost split +sequences in mod-Λ. Then +0 +�� B′ +A +� +g′ +� +� +� g′ +1 +� +� 1 +f +� +� +� +�( A +A )1 ⊕ +� B′ +B +� +fg′ +� [ −1 g′ ] +[ −f 1 ] +� +�( A +B )f +�0, +is an almost split sequence in H. Further, +� B′ +B +� +fg′ is an indecomposable object. +The following theorem should be served as the second main result of this section. +Theorem 6.6. Assume Λ is a self-injective algebra of finite representation type and let A be +its Auslander algebra. Let also S be a simple A-module of projective dimension two. Then there +exists a simple A-module S′ of projective dimension two such that ΩA(S′) ≃ τ −1 +A (S). In this +case, Ext2 +A(S′, S) ≃ DHomA(S, S). + +20 +HOSSEIN ESHRAGHI AND RASOOL HAFEZI +Proof. We may identify the simple module S by the simple functor (−, A)/rad(−, A) lying in +the exact sequence +0 → (−, A′) +(−,f ′) +→ +(−, B′) +(−,g′) +→ +(−, A) → (−, A)/rad(−, A) → 0 +(†) +in mod-(mod-Λ) in such a way that δ : 0 → A′ f ′ +→ B′ g′ +→ A → 0 is an almost split sequence in +mod-Λ. Let also δ′ : 0 → A +f→ B +g→ C → 0 be an almost split sequence in mod-Λ. Then by +Lemma 6.5 there exists an almost split sequence +0 +�� B′ +A +� +g′ +� +� +� g′ +1 +� +� 1 +f +� +� +� +�( A +A )1 ⊕ +� B′ +B +� +fg′ +� [ −1 g′ ] +[ −f 1 ] +� +�( A +B )f +�0 +in H. Thanks to Theorem 4.8, this induces the almost split sequence +0 → Θ(B′ g′ +→ A) → Θ(B′ fg′ +→ B) → Θ(A +f→ B) → 0 +(††) +in mod-(mod-Λ). Note that (†) implies Θ(B′ +g′ +→ A) = (−, A)/rad(−, A) = S. Hence by (††), +τ −1 +A (S) = Θ(A +f→ B). Set now W = (A +f→ B). Then, by definitions, there exists an exact +sequence +(−, A) +� (−, B) +� +� +(−, C) +� +� +Ext1 +Λ(−, A) +� Ext1 +Λ(−, B). +Θ(W) +� +(−, C)/rad(−, C) +� +Set S′ be the simple functor (−, C)/rad(−, C). Then the short exact sequence 0 → Θ(W) → +(−, C) → S′ → 0 proves the claim. +□ +7. Auslander-Reiten components of Auslander algebras +Throughout the section, we assume that Λ is a non-semisimple self-injective algebra of finite +representation type and A denotes its Auslander algebra. In the whole section, we use the iden- +tification mod-A ≃ mod-(mod-Λ) described earlier. Once more, in this section, the quadruple +family of objects in H of types (a), (b), (c), and (d) become important. We aim to identify cer- +tain components of the (stable) Auslander-Reiten quiver of A. To this end, we need firstly study +particular τH-periodic objects in H and their periodicity. +7.1. τH-periodic objects. As we observed in Theorem 4.8, the functor Θ : H → mod-A behaves +well with respect to almost split sequences in the sense that if there exists an almost split sequence +0 → X → Y → Z → 0 in H where Z is not of type (b) or (c), then 0 → Θ(X) → Θ(Y) → Θ(Z) → 0 +is also an almost split sequence in mod-A. Also we have seen in Theorem 4.2 that one is given an +equivalence H/V ≃ mod-A where V is generated by the objects of type (b) or (c). Therefore, the +Auslander-Reiten quiver ΓA of the Auslander algebra A might be computed via the Auslander- +Reiten quiver ΓH of H by removing vertices corresponding to iso-classes of indecomposable +objects of either types (b) or (c). +The following construction is vital for the rest of this section. It is mainly based on an analysis +of various almost split sequences already obtained in [HE]. For the sake of brevity, we prefer not +to rewrite most of them here and suffice to give the precise reference number therein. + +FROM MORPHISM CATEGORIES TO FUNCTOR CATEGORIES +21 +Construction 7.1. Let C be an indecomposable non-projective Λ-module. There exist almost +split sequences ϵ1 : 0 → τ(C) +f→ B +g→ C → 0 and ϵ2 : 0 → τ 2(C) +f ′ +→ B′ g′ +→ τ(C) → 0 in mod-Λ. +Applying Lemmas 6.5 and 4.4 we deduce the almost split sequences +0 +� +� +B′ +τ(C) +� +g′ +� +� +� g′ +1 +� +� 1 +f +� +� +� +� +� +τ(C) +τ(C) +� +1 ⊕ +� B′ +B +� +fg′ +� [ −1 g′ ] +[ −f 1 ] +� +�� τ(C) +B +� +f +�0, +0 +� +� +τ(C) +τ(C) +� +1 +� 1 +f +� +�� τ(C) +B +� +f +� 0 +g +� +�( 0 +C )0 +�0, +and +0 +�� τ(C) +0 +� +0 +� f +0 +� +�( B +C )g +( g +1) +�( C +C )1 +�0 +in H. Evidently, the indecomposable object (B′ fg′ +→ B) is not projective; so let X := τH(B′ fg′ +→ B). +Also, as (B′ g′ +→ τ(C)) is not projective, we let Y := τH(B′ g′ +→ τ(C)) and note that X and Y are +not projective. In view of [HE, Propositions 2.2, 4.1], there exists an almost split sequence +0 +� +� +ν(P ) +ν(Q) +� +ν(h) +�Y ⊕ ( I +0 )0 +� +� +τ 2(C) +0 +� +0 +�0 +in H where P +h→ Q → τ 2(C) → 0 is the minimal projective presentation, and I is an injective +module. On the other hand, by [HE, Propositions 2.4, 4.2], we have the almost split sequence +0 +� +� +0 +τντ 2(C) +� +0 +�τH(Y) ⊕ ( 0 +P )0 +� +� +ν(P ) +ν(Q) +� +ν(h) +�0 +in H where P is projective. Putting all together, one obtains the mesh +τ(X) +� +X +� +� +� B′ +B +� +fg′ +� +� +τ(Y) +� +� +Y +� +� +� +� +B′ +τ(C) +� +g′ +� +� +� +� τ(C) +B +� +f +� +� +� +0 +ντ 3(C) +� +0 +� +� +ν(P ) +ν(Q) +� +ν(h) +� +� +� +τ 2(C) +0 +� +0 +� +� +� +τ(C) +τ(C) +� +1 +� +� +( 0 +C )0 +� +in the Auslander-Reiten quiver ΓA of A in which the vertices (I → 0) and (0 → P) have been +ignored. +Let us recall from [HE, Remark 5.7] that A = ντ 3 defines an auto-equivalence on the stable +category mod-Λ. As is expected, the A -orbit of an indecomposable non-projective Λ-module M +consists of the modules A m(M) where m ranges over the integer numbers. +Proposition 7.2. Suppose every indecomposable non-projective Λ-module possesses a finite A - +orbit. Then any indecomposable non-projective object X in H of either types (a), (b), (c) or (d) +is of τH-periodicity a multiple of 4. + +22 +HOSSEIN ESHRAGHI AND RASOOL HAFEZI +Proof. According to our previous observations, all mentioned objects lie in the τH-orbit of some +indecomposable object of type (a). Hence it suffices to prove the statement only for X = (0 → N) +with N an indecomposable non-projective module. Justified by the hypothesis, choose a least +integer n with A n(N) = N. Considering the particular mesh in ΓA as illustrated in Construction +7.1, we get τ 4 +H(0 → N) ≃ (0 → A (N)) and thus τ 4n +H (0 → N) ≃ (0 → A n(N)) = (0 → N). +□ +Based on previous proposition, we are now able to prove the following theorem which will +prove useful later on. +Theorem 7.3. Assume every indecomposable non-projective Λ-module possesses a finite A - +orbit. Then every simple A-module of projective dimension 2 is τA-periodic of periodicity divided +by 4. +Proof. Recall that such simple A-modules might be identified with simple functors SM = +(−, M)/rad(−, M) where M is an indecomposable non-projective Λ-module lying in an almost +split sequence 0 → τ(M) +g→ N +f→ M → in mod-Λ. By Proposition 7.2, (0 → τ −1(M)) is of +τH-periodicity 4n for a suitable integer n. Since (0 → τ −1(M)) and (M +1→ M) lie in the same +τH-orbit, if it follows that (M +1→ M) is also of the same periodicity 4n. Thus the irreducible +morphism (N +f→ M) → (M +1→ M) in ΓH remains fixed after 4n applications of τH and ac- +cordingly, (N +f→ M) should be of τH-periodicity 4n. Consequently, according to Theorem 4.8, +τ 4n +A (SM) = τ 4n +A Θ(N +f→ M) = Θτ 4n +H (N +f→ M) = Θ(N +f→ M) = SM. +□ +7.2. Modules M with τ(M) = Ω(M). +Definition 7.4. Let M be an indecomposable non-projective module. +We say M has the +property (∗) if 0 → Ω(M) → P(M) → M → 0 is an almost split sequence in mod-Λ where +P(M) is the projective cover of M. +Modules satisfying this property have already been classified in [ARS, Theorem V.3.3]: these +are exactly non-injective simple Λ-modules M that are not a composition factor of rad(I)/soc(I) +for every injective Λ-module I. This clearly yields that such modules are necessarily A-periodic. +Note also that in the situation of the definition, τ(M) = Ω(M). The goal in this subsection is to +see that existence of modules with this property may heavily affect the shape of the AR-quiver +of A and in particular cases may even make it into an algebra of finite representation type. As +a first pace to study modules with property (∗), the following lemma shows that this property +carries over from a module to its (co)syzygies. +Lemma 7.5. Let M be an indecomposable non-projective Λ-module. If M has the property (∗), +then so do all its syzygies (resp. cosyzygies). In particular, the short exact sequences +0 → Ωi+1(M) → P i → Ωi(M) → 0 for i ≥ 0, and +0 → Ωi(M) → Ii → Ωi−1(M) → 0 for i ≤ 0 +in mod-Λ induced by the minimal projective (resp. injective) resolution of M are almost split. +Proof. We prove the lemma for integers i ≥ 0 by using an inductive argument whose basis +i = 0 is satisfied by the assumption; so we put i > 0. +Consider the almost split sequence +0 → τΩi(M) → B → Ωi(M) → 0 in mod-Λ. +We claim that B is projective. +Assume to +the contrary that B has a non-projective indecomposable direct summand C. The induction + +FROM MORPHISM CATEGORIES TO FUNCTOR CATEGORIES +23 +hypothesis then implies that τ −1(C) is a non-projective direct summand of P i−1, which is +absurd. Now use the fact that the morphisms involved in an almost split sequence are minimal +to deduce that τΩi(M) = Ωi+1(M). +□ +The following lemma shows a property of the modules M for which (∗) is satisfied; this will +be used later on in this section. +Lemma 7.6. Under the hypothesis of Lemma 7.5, one has ν(P i+1) ≃ P i for i ≥ 0 and +ν−1(Ii−1) ≃ Ii for i ≤ 0. +Proof. We prove the first assertion. The minimal projective presentation P 1 → P 0 → M → 0 +induces the short exact sequence 0 → τ(M) → ν(P 1) → ν(P 0) → ν(M) → 0 in mod-Λ. +Note that, as ν is an auto-equivalence of mod-Λ, the map τ(M) → ν(P 1) is minimal and +thus defines the injective envelope of τ(M) as ν(P 1) is injective. However, by definition, the +monomorphism τ(M) → P 0 obtained by composing the isomorphism τ(M) ≃ Ω(M) and the +inclusion Ω(M) → P 0 is also minimal with P 0 injective. +Therefore ν(P 1) ≃ P 0 since the +injective envelope is unique up to isomorphism. +Now we deduce the result by applying an +inductive argument in conjunction with Lemma 7.5. +□ +The following theorem is the promised one. +Theorem 7.7. Assume there exists an indecomposable non-projective Λ-module M with the +property (∗). Then the Auslander-Reiten quiver ΓA of A is a finite oriented cycle. In particular, +the Auslander algebra A is of finite representation type. +Proof. The minimal projective presentation +· · · → P n wn +→ P n−1 → · · · P 1 w1 +→ P 0 → M → 0 +of M induces, according to Lemma 7.5, the almost split sequences +ϵi : 0 → Ωi+1(M) +vi +→ P i ui +→ Ωi(M) → 0 +in mod-Λ. Applying Lemma 6.5 on ϵ0 and ϵ1 gives the almost split sequence +0 +� +� +P 1 +Ω(M) +� +u1 +� +� +Ω(M) +Ω(M) +� +1 ⊕ +� +P 1 +P 0 +� +w1 +� +� +Ω(M) +P 0 +� +v0 +�0. +Note that, by Lemma 4.4, τH(( 0 +M )) = +� +Ω(M) +Ω(M) +� +1 and τH +�� +Ω(M) +Ω(M) +� +1 +� += +� +Ω2(M) +0 +� +. Moreover, in +light of [HE, Proposition 2.4], we get τH(P 1 w1 +→ P 0) = (0 → Ω(M)) and so there exists an almost +split sequence +0 +�� +0 +Ω(M) +� +0 +� +� +P 1 +Ω(M) +� +u1 +⊕ +� 0 +P 0 +� +0 +� +� +P 1 +P 0 +� +w1 +�0. +Furthermore, an application of [HE, Proposition 3.5] provides us with another almost split +sequence +0 +� +� +Ω2(M) +P 1 +� +v1 +�� +0 +Ω(M) +� +0 ⊕ +� +P 1 +P 1 +� +1 ⊕ +� +Ω2(M) +0 +� +0 +� +� +P 1 +Ω(M) +� +u1 +�0. +Also [HE, Proposition 2.2] combined to Lemma 7.6 results in the almost split sequence +0 +� +� +ν(P 3) +ν(P 2) +� +ν(w3) +� +� +Ω2(M) +P 1 +� +v1 ⊕ +� +ν(P 3) +0 +� +0 +� +� +Ω2(M) +0 +� +0 +�0. + +24 +HOSSEIN ESHRAGHI AND RASOOL HAFEZI +It is easy to see that, as Ω2(M) satisfies (∗), so does νΩ2(M) and consequently, τνΩ2(M) = +ΩνΩ2(M) ≃ νΩ3(M) = A (M). Therefore, by [HE, Proposition 2.4], we have τH(ν(P 3) +ν(w3) +→ +ν(P 2)) = (0 → A (M)). +Continuing in this manner, one obtains the following mesh in the +Auslander-Reiten quiver of H. +� +0 +P 0 +� +0 +� +� +Ω2(M) +Ω2(M) +� +1 +� +� +0 +Ω(M) +� +0 +� +� +� +� +P 1 +P 0 +� +w1 +� +� +� +P 2 +Ω2(M) +� +u2 +� +� +� +� +ν(P 3) +ν(P 3) +� +1 +� +� +Ω2(M) +P 1 +� +v1 +� +� +� +� +P 1 +P 1 +� +1 +� +� +P 1 +Ω(M) +� +u1 +� +� +� Ω(M) +P 0 +� +v0 +� +� +� +0 +A (M) +� +0 +� +� +ν(P 3) +ν(P 2) +� +ν(w3) +� +� +� +� +Ω2(M) +0 +� +0 +� +� +� Ω(M) +Ω(M) +� +1 +� +� +� 0 +M +� +0 +� +� +ν(P 3) +0 +� +0 +� +Starting then with the vertex (0 → A (M)) and iterating the above arguments, one may +calculate the vertices lying on the left side of (0 → M) in ΓH. Also, by considering the minimal +injective resolution of M, the vertices on the right part appear. Summarizing, it follows that the +component in ΓH containing the vertex (0 → M) is obtained by putting together all parts of the +above shape corresponding to modules in the A -orbit of M. By virtue of previous considerations, +ΓA comes up from ΓH by removing vertices of types (b) and (c). Hence the AR-quiver ΓA is +obtained by gluing together all pieces of the following shape. +� +0 +Ω(M) +� +0 +� +� +P 1 +P 0 +� +w1 +� +� +� +P 2 +Ω2(M) +� +u2 +� +� +Ω2(M) +P 1 +� +v1 +� +� +� +P 1 +Ω(M) +� +u1 +� +� +� Ω(M) +P 0 +� +v0 +� +� +� +0 +A (M) +� +0 +� +� +ν(P 3) +ν(P 2) +� +ν(w3) +� +� +� 0 +M +� +0 +Now, since M is A -periodic, we get a finite oriented cycle as a component in ΓA that by [ARS, +Theorem VII.2.1] must be the whole of ΓA. +□ +7.3. Components of the stable Auslander-Reiten quiver of A. We let Γs +A, the stable +Auslander-Reiten quiver of A, be the subquiver of ΓA obtained by removing projective vertices +and their τA-orbits. +It should be clarified that here, we distinguish with a usual custom in +the corresponding literature where this terminology applies while removing vertices that are +both projective and injective. Also we notice that, generally, this has nothing to do with the +Auslander-Reiten quiver of the stable Auslander algebra A. For instance, despite A which is +self-injective in this case, A can not be self-injective since it is of global dimension 2. +Below, we use results from [AS93] to get a nice intuition of the stable Auslander-Reiten quiver +Γs +A of A in terms of the AR quiver of a triangulated category. +Remark 7.8. Recall from Section 3 that the class X in H consisting of all objects of type +(a), (b), (c), or (d) determines an exact structure HX on H which has enough projectives and +enough injectives. +We claim that P(HX ) = X ∪ proj-H and I(HX ) = τH(X) ∪ inj-H, the +subcategories of projectives and injectives of HX , coincide. +Indeed, as in Construction 7.1, + +FROM MORPHISM CATEGORIES TO FUNCTOR CATEGORIES +25 +τH(X) ⊆ X. Since every indecomposable injective object in H is of type (b) or (c), we get +I(HX ) ⊆ P(HX ). To settle the reverse inclusion, note that by [HE, Proposition 3.6], for an +indecomposable projective Λ-module P there exists a projective Λ-module Q with τH(Q → +0) = (0 → P). Besides that projective objects of type (P +1→ P) lie in inj-H, this ensures that +proj-H ⊆ I(HX ). Now take a non-injective object M of X. If M is of type (a), then M = (0 → M) +for an indecomposable Λ-module M. If M is projective then as above, M = τH(Q → 0) for some +Q; otherwise M is non-injective and Proposition 2.4 of [HE] shows that M = τH(P1 → P0) +where P1 → P0 → τ −1(M) → 0 is the minimal projective presentation. If M = (M +1→ M) is of +type (b) with M non-injective, then Lemma 4.4 gives M ≃ τH(0 → τ −1(M)). Furthermore, if +M = (M → 0) is of type (c), then again Lemma 4.4 shows that M ≃ τH(τ −1(M) +1→ τ −1(M)) +as M is non-injective. Finally if M = (P +f→ Q) is of type (d) then, setting N = Coker(ν(f)), +we deduce from [HE, Proposition 2.2] that M = τH(N → 0). Summarizing, these imply that +X ⊆ I(HX ) and the above claim follows; that is to say, HX is a Frobenius exact category and, +consequently, the stable category HX is triangulated. +On the other hand, according to [AS93, Proposition 1.9], we infer that an almost split sequence +0 → X → Y → Z → 0 in H is an almost split sequence in HX if and only if neither X ∈ I(HX ) +nor Z ∈ P(HX ). Thus, in order to get the Auslander-Reiten quiver of the triangulated category +HX , it is enough to remove the iso-classes of indecomposable objects in X and arrows attached +to them from the Auslander-Reiten quiver ΓH of H. But, as stated before, what remains after +deleting, is exactly the stable Auslander-Reiten quiver Γs +A of A. +The following theorem is the main result in this subsection. Note that if Λ admits a module +M with property (∗) then, according to Theorem 7.7, Γs +A is just a set of single vertices. That’s +why one has to exclude this case from the hypothesis below. +Theorem 7.9. Assume Λ is self-injective of finite representation type and Ξ is a component +of Γs +A containing a simple module SM for an indecomposable non-projective Λ-module M not +fulfilling (∗). Then +(i) If Ξ is finite, then Ξ = Z∆/G, where ∆ is a Dynkin quiver and G is an automorphism +group of Z∆ containing a positive power of the translation. Moreover, Ξ is Γs +A itself if +we further assume that Λ is indecomposable. +(ii) If Ξ is infinite, then it is a stable tube. +Proof. We have seen before that the AR-quiver ΓA of the Auslander algebra A is obtained from +ΓH by removing the vertices of types (b) and (c). Note that indecomposable objects of type (a) +correspond to indecomposable projective A-modules and those of type (d) lie in the τA-orbit of +indecomposable projective A-modules by Construction 7.1. So in fact, Ξ emerges by deleting +vertices of either types (a), (b), (c) and (d) and the arrows attached to them from a component +Ξ′ of ΓH containing the vertex (0 → M). +Note that such a Ξ is connected as M does not +satisfy the property (∗). Note also that all vertices in Γs +A are stable in the sense that τ m +A (−) is +well-defined over them for arbitrary integers m. Therefore Theorem 7.3 implies that the vertex +SM in Ξ is τA-periodic and both the assertions in (i) and (ii) follow from [L, Theorem 5.5]. For +the second statement in (i), note that indecomposability of Λ implies that the lower triangular +matrix algebra T2(Λ) = ( Λ 0 +Λ Λ ) is also indecomposable and recall that H is naturally equivalent to +the category mod-T2(Λ). Now if Ξ is finite, then so is Ξ′. As such, Ξ′ itself is a finite component +of ΓH which, according to [ARS, Theorem VII.2.1], should be the whole of ΓH. Hence Ξ = Γs +A, +as desired. +□ + +26 +HOSSEIN ESHRAGHI AND RASOOL HAFEZI +We conclude this section by quoting an observation from [HE]. Assume M is an indecom- +posable non-projective Λ-module. Denote by [M]A the A -orbit of M. Let also ΓH(M) be the +unique component of ΓH containing the vertex (0 → M). Moreover, we set +T = {ΓH(M) | M is indecomposable non-projective} +E = {[M]A | M is indecomposable non-projective} +Then there exists a well-defined map δ : E → T which is given by sending [M]A to ΓH(M). +Further, if we let T∞ denote the subset of T consisting of all infinite components, and E∞ be the +inverse image of T∞ under δ, then by [HE, Proposition 5.8], δ is surjective and the restricted map +δ |: E∞ → T∞ is a bijection whenever Λ is indecomposable self injective of finite representation +type. +Inspired by this result, we let L be the set of all components of ΓA containing a simple +module. Define λ : T −→ L by λ(ΓH(M)) = ΓA(SM) where ΓA(SM) is the component of ΓA +that contains the simple vertex SM. +Proposition 7.10. Suppose Λ is indecomposable self-injective of finite representation type. +(i) The map λ is well-defined and surjective. +(ii) λ restricts to a bijection λ |: T∞ → L∞, where L∞ is the subset of L consisting of all +infinite components. +(iii) The sets E∞ and L∞ are in bijection. +Proof. (i). Take M0 and M1 to be indecomposable non-projective Λ-modules with ΓH(M0) = +ΓH(M1). Then there exist almost split sequences 0 → A0 +f0 +→ B0 +g0 +→ M0 → 0 and 0 → A1 +f1 +→ +B1 +g′ +1 +→ M1 → 0 in mod-Λ. By Lemma 4.4, there exist almost split sequences +0 +�� Ai +0 +� +0 +� fi +1 +� +�� Bi +Mi +� +gi +� 0 +gi +� +�� 0 +Mi +� +0 +�0 +in H for i = 0, 1. Accordingly, (B0 +g0 +→ M0) and (B1 +g1 +→ M1) lie inside ΓH(M0) and the connected- +ness of ΓH(M0) gives the existence of a walk (B0 +g0 +→ M0) ←→ x1 ←→ · · · xn−1 ←→ (B1 +g1 +→ M1) +in ΓH. If none of the xi is of the form (b) or (c), then using Theorem 4.8 we get a walk in ΓA be- +tween Θ(B0 +g0 +→ M0) = SM0 and Θ(B1 +g1 +→ M1) = SM1. Consequently, λ(ΓH(M0)) = λ(ΓH(M1)). +Otherwise we may, without loss of generality, assume that the xi are all non-projective and apply +the arguments used in Construction 7.1 to obtain a walk in ΓH passing through (Bi +gi +→ Mi), +i = 0, 1, none of the vertices over which are of the forms (b) or (c). +To settle (ii), assume +ΓA(SM) is an infinite component of ΓA for an indecomposable non-projective Λ-module M. By +Theorem 7.9, ΓA(SM) is a stable tube and the τA-orbit of SM generates the mouth of ΓA(SM). +The fact that the mouth of a stable tube is unique reveals that ΓA(SM) is uniquely determined +by ΓH(M). The last statement is a combination of (ii) and [HE, Proposition 5.8]. +□ +References +[AIR] T. Adachi, O. Iyama, and I. Reiten, τ-tilting theory, Compos. Math., 150(3):415–452, 2014. +[A65] M. Auslander, Coherent functors, in Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965), 189231, +Springer, New York, 1966. +[A71] M. Auslander, Representation dimension of artin algebras, Queen Mary College Notes (1971). + +FROM MORPHISM CATEGORIES TO FUNCTOR CATEGORIES +27 +[A76] M. Auslander, Functors and morphisms determined by objects. Representation theory of algebras (Proc. +Conf., Temple Univ., Philadelphia, Pa., 1976), pp. 1-244. Lecture Notes in Pure Appl. Math., Vol. 37, +Dekker, New York, 1978. +[AR74] M. Auslander and I. Reiten, Stable equivalence of dualizing R-varieties, Advances in Math. 12 (1974), +306-366. +[AR76] M. Auslander and I. Reiten, On the representation type of triangular matrix rings, J. London Math. +Soc. (2), 12 (1976), 371-382. +[AR77] M. Auslander and I. Reiten, Representation theory of Artin algebras. IV. Invariants given by almost +split sequences, Comm. Algebra 5 (1977), 443-518. +[AR78] M. Auslander and I. Reiten, Representation theory of Artin algebras. VI. A functorial approach to +almost split sequences, Comm. Algebra 6 (1978), no. 3, 257-300. +[ARS] M. Auslander, I. Reiten and S. O. Smalø, Representation theory of Artin algebras, Cambridge Studies +in Advanced Mathematics, 36. Cambridge University Press, Cambridge, 1995. xiv+423 pp. ISBN: 0-521- +41134-3. +[AS81] M. Auslander, and S. O. Smalø, Almost split sequences in subcategories, Journal of Algebra, 69 (1981), +426-454. +[AS93] M. Auslander and O. Solberg, Relative homology and representation theory I, relative homology and +homologically finite categories, Comm. Algebra 21 (9) (1993), 2995-3031. +[Ba] +R. Bautista, The category of morphisms between projective modules, Comm. Algebra 32 (2004), no. 11, +4303-4331. +[Bu] +A. B. Buan, Closed subbifunctors of the extension functor, J. Algebra 244 (2001), no. 2, 407-428. +[E] +H. Eshraghi, The Auslander-Reiten translation in morphism categories, J. Algebra Appl. 13 (2014), no. +3, 1350119, 12 pp. +[G] +J. Gillespie, The homotopy category of N-complexes is a homotopy category, J. Homotopy Relat. Struct. +10 (2015), 93-106. +[H] +R. Hafezi, From subcategories to the entire module categories, Forum Math. 33 (2021), no. 1, 245-270. +[HE] +R.Hafezi and H. Eshraghi, Determination of some almost split sequences in morphism categories, avail- +able via arXiv:2103.08883 [Math.RT]. +[HM] +R. Hafezi and E. Mahdavi, Covering theory, (mono)morphism categories and stable Auslander algebras, +available via arXiv:2011.08646 [Math.RT]. +[HMV1] L. Angeleri H¨ugel, F. Marks, and J. Vit´oria, silting modules and ring epimorphisms, Adv. Math. +303 (2016), no. 5, 1044-1076. +[HMV2] L. Angeleri H¨ugel, F. Marks, and J. Vit´oria, A characterization of τ-tilting finite algebras, available +via arXiv:1801.04312 [math.RT]. +[HZ] +R. Hafezi and Y. Zhang, Stable Auslander-Reiten components of monomorphism categories, available +via arXiv:2204.11705 [Math.RT]. +[INY] O. Iyama, H. Nakaoka and Yann Palu, Auslander-Reiten theory in extriangulated categories, available +via arXiv:1805.03776 [Math.RT]. +[IPTZ] K. Igusa, M.-I. Platzeck, G. Todorov and D. Zacharia, Auslander algebras of finite representation +type, Comm. Algebra 15 (1987), no. 1-2, 377-424. +[J] +P. Jiao, The generalized Auslander-Reiten duality on an exact category, J. Algebra Appl. 17 (2018), no. +12, 1850227. +[K] +B. Keller, Chain complexes and stable categories, Manuscripta Math. 67 (1990), 379-417. +[L] +S. Liu, Auslander-Reiten theory in a Krull-Schmidt category, S˜ao Paulo J. Math. Sci. 4 (2010), no. 3, +425-472. +[LZ] +X-H. Luo and P. Zhang, Separated monic representations I: Gorenstein-projective modules, J. Algebra +479 (2017), 1-34. +[MO] +R. Mart´ınez-Villa and M. Ortiz-Morales, Tilting theory and functor categories III. The maps category, +Int. J. Algebra 5 (2011), no. 9-12, 529-561. +[MS] +F. Marks and J. ˘S´tov´ı˘cek, Torsion classes, wide subcategories and localisations, Bull. London Math. +Soc. 49 (2017), 405-416. +[Q] +D.Quillen, Higher algebraic K-theory. I, in: Algebraic K-theory, I: Higher K-theories, Proceedings of the +Conference, Battelle Memorial Institute, Seattle, Washington,1972, Lecture Notes in Mathematics, 341, +Springer, Berlin, (1973), 85-147. +[RS] +C. M. Ringel and M. Schmidmeier, The Auslander-Reiten translation in submodule categories, Trans. +Amer. Math. Soc. 360 (2008), no. 2, 691-716. + +28 +HOSSEIN ESHRAGHI AND RASOOL HAFEZI +[RZ] +C. M. Ringel and P. Zhang, From submodule categories to preprojective algebras, Math. Z. 278 (2014), +no. 1-2, 55-73. +[SY] +A. Skowro´nski and K. Yamagata, Frobenius algebras. I. Basic representation theory. EMS Textbooks in +Mathematics. European Mathematical Society (EMS), Z¨urich, 2011. xii+650 pp. ISBN: 978-3-03719-102-6. +[XZZ] B-L. Xiong, P. Zhang and Y-H. Zhang, Auslander-Reiten translations in monomorphism categories, +Forum Math. 26 (2014), no. 3, 863-912. +[Z] +P. Zhang, Monomorphism categories, cotilting theory, and Gorenstein projective modules, J. Algebra 339 +(2011), 181-202. +[ZX] +P. Zhang and B-L. Xiong, Separated monic representations II: Frobenius subcategories and RSS equiv- +alences, Trans. Amer. Math. Soc. 372 (2019), 981-1021. +Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, PO +Box 87317-51167, Kashan, Iran +Email address: eshraghi@kashanu.ac.ir +School of Mathematics and Statistics, Nanjing University of Information Science & Technology, +Nanjing, Jiangsu 210044, P. R. China +Email address: hafezi@nuist.edu.cn + diff --git a/79AyT4oBgHgl3EQfp_gy/content/tmp_files/load_file.txt b/79AyT4oBgHgl3EQfp_gy/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..92b6613a08d621eb04a2bb9b3603920357a9529f --- /dev/null +++ b/79AyT4oBgHgl3EQfp_gy/content/tmp_files/load_file.txt @@ -0,0 +1,1054 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf,len=1053 +page_content='FROM MORPHISM CATEGORIES TO FUNCTOR CATEGORIES HOSSEIN ESHRAGHI AND RASOOL HAFEZI Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' For a nice-enough category C, we construct both the morphism category H(C) of C and the category mod-C of all finitely presented contravariant additive functors over C with values in Abelian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' The main theme of this paper, is to translate some representation- theoretic attributes back and forth from one category to the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' This process is done by using an appropriate functor between these two categories, an approach which seems quite promising in particular when we show that many of almost split sequences are preserved by this functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' We apply our results to the case of wide subcategories of module categories to obtain certain auto-equivalences over them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Another part of the paper deals with Auslander algebras arising from algebras of finite representation type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' In fact, we apply our results to study the Auslander-Reiten translates of simple modules over such algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' In the last parts, we try to recognize particular components in the stable Auslander-Reiten quiver of Auslander algebras arising from self-injective algebras of finite representation type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Introduction As a popular belief, it is said that the introduction of the language of functor categories to the study of categories of modules over rings dates back to Auslander and his colleagues’ works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' These works trace back mainly to the papers [A65, A71, A76, AR74, AR78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' In particular Auslander’s Formulae [A65] that suggests to recover the category mod-Λ of finitely generated modules over an Artin algebra Λ as the quotient mod-Λ ≃ mod-(mod-Λ) {F : F(Λ) = 0} deserves attention;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' here and throughout, mod-(mod-Λ) denotes the category of additive con- travariant coherent functors on mod-Λ with values in Ab, the category of Abelian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' While talking about the exchange between two categories consisting objects that are apparently of different types, one expects to encounter with functors transferring from one category to the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Concerning the morphism categories and the functor categories, such a study has initi- ated probably in [A71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Roughly, the general theme of the current paper is to figure out how some representation-theoretic attributes transfer between functor and morphism categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' However, to be more precise, we prefer to provide a layout of the paper section by section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Prior to this, we want to point out that the morphism category of Λ has on its own right been systematically studied from various aspects: deriving its Auslander-Reiten theory in the language of AR-theory of Λ [RS, XZZ, E, HE], establishing its links to Gorenstein homological algebra [Z, LZ, ZX], and looking at a particular subcategory of it, namely the monomorphism category, in order to study the so-called Auslander algebras [AR76, HM].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' 18A25, 16G70, 16G10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Functor Category, Morphism Category, Auslander-Reiten Components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='00534v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='RT] 2 Jan 2023 2 HOSSEIN ESHRAGHI AND RASOOL HAFEZI First of all, to keep the results as general as possible, we try to deal with the morphism category H(C) of a nice-enough category C (definitions are recalled later on).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Namely, if we assume that C is an idempotent-complete additive category that admits pseudokernels then, in Section 3, we endow H(C) with an exact structure defined by degree-wise split exact sequences in C, denoted Hcw(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Even though such constructions have been considered in some particular cases, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' in [Ba] where the category of morphisms between projective modules over an Artin algebra have come to play, we do it in a most general possible circumstance as declared above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' The motivation behind such considerations comes from two origins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Firstly, we look for a reasonable structure on H(C) with respect to which one may define almost split sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Note, secondly, that if one imposes tougher conditions on C, for instance taking C to be an extension-closed subcategory of mod-Λ, then H(C) inherits an exact structure as an extension-closed subcategory of the morphism category of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' So now a natural question arises: What are intrinsic similarities between these two exact structures on H(C)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' To get more involved with the aforementioned question, we need to take a glance at the contents of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' For, we recall form [A71] that there exists a functor Θ : H(C) −→ mod-C, where mod-C is the category of contravariant additive coherent functors on C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' The objective in Section 4 is to study Θ form the point of view of Auslander-Reiten theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' We show that Θ induces an equivalence H(C)/ � (M → 0), (M 1→ M) � ≃ mod-C where M runs through the objects of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Using this, we show that Hcw(C) admits almost split sequences whenever C is assumed to be a dualizing variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Furthermore, to conquer the question posed above, it is shown that if C is an extension-closed dualizing subvariety of mod-Λ then, in many cases, the almost split sequences in Hcw(C) and H(C) coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Not going off-topic, one more thing will be proved: Θ respects almost split sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' In Section 5, we turn to apply some of the results to the case of wide subcategories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' To illuminate the role and importance of wide subcategories of mod-Λ, we must point out that such subcategories arise naturally in the study of τ-tiling theory of Λ [AIR] and in connection with determination of certain torsion classes in mod-Λ [MS].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' These also play significant role in the study of certain classes of universal localizations over Λ [MS, HMV1, HMV2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Such classes of modules also appear in classification problems for the so-called τ-tilting finite algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Among other things, for a given functorially finite wide subcategory X of mod-Λ we construct, based on our previous results, an auto-equivalence σX : X → X which fulfills the exact sequence 0 → (−, σX τX (X)) → D(P, −) → D(Q, −) → D(X, −) in mod-X for every indecomposable module X which is not projective in X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' here τX denotes the Auslander-Reiten translation of X and P → Q → X → 0 is the minimal projective presentation of X with respect to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' In this regard, recall that for a non-projective Λ-module M with minimal projective presentation P → Q → M → 0, there exists an exact sequence 0 → τ(M) → ν(P) → ν(Q) → ν(M) → 0 where τ and ν stand respectively for the Auslander-Reiten translation and the Nakayama functor over mod-Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Hence the aforementioned exact sequence of functors resembles, and generalizes, the latter one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' This is more clarified by showing that when X is the whole category mod-Λ, then σX is nothing but the identity functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' We believe that this observation is convincing-enough to say that the rich treasury behind functorially finite wide subcategories of mod-Λ might be discovered by applying some instruments from functor categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' In Section 6, we switch to algebras Λ of finite representation type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' The main impetus for such a study comes from the fact that in this case, one may construct the Auslander algebra A of Λ which is, by definition, the endomorphism algebra of a representation-generator M of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Then FROM MORPHISM CATEGORIES TO FUNCTOR CATEGORIES 3 there is a nice interpretation of the category mod-A in terms of the functor category;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' namely, there is a categorical equivalence mod-A ≃ mod-(mod-Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' In the meanwhile, it is known [A76] that simple functors over mod-Λ correspond bijectively to indecomposable Λ-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Hence this categorical equivalence provides a framework in which one tries to understand in more details the simple modules over A and its projectively stable version A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' The results presented in this section come up by analyzing certain almost split sequences mainly provided in [HE] and also in [HZ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' The main results discover a relation between the (inverse) Auslander-Reiten translation of simple A-(resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' A-) modules and the cosyzygies (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' syzygies) of simple A-(resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' A-) modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' The last section is devoted to study certain components in the (stable) Auslander-Reiten quiver ΓA of the Auslander algebra A whenever Λ is self-injective of finite representation type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Note that recognition of such components have already been the subject of some earlier researches [IPTZ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' To this end, we firstly deal with τH-periodic objects by invoking some almost split sequences already obtained in [HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' In this direction, it turns out that the auto-equivalence A = ντ 3 of the stable category mod-Λ, as defined in [HE], plays a significant role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' In fact, we show that the existence of certain A -periodic Λ- modules makes ΓA into a finite oriented cycle, and in particular, makes A into an algebra of finite representation type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Another result asserts that for Λ self-injective of finite representation type, any component Ξ of the stable Auslander-Reiten quiver of A that contains a certain simple module is either infinite or is of the form Z∆/G for a Dynkin quiver ∆ and an automorphism group G of Z∆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' this is based on a structural theorem due to Liu [L].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' preliminaries and notation In this section, we collect very briefly some necessary background material of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' When required, explicit references are provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Functor Categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Let k be a commutative Artinian ring and let C be a k-linear Krull- Schmidt category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' A C-module is a contravariant additive functor from C to the category Ab of Abelian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' We denote by Mod-C the category of all C-modules, and by mod-C the full subcategory of Mod-C consisting of finitely presented modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Recall from [A65] that a C-module M is called finitely presented if there exists an exact sequence HomC(−, A) → HomC(−, B) → M → 0 in Mod-C, for some objects A, B of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Moreover, proj-C and inj-C denote the full subcategories of mod-C consisting of projective and injective objects in mod-C, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' The category mod-C is an abelian category if and only if C admits pseudokernels;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' see page 315 of [AR74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' We shall sometimes write (−, X) instead of the representable functor HomC(−, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Dualizing k-varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Let r be the radical of k and E(k/r) be the injective envelope of the k-module k/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' A Hom-finite k-linear Krull-Schmidt category C is called a dualizing k-variety [AR74] if the k-dual functors D : Mod-C → Mod-(Cop) and D : Mod-(Cop) → Mod-C given by D(F)(C) = Homk(F(C), E(k/r)) for every object C of C and F ∈ Mod-(C) or Mod-(Cop) induce dualities D : mod-C → mod-(Cop) and D : mod-(Cop) → mod-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' In this case, it turns out that mod-C is an abelian subcategory of Mod-C that admits enough pro- jective and enough injective objects [AR74, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' As an example, proj-Λ, the category of finitely generated projective modules over an Artin k-algebra Λ, is a dualizing k-variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' We 4 HOSSEIN ESHRAGHI AND RASOOL HAFEZI note from [AR74, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='6] that if C is a dualizing k-variety then so is mod-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Further- more, any functorially finite subcategory of a dualizing k-variety is itself a dualizing k-variety by [AS81, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Morphism Categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Let C be a category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' The morphism category H(C) of C is a category whose objects are morphisms f : X → Y in C, and whose morphisms are given by commutative diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' If we regard the morphism f : X → Y as an object in H(C), we will usually present it as (X f→ Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' However, due to typographical considerations, we have to use also the vertical notation ( X Y )f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' A morphism between the objects (X f→ Y ) and (X′ f ′ → Y ′) is presented as (σ1, σ2) : (X f→ Y ) → (X′ f ′ → Y ′) or, ( σ1 σ2 ) : ( X Y )f → � X′ Y ′ � f ′, where σ1 : X → X′ and σ2 : Y → Y ′ are morphisms in C with σ2f = f ′σ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Adapting the notation, the morphism category raised from C = mod-Λ, the category of finitely generated right modules over an Artin k-algebra Λ, will be denoted simply by H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' this will cause no ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' The same rule also applies to the monomorphism category S of Λ whose objects are just monic Λ-maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Auslander-Reiten-Serre Duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Let (C, E) be an exact category in the sense of Quillen [Q, K] (see next section for an introduction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Recall that a morphism v: E → Y in C is called right almost split if it is not a retraction and each f : Z → Y which is not a retraction factors through v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Dually, a morphism u: X → E in C is called left almost split if it is not a section and each f : X → Z which is not a section factors through u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' An admissible sequence δ: 0 → X u−→ E v−→ Y → 0 in E is an almost split sequence if u is left almost split and v is right almost split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Since δ determines X and Z in a unique way, we call X the Auslander-Reiten translation X = τC(Y ) of Y in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' A non-zero object X ∈ C is said to be endo-local if its endomorphism ring EndC(X) is local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Following [INY, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='1], we say that C has almost split sequences if endo-local non projective objects of C and endo-local non-injective objects of C are respectively the terminal and the initial terms of some almost split sequence in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Assume now that C is further a k-linear category and let D be the k-dual functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Put C and C denote respectively the projectively and the injectively stable categories of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' An Auslander- Reiten-Serre duality (ARS duality, in brief) is a pair (τC, η) consisting of an equivalence functor τC : C → C together with a bi-natural isomorphism ηX,Y : HomC(X, Y ) ≃ DExt1 C(Y, τC(X)) for any X, Y ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' The following lemma, taken from [INY, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='6] (see also [J]), provides a close connection between the existence of almost split sequences in C and the existence of an ARS-duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Let us recall that under the above hypothesis, C is Ext-finite if the k-modules Ext1 C(X, Y ) are finitely generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Let C be a k-linear Ext-finite Krull-Schmidt exact category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Then the following conditions are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' (1) C has almost split sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' (2) C has an Auslander-Reiten-Serre duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' (3) The stable category C is a dualizing k-variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' (4) The stable category C is a dualizing k-variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Throughout the paper, Λ will stand for a fixed Artin k-algebra and modules are, by default, finitely generated right modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' The Auslander-Reiten translation, the Nakayama functor, FROM MORPHISM CATEGORIES TO FUNCTOR CATEGORIES 5 the syzygy and the cosyzygy functor of Λ are respectively denoted by τ, ν, Ω, and Ω−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' If we deal with an algebra other than Λ or with a category, these functors will be accompanied with necessary subscripts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' The symbols Ker, Coker, and Im, used freely in all contexts, stand respectively for the kernel, cokernel, and the image of morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Exact structures on the morphism category An exact category (C, E) is formed by an additive category C, and a class E of composable pairs of morphisms in C (also called kernel-cokernel pairs) satisfying certain axioms that we refrain to exhibit here and refer the reader e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' to [K].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' The composable pair (i, p) in E is usually denoted by 0 → A′ i→ A p→ A′′ → 0, where i : A′ → A and p : A → A′′ are respectively called an E-admissible monic and an E-admissible epic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Composable pairs, admissible monics and admissible epics are sometimes referred to respectively as conflations, inflations and deflations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' The notion of an exact category was first introduced by Quillen in [Q] and then Keller [K] proved the redundancy of some axioms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Let C be an additive category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' In this section, we shall put an exact structure on the morphism category H(C) of C [Ba].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' For let Ecw be the class of all pairs of composable morphisms δ : � X1 X2 � f � φ1 φ2 � �� Z1 Z2 � h � ψ1 ψ2 � �� Y1 Y2 � g such that the induced composable morphisms Xi φi → Zi ψi → Yi split in C for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' It can be easily seen that any pair of composable morphisms in Ecw is isomorphic to a pair of composable morphisms of the form δ′ : � X1 X2 � f �[ 1 0] [ 1 0] � �� X1⊕Y1 X2⊕Y2 � h � [0 1] [0 1] � �� Y1 Y2 � g where h = � f q 0 g � and q : Y1 → X2 is a possibly non-zero morphism in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Regarding this easy observation, without loss of generality, we usually take all kernel-cokernel pairs in H(C) to be of this form;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' this is justified by the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Any object in Ecw is a kernel-cokernel pair in H(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Take the element δ′ of Ecw and assume that the composite of the morphisms (σ1, σ2) : (X1 ⊕ Y1 h→ X2 ⊕ Y2) → (V s→ W) and ([ 1 0 ], [ 1 0 ]) vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' This means that the restriction of σi on Xi, for i = 1, 2, is the zero map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' This enables us to define the morphisms σ1|Y1 and σ2|Y2 and it readily follows that (σ1, σ2) factors uniquely over ([0 1], [0 1]) via the morphism (σ1|Y1, σ2|Y2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' The remaining axioms are verified similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' □ Recall that an additive category D is called idempotent-complete if every idempotent endo- morphism in D admits a kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Assume C is idempotent-complete and admits pseudokernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Then Ecw de- fines an exact structure on the additive category H(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Since C is idempotent-complete, it is known that the Yoneda functor gives an equivalence C ≃ proj-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' This equivalence is naturally extended to an equivalence between corresponding morphism categories;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=', H(C) ≃ H(proj-C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' One observes that, under this equivalence, the 6 HOSSEIN ESHRAGHI AND RASOOL HAFEZI kernel-cokernel pairs in Ecw provided by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='1 correspond bijectively to the short exact se- quences in the abelian category H(mod-C) whose terms lie inside H(proj-C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' But the subcategory H(proj-C) is closed under extensions and inherits an exact structure from H(mod-C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' To make our arguments work, we had to impose some restrictions on the additive category C to get a suitable exact structure out of H(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' However, it may be the case that the aforementioned set of requirements is not minimal in the sense that the above family of kernel-cokernel pairs may equip H(C) with an exact structure even if some of the hypothesis in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='2 are dropped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' From now on we assume that C is idempotent-complete and admits pseudokernels, and the symbol Hcw(C) stands for the exact category (H(C), Ecw), sometimes also called the cw-exact category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' The following proposition is recorded for future use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Suppose (X1 f→ X2) is an object in Hcw(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' (1) f defines an indecomposable projective object in Hcw(C) if and only if it is isomorphic either to (X 1→ X) or (0 → X) for some indecomposable object X in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' (2) f defines an indecomposable injective object in Hcw(C) if and only if it is isomorphic either to (X 1→ X) or (X → 0) for some indecomposable object X in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Furthermore, Hcw(C) has enough projectives and enough injectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' This should be compared to [Ba, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' We just remark that the last claim follows from the short exact sequences 0 �� 0 X1 � 0 � 0 � f −1 � � �� 0 X2 � 0 ⊕ � X1 X1 � 1 � 1 [ 1 f ] � �� X1 X2 � f �0 and 0 �� X1 X2 � f � � f 1 � 1 � �� X2 X2 � 1 ⊕ � X1 0 � 0 � [ −1 f ] 0 � �� X2 0 � �0 in Ecw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' □ Now assume C is an extension-closed subcategory of mod-Λ for an Artin algebra Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' We may consider C as an exact category through the structure induced by the abelian category mod-Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Then also H(C), as an extension-closed subcategory of the abelian category H is endowed with the canonical exact structure inherited from H, still denoted by H(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' We also keep the cw- exact structure Hcw(C) defined by degree-wise split sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' It will be indicated in the next section that if C is a k-dualizing variety, then Hcw(C) admits almost split sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Further, the canonical exact category H(C) admits almost split sequences provided C is a k-dualizing subvariety of mod-Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' It also becomes clear how the canonical exact category H(C) inherits almost split sequence from Hcw(C) in the latter case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' However, for technical reasons, we have to defer the proofs until next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Suppose for a moment that C is further functorially finite in mod-Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' In this case, another approach one may take to show that H(C) has almost split sequences is to explore when H(C) is functorially finite in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' This seems natural in view of the fact that, by [AS81, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='4], any functorially finite extension-closed subcategory of H admits almost split sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' FROM MORPHISM CATEGORIES TO FUNCTOR CATEGORIES 7 Restricting to the case where C = mod-Λ, in the last part of this section, we put a third exact structure on H that will turn out in Section 7 to be in connection with the stable Auslander- Reiten quiver of Auslander algebras;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' see Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' An indecomposable object in H is said to be of type (a) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' (b), or (c)) provided it is isomorphic to (0 → M) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' (M 1→ M), or (M → 0)) for some Λ-module M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Further, an indecomposable object is said to be of type (d) if it is isomorphic to (P f→ Q) where P, Q are projective Λ-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Let X be the smallest subcategory of H containing all objects of types (a), (b), (c) and (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Let also EX be the class of all short exact sequences 0 → X → Y → Z → 0 in H such that the induced sequence 0 → HomH(V, X) → HomH(V, Y) → HomH(V, Z) → 0 is exact for every V ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' We know from [AS93] and [Bu] that EX induces an exact structure on H denoted by HX = (H, EX ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' One infers from [AS93, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='12] that the exact category HX has enough projectives and enough injectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Denote by P(HX ) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' I(HX )) the subcategory of projective (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' injective) objects in HX .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' In view of [AS93, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='6 and Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='10], we have P(HX ) = X ∪ proj-H and I(HX ) = τH(X) ∪ inj-H, where proj-H and inj-H stand respectively for the subcategories of projective and injective objects in H and τH is the Auslander-Reiten translation of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' We exploit [AS93, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='9] to examine the almost split sequences in HX ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' it turns out that an almost split sequence 0 → X → Y → Z → 0 in H is an almost split sequence in HX if and only if neither X ∈ I(HX ) nor Z ∈ P(HX ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Interplay between morphism and functor categories Until further notice, we assume throughout the section that C is a dualizing k-variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' In this section, we will be involved with a functor going from morphism category to the functor category, originally defined and studied in [A71] and then reconsidered in [HM].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' This functor is our main tool to exchange between these two categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' The construction is based on the Yoneda functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Construction 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Let (X1 f→ X2) be an object of H(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Define (X1 f→ X2) Θ �→ Coker(C(−, X1) C(−,f) −→ C(−, X2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' If h = � h1 h2 � : X = � X1 X2 � f → � X′ 1 X′ 2 � f ′ = X′ is a morphism in H(Λ), then we let Θ(h) be the unique morphism σ that makes the following diagram commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' HomC(−, X1) HomC(−,f) � HomC(−,h1) � HomC(−, X2) � HomC(−,h2) � Θ(X) � σ � 0 HomC(−, X′ 1) HomC(−,f ′) � HomC(−, X′ 2) � Θ(X′) � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' It is routine to verify that this rules introduce a well-defined functor Θ : H(C) → mod-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' The purpose of this section is to study this functor from the perspective of almost split sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' It turns out that Θ behaves well over such sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Firstly, we need to recall some facts on objective functors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' more details are provided in the Appendix of [RZ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Let F : C −→ D be an additive functor between additive categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' F is called an objective functor if any morphism f in C with F(f) = 0 factors through an object K of C with F(K) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' such a K is then called a kernel object of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' We say that the kernel of a functor F is generated by a subcategory X of C if add-X, the additive closure of X in C, is the class of all kernel objects of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' 8 HOSSEIN ESHRAGHI AND RASOOL HAFEZI Let F : C −→ D be a full, dense and objective functor and let the kernel of F be generated by X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Then F induces an equivalence F : C/X −→ D where the additive quotient category C/X of C with respect to X has the same objects as C and the morphisms are defined via the rule C/X(X, Y ) := C(X, Y )/{φ | φ factors through an object in add-X} for any pair of objects X, Y of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' The functor Θ : H(C) −→ mod-C is full, dense and objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Thus, there exists an equivalence Θ of categories that makes the following diagram commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' H(C) Θ � π � mod-C H(C) V Θ � Here, π is the natural quotient map and V is the full subcategory of H(C) generated by all finite direct sums of objects of type (b) or (c), that is to say, objects of the form (M 1 −→ M) and (M −→ 0), where M runs through the objects of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Θ is dense;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' for take F ∈ mod-C with a projective presentation (−, X) (−,g) → (−, Y ) → F → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' It is plain that Θ(X g→ Y ) = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' To see the fullness of Θ, take two objects (X g→ Y ) and (X′ g′ → Y ′) of H(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' As the representable functors (−, Y ) and (−, Y ′) are projective, it follows that any morphism σ : F = Θ(X g→ Y ) → Θ(X′ g′ → Y ′) = F ′ in mod-C might be lifted to a map from the augmented projective presentation (−, X) (−,g) → (−, Y ) → F → 0 to (−, X′) (−,g′) → (−, Y ′) → F ′ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Then using Yoneda’s Lemma and the aforementioned construction, one obtains a morphism h : (X g→ Y ) → (X′ g′ → Y ′) in H(C) with σ = Θ(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Now assume Θ(X g→ Y ) = 0, for some object (X g→ Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Then by the construction, we have the exact sequence 0 → (−, X) (−,g) −→ (−, Y ) → 0 in mod-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' One then observes that the identity map 1 : Y → Y factors over g via, say, h : Y → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Therefore, X = Im(h) ⊕ Ker(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' This leads to the decomposition (X g→ Y ) = (Ker(g) → 0) ⊕ (Im(h) g| → Y ) where g| is the restricted map which must be an isomorphism since gh = 1Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' This settles that the kernel of Θ is generated by V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Finally, suppose Θ(h) = 0, for h = (h, h′) : (X g→ Y ) → (X′ g′ → Y ′) in H(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Setting F = Θ(X g→ Y ) and F ′ = Θ(X′ g′ → Y ′), this induces a chain map between complexes of functors · · � (−, Z0) (−,α0) � � (−, X) (−,h) � (−,g) � (−, Y ) (−,h′) � � F 0 � � 0 · · � (−, Z′ 0) � (−, X′) (−,g′)� (−, Y ′) � F ′ � 0 raised by taking projective presentations of F and F ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Evidently, this chain map is null- homotopic and, according to [G, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='5], factors through a contractible complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' As FROM MORPHISM CATEGORIES TO FUNCTOR CATEGORIES 9 any contractible complex of functors might be imagined to be a direct sum of complexes of the form · · → 0 → (−, W) 1→ (−, W) → 0 → · · · for various objects W of C, this induces a commutative diagram · · � (−, Z0) (−,α0) � � � (−, X) (−,h) � � (−,g) � (−, Y ) (−,h′) � � � 0 · · � (−, Z0 ⊕ X′) � � (−, X′ ⊕ Y ′) � � (−, Y ′) � � 0 · · � (−, Z′ 0) � (−, X′) (−,g′) � (−, Y ′) � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Therefore, there exists a factorization of h through the object (X′ → 0) ⊕ (Y ′ 1→ Y ′), which is a kernel object according to the above paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' This shows that Θ is an objective functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Now the existence of the equivalence Θ comes up from observations prior to the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' □ Let us record here that applying a dual construction to the opposite category Cop results in a contravariant functor Θ′ : H(C) → mod-Cop, (X f→ Y ) �→ Coker(C(Y, −) C(f,−) −→ C(X, −)) which is seen to induce a duality Θ′ that makes the diagram H(C) Θ′ � π′ � mod-Cop H(C) V′ Θ′ � commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Here, V′ is the full subcategory of H(C) generated by all finite direct sums of objects of type (a) or (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Consider the morphism category H(C), endowed with the exact structure given by Ecw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Ac- cording to Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='4, V (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' V′) is nothing but the subcategory of injective (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' pro- jective) objects of the exact category Hcw(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Consequently, the factor categories H(C)/V′ and H(C)/V are equivalent respectively to the projectively and injectively stable categories Hcw(C) and Hcw(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Hence the following proposition emerges to settle that Hcw(C) admits almost split sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' For a dualizing k-variety C, the following statements hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' (1) Hcw(C) admits almost split sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' (2) Hcw(C) has an Auslander-Reiten-Serre duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' The above observations along with Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='2 provide the equivalences H(C)/V ≃ mod-C ≃ Hcw(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Note that by [AR74, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='6], mod-C is a dualizing k-variety as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Hence Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='1 completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' □ We are now in a position to prove the assertion in previous section concerning the existence of almost split sequences in H(C) where C is an extension-closed dualizing subvariety of mod-Λ for an Artin algebra Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' The following lemma which is taken from [MO, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='1] will be 10 HOSSEIN ESHRAGHI AND RASOOL HAFEZI fruitful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' While the special case C = mod-Λ has been dealt with in [MO], the same proof still works for general C as we consider here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Assume δ : 0 → A f→ B g→ C → 0 is an almost split sequence in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Then (1) The almost split sequence in H(C) ending at (0 → C) is of the form 0 �( A A )1 � 1 f � �( A B )f � 0 g � �( 0 C )0 �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' (2) The almost split sequence in H(C) ending at (C 1→ C) is of the form 0 �( A 0 )0 � f 0 � �( B C )g ( g 1) �( C C )1 �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Let C be an extension-closed k-dualizing subvariety of mod-Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Then the canonical exact category H(C) admits almost split sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Let Z be an indecomposable non-projective object in H(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Assume first that Z is of either types (0 → X) or (X 1→ X), for an object X ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Then since C admits almost split sequences by [AS81, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='1], from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='4 we infer that Z is the end term of an almost split sequence in H(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Otherwise, Z is not projective in the exact category Hcw(C) by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Hence, by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='3, there exists an almost split sequence ending at Z in the exact category Hcw(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' However, following the definitions, it is easy to verify that this is an almost split sequence in H(C) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' □ The following corollary is an immediate consequence of the arguments above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Let C be an extension-closed k-dualizing subvariety of mod-Λ and let 0 �� X1 X2 � f � φ1 φ2 � �� Z1 Z2 � h � ψ1 ψ2 � �� Y1 Y2 � g �0 be an almost split sequence in H(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Then the sequences 0 → Xi φi → Zi Ψi → Yi → 0, i = 1, 2, split provided that either of the following situations occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' (1) The terminal term (Y1 g→ Y2) is not of type (a) or (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' (2) The initial term (X1 f→ X2) is not of type (b) or (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' We now turn to show that Θ respects almost split sequences;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' so we return to the setting that C is a dualizing k-variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Let Y = (Y1 g→ Y2) be an indecomposable non-projective object in H(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Take δ : � X1 X2 � f � φ1 φ2 � �� Z1 Z2 � h � ψ1 ψ2 � �� Y1 Y2 � g to be the almost split sequence in Hcw(C) ending at Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' For simplicity, set Z = (Z1 h→ Z2), X = (X1 f→ X2), φ = (φ1, φ2) and ψ = (ψ1, ψ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Note that δ induces degree-wise split sequences FROM MORPHISM CATEGORIES TO FUNCTOR CATEGORIES 11 and, applying Θ, one gets the commutative diagram with exact rows 0 � 0 � 0 � 0 � K1 � � K2 i � η � K3 λ � 0 � (−, X1) (−,f) � � (−, Z1) (−,h) � (−,ψ1)� (−, Y1) (−,g) � � 0 0 � (−, X2) � � (−, Z2) � (−,ψ2)� (−, Y2) � � 0 Θ(X) Θ(φ) � � Θ(Z) Θ(ψ) � � Θ(Y) � � 0 0 0 0 in mod-C whose bottom row is indeed the image Θ(δ) of δ under the functor Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' The map η in the above diagram is an epimorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' As an upshot, Θ(δ) is a short exact sequence in mod-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Let (−, P) σ→ K3 → 0 be an epimorphism in mod-C and let d : P → Y1 be a morphism in C which represents the composite λσ : (−, P) → K3 → (−, Y1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Since gd = 0, Yoneda’s lemma induces a morphism (d, 0) : (P → 0) → Y which is plainly not a retraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Hence it must factor over the right almost split map (ψ1, ψ2) via, say, (a, 0) for some a : P → Z1 in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Consequently, the map (−, a) in mod-C satisfies (−, h)(−, a) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Adding that (−, P) is a projective functor, this gives a map γ : (−, P) → K2 in mod-C with (−, a) = iγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Note that ληγ = (−, ψ1)iγ = (−, ψ1)(−, a) = λσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' But λ is a monomorphism;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' thus ηγ = σ whence the surjectivity of η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' The second claim comes up immediately from the Snake Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' □ The following theorem is another main result of the section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Under the above notation, Θ(δ) is an almost split sequence in mod-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' The indecomposability of X and Y imply that Θ(X) and Θ(Y) are indecomposable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' By previous lemma, Θ(δ) is an exact sequence that, moreover, does not split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Indeed, if it did, then (−, f) ⊕ (−, g) would be a minimal projective presentation for Θ(Z) which should comply with the one provided by the middle column of the above diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' In view of the form of kernel elements of the functor Θ declared by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='2, an application of Yoneda’s lemma gives that, for some objects A, B of C, there should exist an isomorphism (Z1 h→ Z2) = (X1 f→ X2) ⊕ (Y1 g→ Y2) ⊕ (A 1→ A) ⊕ (B → 0) of objects in H(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' As stated before, we may assume Zi ≃ Xi ⊕ Yi, i = 1, 2 since δ belongs to Ecw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' However, C being a Krull-Schmidt category implies A = B = 0 which makes δ split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' This contradiction shows that Θ(δ) does not split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' 12 HOSSEIN ESHRAGHI AND RASOOL HAFEZI Now, as [AR74, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='4] guarantees that mod-C is abelian in this case, invoking [AR77, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='14], it suffices to show that Θ(ψ) is right almost split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' For let q : F → Θ(Y) be a non-retraction in mod-C and take a projective presentation (−, W1) (−,d) → (−, W2) → F → 0 of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Note that by definition, Θ(W1 d→ W2) = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' The morphism q lifts to a morphism between the projective presentations (−, W1) (−,d) → (−, W2) → F → 0 and (−, Y1) (−,g) → (−, Y2) → Θ(Y ) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' The lifted morphism induces, again by the Yoneda lemma, a map ( σ1 σ2 ) : � W1 W2 � d → � Y1 Y2 � g in H(C) such that Θ(σ1, σ2) = q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Then (σ1, σ2) is not a retraction since otherwise q would be so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Now, δ being an almost split sequence in Hcw(C), (σ1, σ2) factors over ψ via some (η1, η2) : (W1 d→ W2) → (Z1 h→ Z2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Then applying Θ, we see that the morphism q factors over Θ(ψ) via Θ(η1, η2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' The case of wide subcategories Our objective in this section is to study the morphism categories raised by functorially finite wide subcategories of mod-Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Some results from previous section will come to play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' After- wards, we shall switch to functor categories and obtain some results in this direction that extend others from the module category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' So let firstly X be a functorially finite idempotent-complete subcategory of mod-Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' By [AS81, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='3], X itself is a dualizing variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Following Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='3, for a dualizing k-variety C, there is an equivalence τH(C) : Hcw(C) → Hcw(C) that, based on what we said in previous section, might be considered as an equivalence from H(C)/V′ to H(C)/V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Pictorially, there exists a composition of equivalences H(C)/V τ −1 H(C) � H(C)/V′ Θ′ � mod-C (Θ)−1 � � mod-Cop D � mod-C denoted throughout by ∆C, or simply by ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Applied to the functorially finite subcategory X of mod-Λ, this yields an equivalence ∆X : mod-X → mod-X which is restricted to the category proj-X of projective functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Since X is idempotent-complete, the Yoneda functor induces an equivalence proj-X ≃ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Summing up, one obtains an equivalence σX : X → X which agrees with the restricted equivalence ∆X via the latter identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' We notice that, going through the definitions, one figures out that for an object X of X, there are A, B ∈ X and an exact sequence (B, −) (f,−) → (A, −) → D(−, σX (X)) → 0 in mod-X such that τ −1 H(X)(0 → X) = (A f→ B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' A minimal projective presentation of an object C ∈ X with respect to X is an exact sequence P1 f→ P0 h→ C with P1, P0 ∈ P(X), the class of projective objects of X, and is computed by taking minimal right P(X)-approximations consecutively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Minimal injective presentations with respect to X are defined dually via I(X), the class of injective objects of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Recall that a subcategory M of mod-Λ is said to be closed under kernels (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' cokernels, images) if for every morphism X f→ Y in M also Ker(f) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Coker(f), Im(f)) belongs to FROM MORPHISM CATEGORIES TO FUNCTOR CATEGORIES 13 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Further, M is called a wide subcategory of mod-Λ if it is closed under extensions, kernels and cokernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' It is clear that a wide subcategory is closed under images and is automatically idempotent-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' The following couple of propositions are quite useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Assume X is a functorially finite wide subcategory of mod-Λ and δ : 0 → A f→ B g→ C → 0 is an almost split sequence in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Let also A d→ I0 q→ I1 be a minimal injective presentation with respect to X, where b : Coker(d) → I1 is a minimal left I(X)-approximation, a : I0 → Coker(d) is the canonical quotient map and q = ba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Then the exact sequence 0 �� I0 I1 � q ( u 1 ) �� W I1 � br ( v 0 ) �( C 0 )0 �0 in H(X) raised by forming the push out diagram A d � f � B h � g � C I0 a � u � W r � v � C Coker(d) Coker(d) in the exact category X, is almost split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Using that A d→ I0 s→ I1 is a minimal injective presentation and A is indecomposable, we deduce that (I0 s→ I1) is indecomposable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Hence it suffices to show that any non-retraction (φ, 0) : (M p→ N) → (C → 0) in X factors over (v, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' If φ is a non-retraction, then, since δ is an almost split sequence, it factors in X over g via, say, w : M → B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Then it is easy that the morphism (hw, 0) : (M p→ N) → (W br → I1) factors the morphism (φ, 0) over (v, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' So now take φ to be a retraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Without loss of generality, we may assume M = C and φ = 1C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Two cases might be distinguished: Case 1: p is a monomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Since v is a retraction in X, there exists s : C → W such that vs = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' As p : C → N is a monomorphism in X, there exists an extension of brs : C → I1 to a map z : N → I1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' that is to say, zp = brs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' It follows then that (s, z) : (C p→ N) → (W br → I1) produces the desired factorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Case 2: Assume Ker(p) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Note that since X is a wide subcategory, Ker(p) lies in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' The fact that (φ, 0) is a non-retraction implies that Ker(p) is a proper submodule of C and thus the canonical inclusion i : Ker(p) → C is a non-retraction in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' According to the hypothesis, we infer the existence of a map y : Ker(p) → B such that gy = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Note further that, since v is retraction, one may write W = Im(s)⊕Ker(v) and, consequently, present h as h = [l1, l2]t, where l1 : B → Im(s) and l2 : B → Ker(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Using the injectivity of Ker(v) in X yields an extension of l2y : Ker(p) → Ker(v) to C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' that is, there exists y′ : C → Ker(v) such that y′i = l2y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Putting together, we get a diagram 14 HOSSEIN ESHRAGHI AND RASOOL HAFEZI Ker(p) y � i � C [s y′]t � � Im(p) � 0 B h � W r � Coker(h) � 0 with commutative left part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' This induces a map y′′ : Im(p) → Cok(h) completing the diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Again, as X is wide, the monomorphism Im(p) i′ → N lies inside X and, hence, the injectivity of I1 in X gives a map z′ : N → I1 with z′i′ = by′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Finally, one verifies that the morphism ([s y′]t, z′) : (C p→ N) → (W br → I1) gives the required factorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' □ As a dual statement, we record the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Assume X is a functorially finite wide subcategory of mod-Λ and δ : 0 → A f→ B g→ C → 0 is an almost split sequence in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Let also P1 ℓ→ P0 h→ C be a minimal projective presentation with respect to X, where k : P1 → Ker(h) is a minimal right P(X)-approximation, i : Ker(h) → P0 is the canonical inclusion and ℓ = ik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Then the exact sequence 0 �( 0 A ) ( 0 u) �� P1 Z � wk ( 1 v ) �� P1 P0 � ℓ �0 in H(X) raised by forming the pull back diagram Ker(h) w � Ker(h) i � A u � Z r � v � P0 h � A f � B g � C in the exact category X, is almost split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Any functorially finite wide subcategory X of mod-Λ admits almost split sequences by [AS81, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Hence, following Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='1, we let τX denote the Auslander-Reiten translation over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Let X be a functorially finite wide subcategory of mod-Λ and consider the auto- equivalence σX : X → X introduced earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Assume that X ∈ X is an indecomposable module not belonging to P(X), and that P f→ Q → X is a minimal projective presentation with respect to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Then there is an exact sequence 0 → (−, σX τX (X)) → D(P, −) → D(Q, −) → D(X, −) in mod-X Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='3 yields that the inverse Auslander-Reiten translation τ −1 H(X)(0 → τX (X)) of (0 → τX (X)) in H(X) coincides with (P f→ Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Taking into account our previous observations on the functor σX gives the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' □ FROM MORPHISM CATEGORIES TO FUNCTOR CATEGORIES 15 Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' For every non-projective indecomposable Λ-module M, we know that there exists an exact sequence 0 → τ(M) → ν(P) → ν(Q) → ν(M) → 0 where P → Q → M → 0 is the minimal projective presentation of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' In some sense, the sequence presented by Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='4 goes parallel to, and generalizes this observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' This is more justified as we show in the sequel that for the case X = mod-Λ, σX is just the identity functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' However, it would be interesting to explore σX further by considering other functorially finite wide subcategories X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Denote by σ := σX : mod-Λ → mod-Λ the auto-equivalence obtained above in the case where X = mod-Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' We refer e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' to [HE] for further explanation on how the Auslander-Reiten translation τH and its inverse τ −1 H work in this case and suffice to recall that the standard duality functor DH might be computed in a local manner in terms of the standard duality D of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' That is to say, DH(X f→ Y ) = (D(Y ) D(f) → D(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Note that since σ is an equivalence, it clearly restricts to an equivalence σ′ : inj-Λ → inj-Λ on the subcategory of injective modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' In the sequel, it will be shown that σ′, and consequently σ, are nothing but the identity functor on the corresponding categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' The restricted equivalence σ′ is isomorphic to the identity functor on inj-Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Let I be an injective Λ-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' There exists a minimal injective resolution in H 0 → ( 0 I )0 → ( I I )1 → ( I 0 )0 → 0 of the object (0 → I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Applying the duality DH leads to the projective presentation in Hop 0 → ( 0 DI )0 → ( DI DI )1 → ( DI 0 )0 → 0 of the object DH(0 → I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Then, we compute the transpose and deduce that τ −1 H (0 → I) ≃ (ν−1(I) → 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' As we pointed out earlier in this section, this results in an equivalence (ν−1(I), −) ≃ D(−, σ(I)) in mod-(mod-Λ)op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Hence, evaluating on the regular module Λ, yields a natural iso- morphism σ(I) ≃ νν−1(I) ≃ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' □ Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' The equivalence σ is isomorphic to the identity functor on mod-Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' According to Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='6, the restricted equivalence σ′ is naturally isomorphic to the identity functor on the subcategory of injective modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Using injective resolutions, it is then straightforward to see that the same holds for σ itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' □ In the rest of this section, we will provide some applications of the aforementioned theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Let F be a functor in mod-(mod-Λ) with a minimal projective presentation (−, X) (−,f) → (−, Y ) → F → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Then there is an exact sequence (Y ′, −) (g,−) → (X′, −) → DF → 0 in mod-(mod-Λ)op where (X′ g→ Y ′) is the inverse Auslander-Reiten translation of (X f→ Y ) in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Again we specify our constructions to the dualizing variety C = mod-Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' By virtue of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='7, the functor ∆ := ∆mod-Λ acts identically on projective functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Using projective presentations, it follows that ∆ is isomorphic to the identity functor on the whole mod-(mod-Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Definition of ∆ then implies that the duality functor D : mod-(mod-Λ) → mod-(mod-Λ)op is 16 HOSSEIN ESHRAGHI AND RASOOL HAFEZI isomorphic to Θ′ ◦τ −1 H ◦(Θ)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' This proves the claim by following the definitions of the functors involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' □ Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Let M be an indecomposable Λ-module with a minimal projective presentation P f→ Q → M → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Then there is an exact sequence 0 → (−, τ(M)) → D(P, −) → D(Q, −) → D(M, −) → 0 in mod-(mod-Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' If M is projective, then such a sequence exists trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Otherwise, applying Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='4 for X = mod-Λ, there exists an exact sequence 0 → (−, σX τX (M)) → D(P, −) → D(Q, −) → D(M, −) → 0 in mod-(mod-Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Now Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='7 settles the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' □ Let us exploit Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='9 to observe a connection between the inverse Auslander-Reiten translation of an indecomposable non-projective Λ-module M with the second syzygies of injec- tive functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' For, replace M in Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='9 by τ −1M to get the exact sequence 0 → (−, M) → D(P, −) → D(Q, −) → D(τ −1(M), −) → 0 in mod-(mod-Λ)op in which P → Q → τ −1(M) → 0 is a minimal projective presentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Applying the duality D : mod-(mod-Λ)op → mod-(mod-Λ) gives the exact sequence 0 → (τ −1(M), −) → (Q, −) → (P, −) → D(−, M) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' This shows that the functor (τ −1(M), −) might be interpreted as a second syzygy of the injective functor D(−, M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' simple modules over (stable) Auslander algebra Assume Λ is of finite representation type and let M be a basic representation generator of mod-Λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' that is, M is the direct sum of all pairwise non-isomorphic indecomposable finitely generated Λ-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' The endomorphism algebra A(Λ) = EndΛ(M), simply denoted by A throughout the section, is called the Auslander algebra of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Moreover, the stable Auslander algebra of Λ is by definition A = EndΛ(M)/P, where P is the ideal in EndΛ(M) consisting of those endomorphisms factoring through a projective module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' In this case, we can identify mod-A with mod-(mod-Λ) via the equivalence induced by the evaluation functor eM : mod-(mod-Λ) → mod-A, F �→ F(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' It is also easy to see that eM induces an equivalence between mod-(mod-Λ) and mod-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' It is known [A76] that indecomposable modules in mod-Λ correspond bijectively to sim- ple functors in mod-(mod-Λ) by sending an indecomposable module M to the simple functor SM := (−, M)/rad(−, M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Further, for any indecomposable non-projective module M, there is a minimal projective resolution 0 → (−, N) (−,f) → (−, K) (−,g) → (−, M) → SM → 0 of SM such that 0 → N f→ K g→ M → 0 is an almost split sequence in mod-Λ ([A76, §2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Combined to the above observations on the Auslander algebra A, one may identify simple A- modules (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' simple A-modules) and indecomposable (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' indecomposable non-projective) Λ-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' FROM MORPHISM CATEGORIES TO FUNCTOR CATEGORIES 17 Specializing [H, Construction 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='1] to the module category mod-Λ gives a functor Ψ : S → mod-(mod-Λ), S being the monomorphism category of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' This is defined by sending (X f→ Y ) in S to the functor F ∈ mod-(mod-Λ) lying in the exact sequence 0 → (−, X) (−,f) → (−, Y ) → (−, Coker(f)) → F → 0 in mod-(mod-Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' As the following result says, Ψ behaves well with respect to almost split sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' ([H, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='7]) Let 0 → U → V → W → 0 be an almost split sequence in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Assume W is neither of types (a) or (b), nor of the form (Ω(X) → P), where X is a non-projective indecomposable Λ-module with projective cover P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Then 0 → Ψ(U) → Ψ(V) → Ψ(W) → 0 is an almost split sequence in mod-(mod-Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' The following theorem is one of the main results in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Assume Λ is of finite representation type and A is its stable Auslander algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Let S be a simple non-projective A-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Then, exactly one of the followings hold: (1) the Auslander-Reiten translate τA(S) is projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' (2) there exists a simple A-module S′ such that τA(S) ≃ Ω−1 A (S′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' In this case, Ext2 A(S, S′) ≃ DHomA(S, S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' According to aforementioned remarks, the simple non-projective module S corresponds to a simple functor (−, C)/rad(−, C) lying in the exact sequence 0 → (−, A) (−,f) → (−, B) (−,g) → (−, C) → (−, C)/rad(−, C) → 0 in mod-(mod-Λ) in such a way that λ : 0 → A f→ B g→ C → 0 is an almost split sequence in mod-Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Note that the middle term B may not be projective since otherwise there exists an isomorphism (−, C)/rad(−, C) ≃ (−, C) which is against non-projectivity of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' We distinguish two cases: Assume first that A is projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' So by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='3 of [HE], there exists an almost split sequence 0 �� rad(A) A � i �( A A )1 ⊕ � rad(A) B � fi �( A B )f �0 in S(Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Hence, in view of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='1, we get the almost split sequence 0 → Ψ � rad(A) A � i → Ψ � rad(A) B � fi → Ψ( A B )f → 0 in mod-(mod-Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Since A is projective, the definition of Ψ shows that Ψ(rad(A) i→ A) ≃ (−, A/rad(A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Likewise, as λ does not split, we have Ψ(A f→ B) ≃ (−, C)/rad(−, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Hence τA((−, C)/rad(−, C)) ≃ (−, A/rad(A)) that proves the claim in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Assume next that A is not projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Then there exists an almost split sequence ϵ : 0 → A′ → B′ → A → 0 in mod-Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Applying [H, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='3] on λ and ϵ, one infers the almost split sequences 0 �( A A )1 �( A B )f �( 0 C )0 �0 and 18 HOSSEIN ESHRAGHI AND RASOOL HAFEZI 0 �� A′ I � e �� B′ I⊕A � h �( A A )1 �0 in S where the second one is obtained from the push-out diagram A′ e � � B′ h � � A (†) I e � � I ⊕ A d � � A Ω−1 Λ (A) Ω−1 Λ (A) in which e : A′ → I is the injective envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' From [HZ, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='3], we can write (B′ h→ I ⊕ A) ≃ X ⊕ (J 1→ J), where X is an indecomposable non-projective object and J is either zero or isomorphic to I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' It follows then that τS(A f→ B) ≃ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Accordingly, by taking into account that (−, C)/rad(−, C) ≃ Ψ(A f→ B) by the exact sequence mentioned at the beginning of the proof, another application of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='1 shows that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='1) F := τmod-(mod-Λ)((−, C)/rad(−, C)) ≃ Ψ(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' However, the definition of Ψ yields F = Ψ(B′ h→ I ⊕ A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Therefore, abusing the notation, we may write τA(S) ≃ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Regarding the definition of Ψ, the middle column of (†) gives the long exact sequence 0 � (−, B′) � (−, I ⊕ A) � (−, Ω−1 Λ (A)) � � Ext1 Λ(−, B′) � Ext1 Λ(−, I ⊕ A) F � in mod-(mod-Λ) that implies F = Ker(Ext1 Λ(−, B′) → Ext1 Λ(−, A)) because I is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' On the other hand, since ϵ is an almost split sequence, our previous considerations show that there exists an exact sequence 0 � (−, A′) � (−, B′) � (−, A) � � Ext1 Λ(−, A′) � � Ext1 Λ(−, B′) � Ext1 Λ(−, A) (−, A)/rad(−, A) � F � of functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Invoking [AR74, Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='4], we see that Ext1 Λ(−, A′) is an injective functor in mod-(mod-Λ) and so the induced short exact sequence 0 → (−, A)/rad(−, A) → Ext1 Λ(−, A′) → F → 0 gives F = Ω−1 A ((−, A)/rad(−, A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Now it suffices to set S′ = (−, A)/rad(−, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Notice that the last assertion in the theorem is an upshot of the Auslander-Reiten formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' □ Based on previous theorem, in the following result we establish a bijection between certain simple modules over A and Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' This provides an interesting application concerning the stable equivalences of Artin algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Let Λ be of finite representation type and A be its stable Auslander algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' There exists a bijection between FROM MORPHISM CATEGORIES TO FUNCTOR CATEGORIES 19 (1) the set of isomorphism classes of non-projective simple modules S ∈ mod-A whose Auslander-Reiten translate τA(S) is projective;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' and (2) the set of isomorphism classes of indecomposable non-injective projective modules P ∈ mod-Λ such that the middle term of the almost split sequence starting from P is not projective;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' and (3) the set of isomorphism classes of simple modules S ∈ mod-Λ whose projective cover P(S) is non-injective, and the middle term of the almost split sequence starting from P(S) is not projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' The bijection between (2) and (3) might be shown by restricting the well-known bijection between simple and indecomposable projective modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' The map from (2) to (1) is given by sending P to (−, τ −1(P))/rad(−, τ −1(P)), which is well-defined due to the argument given in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Let now S be a simple non-projective module in mod-A with τA(S) projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' We already know that there is an indecomposable non-projective Λ-module C such that S ≃ (−, C)/rad(−, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' We claim that τ(C) is projective;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' otherwise, as in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='2, there exists a simple A-module S′ such that τA(S) ≃ Ω−1 A (S′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Hence the short exact sequence 0 → S′ → I → Ω−1 A (S′) → 0, in which I′ is the injective envelop of S′, splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' This means that Ω−1 A (S′) = 0 and so τA(S) = 0 which is against non-projectivity of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Thus τ(C) is projective and setting P := τ(C) completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' □ Recall that two Artin algebras Λ and Λ′ are said to be stably equivalent if there is an equiv- alence of categories mod-Λ ≃ mod-Λ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Denote by n(Λ) the number of iso classes of simple Λ-modules satisfying the third condition of the above corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' As a byproduct, we show that n(Λ) is an invariant of the stable equivalences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Let Λ and Λ′ be of finite representation type and stably equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Then n(Λ) = n(Λ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Since Λ and Λ′ are stably equivalent, it follows that the corresponding stable Auslander algebras A and A′ are Morita equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' By Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='3, we see that simple modules in mod-Λ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' mod-Λ′) that satisfy condition (3) correspond bijectively to non-projective simple modules over the stable Auslander algebra A (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' A′) with projective Auslander-Reiten translates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' We are done since the modules of latter type are preserved under Morita equivalences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' □ The following lemma is taken from [HE, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Let δ : 0 → A f→ B g→ C → 0 and δ′ : 0 → A′ f ′ → B′ g′ → A → 0 be almost split sequences in mod-Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Then 0 �� B′ A � g′ � � � g′ 1 � � 1 f � � � �( A A )1 ⊕ � B′ B � fg′ � [ −1 g′ ] [ −f 1 ] � �( A B )f �0, is an almost split sequence in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Further, � B′ B � fg′ is an indecomposable object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' The following theorem should be served as the second main result of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Assume Λ is a self-injective algebra of finite representation type and let A be its Auslander algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Let also S be a simple A-module of projective dimension two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Then there exists a simple A-module S′ of projective dimension two such that ΩA(S′) ≃ τ −1 A (S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' In this case, Ext2 A(S′, S) ≃ DHomA(S, S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' 20 HOSSEIN ESHRAGHI AND RASOOL HAFEZI Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' We may identify the simple module S by the simple functor (−, A)/rad(−, A) lying in the exact sequence 0 → (−, A′) (−,f ′) → (−, B′) (−,g′) → (−, A) → (−, A)/rad(−, A) → 0 (†) in mod-(mod-Λ) in such a way that δ : 0 → A′ f ′ → B′ g′ → A → 0 is an almost split sequence in mod-Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Let also δ′ : 0 → A f→ B g→ C → 0 be an almost split sequence in mod-Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Then by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='5 there exists an almost split sequence 0 �� B′ A � g′ � � � g′ 1 � � 1 f � � � �( A A )1 ⊕ � B′ B � fg′ � [ −1 g′ ] [ −f 1 ] � �( A B )f �0 in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Thanks to Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='8, this induces the almost split sequence 0 → Θ(B′ g′ → A) → Θ(B′ fg′ → B) → Θ(A f→ B) → 0 (††) in mod-(mod-Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Note that (†) implies Θ(B′ g′ → A) = (−, A)/rad(−, A) = S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Hence by (††), τ −1 A (S) = Θ(A f→ B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Set now W = (A f→ B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Then, by definitions, there exists an exact sequence (−, A) � (−, B) � � (−, C) � � Ext1 Λ(−, A) � Ext1 Λ(−, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Θ(W) � (−, C)/rad(−, C) � Set S′ be the simple functor (−, C)/rad(−, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Then the short exact sequence 0 → Θ(W) → (−, C) → S′ → 0 proves the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' □ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Auslander-Reiten components of Auslander algebras Throughout the section, we assume that Λ is a non-semisimple self-injective algebra of finite representation type and A denotes its Auslander algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' In the whole section, we use the iden- tification mod-A ≃ mod-(mod-Λ) described earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Once more, in this section, the quadruple family of objects in H of types (a), (b), (c), and (d) become important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' We aim to identify cer- tain components of the (stable) Auslander-Reiten quiver of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' To this end, we need firstly study particular τH-periodic objects in H and their periodicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' τH-periodic objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' As we observed in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='8, the functor Θ : H → mod-A behaves well with respect to almost split sequences in the sense that if there exists an almost split sequence 0 → X → Y → Z → 0 in H where Z is not of type (b) or (c), then 0 → Θ(X) → Θ(Y) → Θ(Z) → 0 is also an almost split sequence in mod-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Also we have seen in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='2 that one is given an equivalence H/V ≃ mod-A where V is generated by the objects of type (b) or (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Therefore, the Auslander-Reiten quiver ΓA of the Auslander algebra A might be computed via the Auslander- Reiten quiver ΓH of H by removing vertices corresponding to iso-classes of indecomposable objects of either types (b) or (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' The following construction is vital for the rest of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' It is mainly based on an analysis of various almost split sequences already obtained in [HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' For the sake of brevity, we prefer not to rewrite most of them here and suffice to give the precise reference number therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' FROM MORPHISM CATEGORIES TO FUNCTOR CATEGORIES 21 Construction 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Let C be an indecomposable non-projective Λ-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' There exist almost split sequences ϵ1 : 0 → τ(C) f→ B g→ C → 0 and ϵ2 : 0 → τ 2(C) f ′ → B′ g′ → τ(C) → 0 in mod-Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Applying Lemmas 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='5 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='4 we deduce the almost split sequences 0 � � B′ τ(C) � g′ � � � g′ 1 � � 1 f � � � � � τ(C) τ(C) � 1 ⊕ � B′ B � fg′ � [ −1 g′ ] [ −f 1 ] � �� τ(C) B � f �0, 0 � � τ(C) τ(C) � 1 � 1 f � �� τ(C) B � f � 0 g � �( 0 C )0 �0, and 0 �� τ(C) 0 � 0 � f 0 � �( B C )g ( g 1) �( C C )1 �0 in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Evidently, the indecomposable object (B′ fg′ → B) is not projective;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' so let X := τH(B′ fg′ → B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Also, as (B′ g′ → τ(C)) is not projective, we let Y := τH(B′ g′ → τ(C)) and note that X and Y are not projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' In view of [HE, Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='2, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='1], there exists an almost split sequence 0 � � ν(P ) ν(Q) � ν(h) �Y ⊕ ( I 0 )0 � � τ 2(C) 0 � 0 �0 in H where P h→ Q → τ 2(C) → 0 is the minimal projective presentation, and I is an injective module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' On the other hand, by [HE, Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='4, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='2], we have the almost split sequence 0 � � 0 τντ 2(C) � 0 �τH(Y) ⊕ ( 0 P )0 � � ν(P ) ν(Q) � ν(h) �0 in H where P is projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Putting all together, one obtains the mesh τ(X) � X � � � B′ B � fg′ � � τ(Y) � � Y � � � � B′ τ(C) � g′ � � � � τ(C) B � f � � � 0 ντ 3(C) � 0 � � ν(P ) ν(Q) � ν(h) � � � τ 2(C) 0 � 0 � � � τ(C) τ(C) � 1 � � ( 0 C )0 � in the Auslander-Reiten quiver ΓA of A in which the vertices (I → 0) and (0 → P) have been ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Let us recall from [HE, Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='7] that A = ντ 3 defines an auto-equivalence on the stable category mod-Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' As is expected, the A -orbit of an indecomposable non-projective Λ-module M consists of the modules A m(M) where m ranges over the integer numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Suppose every indecomposable non-projective Λ-module possesses a finite A - orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Then any indecomposable non-projective object X in H of either types (a), (b), (c) or (d) is of τH-periodicity a multiple of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' 22 HOSSEIN ESHRAGHI AND RASOOL HAFEZI Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' According to our previous observations, all mentioned objects lie in the τH-orbit of some indecomposable object of type (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Hence it suffices to prove the statement only for X = (0 → N) with N an indecomposable non-projective module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Justified by the hypothesis, choose a least integer n with A n(N) = N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Considering the particular mesh in ΓA as illustrated in Construction 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='1, we get τ 4 H(0 → N) ≃ (0 → A (N)) and thus τ 4n H (0 → N) ≃ (0 → A n(N)) = (0 → N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' □ Based on previous proposition, we are now able to prove the following theorem which will prove useful later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Assume every indecomposable non-projective Λ-module possesses a finite A - orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Then every simple A-module of projective dimension 2 is τA-periodic of periodicity divided by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Recall that such simple A-modules might be identified with simple functors SM = (−, M)/rad(−, M) where M is an indecomposable non-projective Λ-module lying in an almost split sequence 0 → τ(M) g→ N f→ M → in mod-Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' By Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='2, (0 → τ −1(M)) is of τH-periodicity 4n for a suitable integer n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Since (0 → τ −1(M)) and (M 1→ M) lie in the same τH-orbit, if it follows that (M 1→ M) is also of the same periodicity 4n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Thus the irreducible morphism (N f→ M) → (M 1→ M) in ΓH remains fixed after 4n applications of τH and ac- cordingly, (N f→ M) should be of τH-periodicity 4n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Consequently, according to Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='8, τ 4n A (SM) = τ 4n A Θ(N f→ M) = Θτ 4n H (N f→ M) = Θ(N f→ M) = SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' □ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Modules M with τ(M) = Ω(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Let M be an indecomposable non-projective module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' We say M has the property (∗) if 0 → Ω(M) → P(M) → M → 0 is an almost split sequence in mod-Λ where P(M) is the projective cover of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Modules satisfying this property have already been classified in [ARS, Theorem V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='3]: these are exactly non-injective simple Λ-modules M that are not a composition factor of rad(I)/soc(I) for every injective Λ-module I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' This clearly yields that such modules are necessarily A-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Note also that in the situation of the definition, τ(M) = Ω(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' The goal in this subsection is to see that existence of modules with this property may heavily affect the shape of the AR-quiver of A and in particular cases may even make it into an algebra of finite representation type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' As a first pace to study modules with property (∗), the following lemma shows that this property carries over from a module to its (co)syzygies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Let M be an indecomposable non-projective Λ-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' If M has the property (∗), then so do all its syzygies (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' cosyzygies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' In particular, the short exact sequences 0 → Ωi+1(M) → P i → Ωi(M) → 0 for i ≥ 0, and 0 → Ωi(M) → Ii → Ωi−1(M) → 0 for i ≤ 0 in mod-Λ induced by the minimal projective (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' injective) resolution of M are almost split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' We prove the lemma for integers i ≥ 0 by using an inductive argument whose basis i = 0 is satisfied by the assumption;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' so we put i > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Consider the almost split sequence 0 → τΩi(M) → B → Ωi(M) → 0 in mod-Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' We claim that B is projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Assume to the contrary that B has a non-projective indecomposable direct summand C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' The induction FROM MORPHISM CATEGORIES TO FUNCTOR CATEGORIES 23 hypothesis then implies that τ −1(C) is a non-projective direct summand of P i−1, which is absurd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Now use the fact that the morphisms involved in an almost split sequence are minimal to deduce that τΩi(M) = Ωi+1(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' □ The following lemma shows a property of the modules M for which (∗) is satisfied;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' this will be used later on in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Under the hypothesis of Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='5, one has ν(P i+1) ≃ P i for i ≥ 0 and ν−1(Ii−1) ≃ Ii for i ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' We prove the first assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' The minimal projective presentation P 1 → P 0 → M → 0 induces the short exact sequence 0 → τ(M) → ν(P 1) → ν(P 0) → ν(M) → 0 in mod-Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Note that, as ν is an auto-equivalence of mod-Λ, the map τ(M) → ν(P 1) is minimal and thus defines the injective envelope of τ(M) as ν(P 1) is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' However, by definition, the monomorphism τ(M) → P 0 obtained by composing the isomorphism τ(M) ≃ Ω(M) and the inclusion Ω(M) → P 0 is also minimal with P 0 injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Therefore ν(P 1) ≃ P 0 since the injective envelope is unique up to isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Now we deduce the result by applying an inductive argument in conjunction with Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' □ The following theorem is the promised one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Assume there exists an indecomposable non-projective Λ-module M with the property (∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Then the Auslander-Reiten quiver ΓA of A is a finite oriented cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' In particular, the Auslander algebra A is of finite representation type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' The minimal projective presentation · · → P n wn → P n−1 → · · · P 1 w1 → P 0 → M → 0 of M induces, according to Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='5, the almost split sequences ϵi : 0 → Ωi+1(M) vi → P i ui → Ωi(M) → 0 in mod-Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Applying Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='5 on ϵ0 and ϵ1 gives the almost split sequence 0 � � P 1 Ω(M) � u1 � � Ω(M) Ω(M) � 1 ⊕ � P 1 P 0 � w1 � � Ω(M) P 0 � v0 �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Note that, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='4, τH(( 0 M )) = � Ω(M) Ω(M) � 1 and τH �� Ω(M) Ω(M) � 1 � = � Ω2(M) 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Moreover, in light of [HE, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='4], we get τH(P 1 w1 → P 0) = (0 → Ω(M)) and so there exists an almost split sequence 0 �� 0 Ω(M) � 0 � � P 1 Ω(M) � u1 ⊕ � 0 P 0 � 0 � � P 1 P 0 � w1 �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Furthermore, an application of [HE, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='5] provides us with another almost split sequence 0 � � Ω2(M) P 1 � v1 �� 0 Ω(M) � 0 ⊕ � P 1 P 1 � 1 ⊕ � Ω2(M) 0 � 0 � � P 1 Ω(M) � u1 �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Also [HE, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='2] combined to Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='6 results in the almost split sequence 0 � � ν(P 3) ν(P 2) � ν(w3) � � Ω2(M) P 1 � v1 ⊕ � ν(P 3) 0 � 0 � � Ω2(M) 0 � 0 �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' 24 HOSSEIN ESHRAGHI AND RASOOL HAFEZI It is easy to see that, as Ω2(M) satisfies (∗), so does νΩ2(M) and consequently, τνΩ2(M) = ΩνΩ2(M) ≃ νΩ3(M) = A (M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Therefore, by [HE, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='4], we have τH(ν(P 3) ν(w3) → ν(P 2)) = (0 → A (M)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Continuing in this manner, one obtains the following mesh in the Auslander-Reiten quiver of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' � 0 P 0 � 0 � � Ω2(M) Ω2(M) � 1 � � 0 Ω(M) � 0 � � � � P 1 P 0 � w1 � � � P 2 Ω2(M) � u2 � � � � ν(P 3) ν(P 3) � 1 � � Ω2(M) P 1 � v1 � � � � P 1 P 1 � 1 � � P 1 Ω(M) � u1 � � � Ω(M) P 0 � v0 � � � 0 A (M) � 0 � � ν(P 3) ν(P 2) � ν(w3) � � � � Ω2(M) 0 � 0 � � � Ω(M) Ω(M) � 1 � � � 0 M � 0 � � ν(P 3) 0 � 0 � Starting then with the vertex (0 → A (M)) and iterating the above arguments,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' one may calculate the vertices lying on the left side of (0 → M) in ΓH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Also, by considering the minimal injective resolution of M, the vertices on the right part appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Summarizing, it follows that the component in ΓH containing the vertex (0 → M) is obtained by putting together all parts of the above shape corresponding to modules in the A -orbit of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' By virtue of previous considerations, ΓA comes up from ΓH by removing vertices of types (b) and (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Hence the AR-quiver ΓA is obtained by gluing together all pieces of the following shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' � 0 Ω(M) � 0 � � P 1 P 0 � w1 � � � P 2 Ω2(M) � u2 � � Ω2(M) P 1 � v1 � � � P 1 Ω(M) � u1 � � � Ω(M) P 0 � v0 � � � 0 A (M) � 0 � � ν(P 3) ν(P 2) � ν(w3) � � � 0 M � 0 Now, since M is A -periodic, we get a finite oriented cycle as a component in ΓA that by [ARS, Theorem VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='1] must be the whole of ΓA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' □ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Components of the stable Auslander-Reiten quiver of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' We let Γs A, the stable Auslander-Reiten quiver of A, be the subquiver of ΓA obtained by removing projective vertices and their τA-orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' It should be clarified that here, we distinguish with a usual custom in the corresponding literature where this terminology applies while removing vertices that are both projective and injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Also we notice that, generally, this has nothing to do with the Auslander-Reiten quiver of the stable Auslander algebra A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' For instance, despite A which is self-injective in this case, A can not be self-injective since it is of global dimension 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Below, we use results from [AS93] to get a nice intuition of the stable Auslander-Reiten quiver Γs A of A in terms of the AR quiver of a triangulated category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Recall from Section 3 that the class X in H consisting of all objects of type (a), (b), (c), or (d) determines an exact structure HX on H which has enough projectives and enough injectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' We claim that P(HX ) = X ∪ proj-H and I(HX ) = τH(X) ∪ inj-H, the subcategories of projectives and injectives of HX , coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Indeed, as in Construction 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='1, FROM MORPHISM CATEGORIES TO FUNCTOR CATEGORIES 25 τH(X) ⊆ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Since every indecomposable injective object in H is of type (b) or (c), we get I(HX ) ⊆ P(HX ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' To settle the reverse inclusion, note that by [HE, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='6], for an indecomposable projective Λ-module P there exists a projective Λ-module Q with τH(Q → 0) = (0 → P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Besides that projective objects of type (P 1→ P) lie in inj-H, this ensures that proj-H ⊆ I(HX ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Now take a non-injective object M of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' If M is of type (a), then M = (0 → M) for an indecomposable Λ-module M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' If M is projective then as above, M = τH(Q → 0) for some Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' otherwise M is non-injective and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='4 of [HE] shows that M = τH(P1 → P0) where P1 → P0 → τ −1(M) → 0 is the minimal projective presentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' If M = (M 1→ M) is of type (b) with M non-injective, then Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='4 gives M ≃ τH(0 → τ −1(M)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Furthermore, if M = (M → 0) is of type (c), then again Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='4 shows that M ≃ τH(τ −1(M) 1→ τ −1(M)) as M is non-injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Finally if M = (P f→ Q) is of type (d) then, setting N = Coker(ν(f)), we deduce from [HE, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='2] that M = τH(N → 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Summarizing, these imply that X ⊆ I(HX ) and the above claim follows;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' that is to say, HX is a Frobenius exact category and, consequently, the stable category HX is triangulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' On the other hand, according to [AS93, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='9], we infer that an almost split sequence 0 → X → Y → Z → 0 in H is an almost split sequence in HX if and only if neither X ∈ I(HX ) nor Z ∈ P(HX ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Thus, in order to get the Auslander-Reiten quiver of the triangulated category HX , it is enough to remove the iso-classes of indecomposable objects in X and arrows attached to them from the Auslander-Reiten quiver ΓH of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' But, as stated before, what remains after deleting, is exactly the stable Auslander-Reiten quiver Γs A of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' The following theorem is the main result in this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Note that if Λ admits a module M with property (∗) then, according to Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='7, Γs A is just a set of single vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' That’s why one has to exclude this case from the hypothesis below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Assume Λ is self-injective of finite representation type and Ξ is a component of Γs A containing a simple module SM for an indecomposable non-projective Λ-module M not fulfilling (∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Then (i) If Ξ is finite, then Ξ = Z∆/G, where ∆ is a Dynkin quiver and G is an automorphism group of Z∆ containing a positive power of the translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Moreover, Ξ is Γs A itself if we further assume that Λ is indecomposable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' (ii) If Ξ is infinite, then it is a stable tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' We have seen before that the AR-quiver ΓA of the Auslander algebra A is obtained from ΓH by removing the vertices of types (b) and (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Note that indecomposable objects of type (a) correspond to indecomposable projective A-modules and those of type (d) lie in the τA-orbit of indecomposable projective A-modules by Construction 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' So in fact, Ξ emerges by deleting vertices of either types (a), (b), (c) and (d) and the arrows attached to them from a component Ξ′ of ΓH containing the vertex (0 → M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Note that such a Ξ is connected as M does not satisfy the property (∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Note also that all vertices in Γs A are stable in the sense that τ m A (−) is well-defined over them for arbitrary integers m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Therefore Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='3 implies that the vertex SM in Ξ is τA-periodic and both the assertions in (i) and (ii) follow from [L, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' For the second statement in (i), note that indecomposability of Λ implies that the lower triangular matrix algebra T2(Λ) = ( Λ 0 Λ Λ ) is also indecomposable and recall that H is naturally equivalent to the category mod-T2(Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Now if Ξ is finite, then so is Ξ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' As such, Ξ′ itself is a finite component of ΓH which, according to [ARS, Theorem VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='1], should be the whole of ΓH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Hence Ξ = Γs A, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' □ 26 HOSSEIN ESHRAGHI AND RASOOL HAFEZI We conclude this section by quoting an observation from [HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Assume M is an indecom- posable non-projective Λ-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Denote by [M]A the A -orbit of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Let also ΓH(M) be the unique component of ΓH containing the vertex (0 → M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Moreover, we set T = {ΓH(M) | M is indecomposable non-projective} E = {[M]A | M is indecomposable non-projective} Then there exists a well-defined map δ : E → T which is given by sending [M]A to ΓH(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Further, if we let T∞ denote the subset of T consisting of all infinite components, and E∞ be the inverse image of T∞ under δ, then by [HE, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='8], δ is surjective and the restricted map δ |: E∞ → T∞ is a bijection whenever Λ is indecomposable self injective of finite representation type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Inspired by this result, we let L be the set of all components of ΓA containing a simple module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Define λ : T −→ L by λ(ΓH(M)) = ΓA(SM) where ΓA(SM) is the component of ΓA that contains the simple vertex SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Suppose Λ is indecomposable self-injective of finite representation type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' (i) The map λ is well-defined and surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' (ii) λ restricts to a bijection λ |: T∞ → L∞, where L∞ is the subset of L consisting of all infinite components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' (iii) The sets E∞ and L∞ are in bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Take M0 and M1 to be indecomposable non-projective Λ-modules with ΓH(M0) = ΓH(M1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Then there exist almost split sequences 0 → A0 f0 → B0 g0 → M0 → 0 and 0 → A1 f1 → B1 g′ 1 → M1 → 0 in mod-Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='4, there exist almost split sequences 0 �� Ai 0 � 0 � fi 1 � �� Bi Mi � gi � 0 gi � �� 0 Mi � 0 �0 in H for i = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Accordingly, (B0 g0 → M0) and (B1 g1 → M1) lie inside ΓH(M0) and the connected- ness of ΓH(M0) gives the existence of a walk (B0 g0 → M0) ←→ x1 ←→ · · · xn−1 ←→ (B1 g1 → M1) in ΓH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' If none of the xi is of the form (b) or (c), then using Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='8 we get a walk in ΓA be- tween Θ(B0 g0 → M0) = SM0 and Θ(B1 g1 → M1) = SM1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Consequently, λ(ΓH(M0)) = λ(ΓH(M1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Otherwise we may, without loss of generality, assume that the xi are all non-projective and apply the arguments used in Construction 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='1 to obtain a walk in ΓH passing through (Bi gi → Mi), i = 0, 1, none of the vertices over which are of the forms (b) or (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' To settle (ii), assume ΓA(SM) is an infinite component of ΓA for an indecomposable non-projective Λ-module M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' By Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='9, ΓA(SM) is a stable tube and the τA-orbit of SM generates the mouth of ΓA(SM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' The fact that the mouth of a stable tube is unique reveals that ΓA(SM) is uniquely determined by ΓH(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' The last statement is a combination of (ii) and [HE, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' □ References [AIR] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Adachi, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Iyama, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Reiten, τ-tilting theory, Compos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=', 150(3):415–452, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' [A65] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Auslander, Coherent functors, in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Categorical Algebra (La Jolla, Calif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=', 1965), 189231, Springer, New York, 1966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' [A71] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Auslander, Representation dimension of artin algebras, Queen Mary College Notes (1971).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' FROM MORPHISM CATEGORIES TO FUNCTOR CATEGORIES 27 [A76] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Auslander, Functors and morphisms determined by objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Representation theory of algebras (Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=', Temple Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=', Philadelphia, Pa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=', 1976), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' 1-244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Lecture Notes in Pure Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' 37, Dekker, New York, 1978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' [AR74] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Auslander and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Reiten, Stable equivalence of dualizing R-varieties, Advances in Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' 12 (1974), 306-366.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' [AR76] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Auslander and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Reiten, On the representation type of triangular matrix rings, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' London Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' (2), 12 (1976), 371-382.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' [AR77] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Auslander and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Reiten, Representation theory of Artin algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Invariants given by almost split sequences, Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Algebra 5 (1977), 443-518.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' [AR78] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Auslander and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Reiten, Representation theory of Artin algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' A functorial approach to almost split sequences, Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Algebra 6 (1978), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' 3, 257-300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' [ARS] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Auslander, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Reiten and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Smalø, Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics, 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Cambridge University Press, Cambridge, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' xiv+423 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' ISBN: 0-521- 41134-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' [AS81] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Auslander, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Smalø, Almost split sequences in subcategories, Journal of Algebra, 69 (1981), 426-454.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' [AS93] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Auslander and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Solberg, Relative homology and representation theory I, relative homology and homologically finite categories, Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Algebra 21 (9) (1993), 2995-3031.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' [Ba] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Bautista, The category of morphisms between projective modules, Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Algebra 32 (2004), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' 11, 4303-4331.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' [Bu] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Buan, Closed subbifunctors of the extension functor, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Algebra 244 (2001), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' 2, 407-428.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' [E] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Eshraghi, The Auslander-Reiten translation in morphism categories, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Algebra Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' 13 (2014), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' 3, 1350119, 12 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' [G] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Gillespie, The homotopy category of N-complexes is a homotopy category, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Homotopy Relat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Struct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' 10 (2015), 93-106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' [H] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Hafezi, From subcategories to the entire module categories, Forum Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' 33 (2021), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' 1, 245-270.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' [HE] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='Hafezi and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Eshraghi, Determination of some almost split sequences in morphism categories, avail- able via arXiv:2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='08883 [Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='RT].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' [HM] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Hafezi and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Mahdavi, Covering theory, (mono)morphism categories and stable Auslander algebras, available via arXiv:2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='08646 [Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='RT].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' [HMV1] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Angeleri H¨ugel, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Marks, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Vit´oria, silting modules and ring epimorphisms, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' 303 (2016), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' 5, 1044-1076.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' [HMV2] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Angeleri H¨ugel, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Marks, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Vit´oria, A characterization of τ-tilting finite algebras, available via arXiv:1801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='04312 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='RT].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' [HZ] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Hafezi and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Zhang, Stable Auslander-Reiten components of monomorphism categories, available via arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='11705 [Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='RT].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' [INY] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Iyama, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Nakaoka and Yann Palu, Auslander-Reiten theory in extriangulated categories, available via arXiv:1805.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='03776 [Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='RT].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' [IPTZ] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Igusa, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='-I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Platzeck, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Todorov and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Zacharia, Auslander algebras of finite representation type, Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Algebra 15 (1987), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' 1-2, 377-424.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' [J] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Jiao, The generalized Auslander-Reiten duality on an exact category, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Algebra Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' 17 (2018), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' 12, 1850227.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' [K] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Keller, Chain complexes and stable categories, Manuscripta Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' 67 (1990), 379-417.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' [L] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Liu, Auslander-Reiten theory in a Krull-Schmidt category, S˜ao Paulo J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' 4 (2010), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' 3, 425-472.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' [LZ] X-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Luo and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Zhang, Separated monic representations I: Gorenstein-projective modules, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Algebra 479 (2017), 1-34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' [MO] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Mart´ınez-Villa and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Ortiz-Morales, Tilting theory and functor categories III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' The maps category, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Algebra 5 (2011), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' 9-12, 529-561.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' [MS] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Marks and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' ˘S´tov´ı˘cek, Torsion classes, wide subcategories and localisations, Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' London Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' 49 (2017), 405-416.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' [Q] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='Quillen, Higher algebraic K-theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' I, in: Algebraic K-theory, I: Higher K-theories, Proceedings of the Conference, Battelle Memorial Institute, Seattle, Washington,1972, Lecture Notes in Mathematics, 341, Springer, Berlin, (1973), 85-147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' [RS] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Ringel and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Schmidmeier, The Auslander-Reiten translation in submodule categories, Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' 360 (2008), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' 2, 691-716.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' 28 HOSSEIN ESHRAGHI AND RASOOL HAFEZI [RZ] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Ringel and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Zhang, From submodule categories to preprojective algebras, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' 278 (2014), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' 1-2, 55-73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' [SY] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Skowro´nski and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Yamagata, Frobenius algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Basic representation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' EMS Textbooks in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' European Mathematical Society (EMS), Z¨urich, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' xii+650 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' ISBN: 978-3-03719-102-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' [XZZ] B-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Xiong, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Zhang and Y-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Zhang, Auslander-Reiten translations in monomorphism categories, Forum Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' 26 (2014), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' 3, 863-912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' [Z] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Zhang, Monomorphism categories, cotilting theory, and Gorenstein projective modules, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Algebra 339 (2011), 181-202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' [ZX] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Zhang and B-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Xiong, Separated monic representations II: Frobenius subcategories and RSS equiv- alences, Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' 372 (2019), 981-1021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, PO Box 87317-51167, Kashan, Iran Email address: eshraghi@kashanu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='ir School of Mathematics and Statistics, Nanjing University of Information Science & Technology, Nanjing, Jiangsu 210044, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content=' China Email address: hafezi@nuist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} +page_content='cn' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfp_gy/content/2301.00534v1.pdf'} diff --git a/8NFJT4oBgHgl3EQfnSyO/content/tmp_files/2301.11591v1.pdf.txt b/8NFJT4oBgHgl3EQfnSyO/content/tmp_files/2301.11591v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..236471edf9593bfbfb8ff30df0c425dfe1d84a0b --- /dev/null +++ b/8NFJT4oBgHgl3EQfnSyO/content/tmp_files/2301.11591v1.pdf.txt @@ -0,0 +1,1867 @@ +Information Entropy-based Camera Path Estimation for In-Situ +Visualization +Ken Iwata* +Kobe University +Naohisa Sakamoto† +Kobe University +Jorji Nonaka‡ +RIKEN R-CCS +Chongke Bi§ +Tianjin University +Information Entropy +(for viewpoint selection) +Depth and Lightness Information +𝑑!𝑑"𝑑#⋯ 𝑑"$% +⋯ +𝐷𝑒𝑝𝑡ℎ + 𝑙! 𝑙" 𝑙# ⋯ 𝑙"$% +⋯ +𝐿𝑖𝑔ℎ𝑡𝑛𝑒𝑠𝑠 +Selected Viewpoint “A” +Camera Path +Selected Viewpoint “A” +Selected Viewpoint “B” +Intermediate Images +Selected +Image “A” +Selected +Image “B” +Selected Image “A” +︓Viewpoints +Rendering Image +(Information Source) +Figure 1: Depth and lightness entropy-based viewpoint selection and camera path estimation for generating a smooth video, with as +much information as possible, to assist the rapid understanding of the underlying simulation phenomena. +ABSTRACT +In-situ processing has widely been recognized as an effective ap- +proach for the visualization and analysis of large-scale simulation +outputs from modern HPC systems. One of the most common +approaches for batch-based in-situ visualization is the image- or +video-based approach. In this kind of approach, a large number of +rendered images are generated from different viewpoints at each +time step and has proven useful for detailed analysis of the main +simulation results. However, during test runs and model calibration +runs before the main simulation run, a quick overview might be +sufficient and useful. In this work, we focused on selecting the +viewpoints which provide as much information as possible by using +information entropy to maximize the subsequent visual analysis task. +However, by simply following the selected viewpoints at each of +the visualization time steps will probably lead to a rapidly changing +video, which can impact the understanding. Therefore, we have +also worked on an efficient camera path estimation approach for +connecting selected viewpoints, at regular intervals, to generate a +smooth video. This resulting video is expected to assist in rapid +understanding of the underlying simulation phenomena and can +be helpful to narrow down the temporal region of interest to min- +imize the turnaround time during detailed visual exploration via +image- or video-based visual analysis of the main simulation run. +We implemented and evaluated the proposed approach using the +OpenFOAM CFD application, on an x86-based Server and an ARM +A64FX-based supercomputer (Fugaku), and we obtained positive +evaluations from domain scientists. +Index Terms: +Human-centered computing—Visualization— +*e-mail: 228x202x@stu.kobe-u.ac.jp +†e-mail: naohisa.sakamoto@people.kobe-u.ac.jp +‡e-mail: jorji@riken.jp +§e-mail: bichongke@tju.edu.cn +Visualization systems and tools—Visualization toolkits +1 +INTRODUCTION +High-end high performance computing (HPC) systems have contin- +uously become more and more capable with higher computational +capacity with every new system replacement. This was the case for +the replacement of the K computer to the supercomputer Fugaku at +the RIKEN R-CCS. The increased number of CPUs and computa- +tional cores have been applied for Capability Computing to tackle +even larger numerical simulations with higher spatio-temporal reso- +lutions. In addition, this also has been used for Capacity Computing +to handle an even larger number of parameters and members during +parametric sweep and ensemble simulations. On the other hand, +this proportionately generates even larger simulation outputs, thus, +making the visualization and analysis tasks even more challenging. +As a result, the importance of in-situ visualization and analysis has +continuously become even more evident. +A variety of approaches have already been proposed and applied +for the in-situ visualization and analysis as discussed in [6]. We can +also verify that there are also a variety of existing applications and +libraries for realizing in-situ visualization and analysis. However, +since in-situ processing is executed simultaneously with the simu- +lation, it becomes highly important to collaborate with the domain +scientists . We have been working with domain scientists working +with computational fluid dynamics (CFD) simulation of the sound +generation mechanisms [27], and we already worked on an adaptive +in situ time-step sampling approach [26]. In this work, we have +used the same OpenFOAM CFD application and simulation model +and obtained assistance from them for necessary technical feedback +during the developments. +Probably the most widely used image-based in-situ visualization +approach is ParaView Cinema [1]. In that approach, a large set of +pre-computed images are generated in-situ on the HPC system side +for the interactive post-hoc visual exploration on a local machine +such as desktop PC and laptop. There is also an image-based in-situ +visualization approach that generates a set of images from omni- +directional camera positions [9], and its extension for video-based +arXiv:2301.11591v1 [cs.GR] 27 Jan 2023 + +in-situ visualization [8]. These image- or video-based in-situ visual- +ization approaches have proven useful for detailed analysis of the +main simulation results. In this work, we have focused on rapid un- +derstanding of the underlying simulation during test runs and model +calibration runs before the main simulation run. For this purpose, +we focused on selecting the most appropriate viewpoints, based on +information entropy, at regular time intervals of the simulation in +order to obtain as much information as possible trying to facilitate +the rapid understanding of such kinds of simulations. +2 +RELATED WORK +There is an extensive work that culminated in the creation of a classi- +fication and terminology for the in-situ visualization approaches [6]. +Here, we will only focus on related works for realizing tightly cou- +pled in-situ visualization, and techniques for selecting time steps +and viewpoints that can be used for minimizing the amount of im- +ages for the image- or video-based in-situ visualization. VTK-based +ParaView and VisIt are probably the most widely used visualization +application for large data visualization. Both applications provide in- +situ visualization APIs, ParaView Catalyst [2] and VisIt LibSim [11], +for integrating to the simulation code. In a batch-based in-situ visu- +alization, a large amount of images can be generated for assisting +the post-hoc visualization [9]. To facilitate this post-hoc visual anal- +ysis, Ahrens et al. [1] proposed an image-based approach for the +in-situ visualization and analysis, and was implemented as Paraview +Cinema. In this approach a large set of images are generated in- +situ, and a custom visualization application is used, on the local +machine, to perform interactive visual analysis by automatically +switching between the generated set of images. Similar to this ap- +proach, Kageyama et al. [8] proposed a video-based approach by +generating an omnidirectional animated video, from the set of in-situ +generated images, which are explorable from a custom visualization +application. Although these approaches have proven efficient, most +of the generated images may have small or even no contribution to +the visual analysis, thus it may be unnecessarily increasing the time +spent on the post-hoc visual analysis task. +An approach to minimize the aforementioned amount of gen- +erated images is the selection of the most valuable time steps for +rendering the images. For this purpose, Ling et al. [12] proposed a +method to estimate the probability density function of the simulation +field, at each time step, by using the kernel density estimation. They +also applied machine learning for extracting feature quantity from +the obtained estimation, and detected potentially valuable time steps +where an important phenomena may occur. However, this method +can cause false detections depending on the high correlation among +the physical quantities on the simulation field as mentioned by the +authors. Yamaoka et al. [26] extend the aforementioned work, and +proposed an adaptive time sampling method for in-situ visualization. +In this method, kernel density function and Kullback–Leibler diver- +gence is applied to estimate the amount of change on the simulation +field. The sampling intervals are adaptively changed according to +the estimated amount of change in the simulation. +Another approach for reducing the amount of images is the se- +lection of viewpoints for generating the images. For this purpose, +Kamada et al. [10] considered the viewpoints capable of minimizing +the number of degenerated face as being the optimal viewpoints. +However, they did not extend their work for the case when there +exist multiple viewpoints with the same number of degenerated +faces. Barral et al. [3] solved this problem by adding the projected +area as a weight to the number of degenerated faces. However, +there still remains a problem on how to properly set these weights. +Vazquez et al. [23] proposed a method to select the optimal view- +point defined by the viewpoint entropy based on the information +entropy. Since this method cannot handle the movement of view- +points, the authors improved the viewpoint entropy and applied it +to molecular objects [25] as well as to image-based modeling [24]. +Page et al. [18] proposed a method to analyze the object shape by +calculating the entropy for the silhouette and surface curvature of +the model. Polonsky et al. [19] discussed evaluation indices for the +viewpoint selection, and concluded that none of them could make +the best choice in any situation. However, they also said that by +improving each of these indices, it will become possible to make a +better choice by using a combination of them. +Secord et al. [21] proposed some evaluation indices for the view- +point selection, and showed that optimal viewpoints can be selected +by combining these metrics. Takahashi et al. [22] proposed a method +for estimating the optimal viewpoint for volume data by using in- +formation entropy. Bordoloi et al. [5] proposed an information +entropy-based evaluation metric for the viewpoints during volume +rendering by using the transfer function, data distribution, and voxel +visibility information. Zhang et al. [28] also proposed an evaluation +metric for volume rendering based on the opacity, brightness, and +structural features. Ji et al. [7] proposed a method to find the optimal +time-varying views by using the viewpoint selection method to max- +imize the amount of information for time-series volume data. They +showed that it is possible to create an animation with the largest +amounts of information. This was realized by searching for a move- +ment route with the largest amounts of information using dynamic +programming. Marsaglia et al. [14] proposed a viewpoint quality +evaluation metric based on information entropy involving the visible +field data, depth, and shading values belonging to each of the pixels +in the image. In another work, they also utilized a trigger-based ap- +proach in combination with information entropy to determine when +to search for a new camera position as a simulation evolves [15]. +Our work was inspired in their viewpoint quality evaluation metric, +which we extended with the lightness information for evaluating +the viewpoint quality. We will detail the methodology behind our +proposed method in the next section. +3 +METHODOLOGY +3.1 +Overview +In this work, we focused on a viewpoint selection approach, based +on information entropy, and on a camera path estimation approach, +based on quaternion interpolation. The viewpoints selected at regular +intervals are used as markers to estimate the smooth camera path. +Following are the necessary requirements to meet this goal: +R1. Images from the selected viewpoints should capture important +phenomenon from the underlying simulation. +R2. The resulting video generated from the rendered outputs should +be smooth for post-hoc analysis. +Below is the adopted approach to satisfy the aforementioned +requirements, and they are divided into the following three parts: +A. Viewpoint evaluation +The viewpoint quality will be evaluated using information +entropy and will be used to select the most appropriate image +for each evaluated time step. Only images from the selected +viewpoints will be output (R1). +B. Camera path estimation +The camera path connecting these selected viewpoints will be +estimated, and the rendered images through this camera path +will also be output as intermediate images (R2). +C. Video generation +At the end of the simulation, these output images will be se- +quentially concatenated to produce a video (R2). +Regarding part A, the simulation state usually does not often +significantly change within a single simulation time step. Therefore, + +Simulation time step +• Execute simulation +Visualization time step +• Store simulation data +Entropy evaluation time step +• Select optimal viewpoint +• Calculate camera path +Times and Intervals +• Simulation time +𝑇! = 𝑖∆𝑇 +• Visualization time +𝑇" +! = 𝑖∆𝑇" = 𝑖𝑁"∆𝑇 +• Entropy evaluation time +𝑇# +! = 𝑖∆𝑇# = 𝑖𝑁"𝑁#∆𝑇 +data1 +time [ 𝑇" ] +∆𝑇" +time [ 𝑇# ] +∆𝑇# +time [ 𝑇 ] +∆𝑇 +data2 +data3 +Rendering +Selected Viewpoint “A” +Selected Viewpoint “B” +Intermediate Images +Figure 2: Different time step intervals for the simulation, visualization and entropy evaluation. +there is usually no need to visualize at every simulation time step, +and the visualization can be performed at every set of simulation time +steps. In the same manner, the viewpoint evaluation for viewpoint +selection will be performed at every set of visualization time steps to +satisfy R1. In this paper, as shown in Fig. 2, we use T to represent +the simulation time step, TV to represent the visualization time step, +and TE to represent the entropy evaluation time step, ∆T to represent +the simulation time step interval, ∆TV to represent the visualization +time step interval, and ∆TE to represent the entropy evaluation time +step interval. The elapsed time for the ith simulation time step T i can +be expressed as T i = i∆T. In the same manner, the visualization time +step and entropy evaluation time steps can be expressed respectively +as T i +V = i∆TV and T i +E = i∆TE. Considering that the simulation is +performed NV times for every visualization, and the visualization is +performed NE times for every optimal viewpoint selection, then we +can express these intervals as ∆TV = NV ∆T and ∆TE = NENV ∆T. +Regarding part B, a camera path connecting viewpoints selected at +every ∆TE entropy evaluation time step will be estimated. Although +visualization is not performed during the visualization time steps +in between the entropy evaluation time steps, the simulation data +for each ∆TV visualization time step is stacked. From the obtained +camera path, the rendered images at the intermediate visualization +time steps will be output as the intermediate images for generating a +smooth video, and this satisfies R2. It is worth noting that it is also +possible to generate the full set of images from the entire viewpoints +for the detailed post-hoc analysis when necessary. +Regarding part C, the set of output images generated at each ∆TV +visualization time step will be joined sequentially to create a video +file. For this purpose, we can use existing tools such as the well- +known FFmpeg available to a variety of platforms. In the resulting +video, the camera will automatically move and capture the important +phenomenon, and this allows the R2 to be satisfied. Traditional +approach requires the user to search for the best location, in the +trial-and-error manner, to visually explore when searching for an +important phenomenon during the simulation. However, by using +the proposed method, this search for the best camera position may +be alleviated and may facilitate narrowing down the spatio-temporal +region of interest for the detailed visual analysis. +3.2 +Viewpoint Selection +In this section, we will detail the utilized viewpoint selection ap- +proach. The evaluation of the viewpoints is based on information +entropy. We used depth and lightness values from the rendered +images for calculating the associated information entropy, that is, +the depth entropy and lightness entropy. +3.2.1 +Information Entropy +Information entropy used in this work can be defined as the expected +value for the amount of information obtained from a certain infor- +mation source [4]. The information entropy H(X) from a source +X given by the set of probabilities P(x1),P(x2),··· ,P(xn) corre- +sponding respectively to the set of information x1,x2,··· ,xn can be +represented as follows: +H(X) = − +n +∑ +i=1 +P(xi)logP(xi) +(1) +Here, when P(xi) = 0, we will consider P(xi)logP(xi) = 0. Re- +garding the selection of the logarithm’s base, the base influences +the multiplication factor and, thus, is arbitrary. Base 2 is commonly +used in information theory, and was used in this work. Taking into +consideration the probability distribution of the information source, +the larger the information bias, the smaller the value of information +entropy, and vice versa. +3.2.2 +Depth Entropy +Depth entropy used in this work is based on the viewpoint quality +evaluation metric proposed by Marsaglia et al. [14]. The information +entropy is calculated by considering the image as the source of +information, and by using the depth values belonging to each of the +pixels in the image. The depth values can vary in the range of 0 ∼ 1, +and the closer the distance to the object, the smaller the value. The +background portion in the image where there is no object is set to +infinity and will have their depth values corresponding to 1. +The depth values from all pixels of the rendered image are binned +into 256 groups d0,d1,··· ,d255 for creating a discrete probability +distribution D, which will be used to calculate the information en- +tropy. At this time, the background portion in the image is considered +to have no information, and only the pixels with some information +will be used in the calculation. By using the discrete probability +distribution, the depth entropy Hd can be calculated as follows: +Hd = − +255 +∑ +i=0 +D(di)log2 D(di) +(2) +Here, D(di) corresponds to the probability for a given value, +selected based on the probability distribution D, being di. When +evaluating a viewpoint using depth entropy, the resulting value will +be larger for images with large dispersion in the distribution of depth +values. Therefore, the viewpoints of images showing objects with +high undulations will be highly evaluated. + +3.2.3 +Lightness Entropy +In this work, in addition to the depth entropy, we propose the use of +lightness entropy to also take into consideration the color informa- +tion in the image. Diverging color maps, proposed by Moreland [16], +have become prevalent in scientific visualization as the substitute +for the traditional but problematic rainbow color map. Although the +change in color values, such as RGB values, in a color map may +follow different behavior depending on the color map, diverging +color maps usually show similar behavior in the lightness values in +CIELAB color space as shown in the Fig. 3. Therefore, lightness +entropy is expected to work robustly for the diverging color maps. +The proposed lightness entropy can be defined as an information en- +tropy using the lightness values from the target image as the source +of information. The lightness value is calculated from RGB values +and can vary in the range of 0 ∼ 100. +PiYG +RdBu +PuOr +Lightness L* +Figure 3: Lightness values for different diverging color maps. +The lightness values, as well as the depth values, obtained +from all pixels of the rendered image, are binned into 256 groups +l0,l1,··· ,l255 for creating a discrete probability distribution L, which +will be used to calculate the information entropy. At this time, the +background portion in the image is considered to have no infor- +mation, and only the pixels with some information will be used in +the calculation. By using the discrete probability distribution, the +lightness entropy Hl can be calculated as follows: +Hl = − +255 +∑ +i=0 +L(li)log2 L(li) +(3) +Here, L(li) corresponds to the probability for a given value, se- +lected based on the probability distribution L, being li. When a +viewpoint is evaluated using the lightness entropy, the resulting +value will be larger for images with large dispersion in the distribu- +tion of lightness values. Therefore, the viewpoints of images with +clear brightness and darkness will be highly evaluated. +3.3 +Path Estimation between Selected Viewpoints +In this section, we will detail the utilized camera path estimation +between the viewpoints selected by using the depth and/or lightness +entropy. In this work, we considered that the viewpoints are pre- +arranged in a spherical surface as shown in Fig. 4, and the camera +path from one viewpoint to another will move over this spherical +surface. More specifically, the position and orientation of a given +viewpoint will be represented as a quaternion, and the movement +from one to another viewpoint will be obtained by using quaternion +interpolation. In this work, we investigated two quaternion interpola- +tion methods: spherical linear interpolation (SLERP) and spherical +quadrangle interpolation (SQUAD). In the following subsections, +we will explain about spherical linear interpolation and spherical +quadrangle interpolation. +3.3.1 +Spherical Linear Interpolation (SLERP) +SLERP is an abbreviation for spherical linear interpolation, and is +a quaternion interpolation method for connecting two points over a +sphere in the straight line direction, or the shortest path, as shown +in Fig. 5. SLERP-based interpolation from a quaternion qA to the +quaternion qB can be calculated by using time t ∈ [0, 1] as follows: +: Viewpoints +Figure 4: Viewpoint distribution over spherical surface. +slerp(qA, qB, t) = sin(1−t)φ +sinφ +qA + sintφ +sinφ qB +(4) +Here, φ = arccos⟨qA, qB⟩. In addition, in the case of ⟨qA, qB⟩ < 0, +the interpolation will be interpolated in the contrary direction over +the sphere surface, that is by the longest path in the straight-line +direction. To interpolate by the shortest path, then either qA or qB +should be replaced with a quaternion with same rotation but in the +opposite direction. For instance, by replacing qA with −qA. +3.3.2 +Spherical Quadrangle Interpolation (SQUAD) +SQUAD is an abbreviation for spherical quadrangle interpolation, +and is a quaternion interpolation method to connect multiple points +in a smoothness way so that the derivatives are continuous in the +neighborhood of the points (Fig. 5). Considering a quaternion list +{q1, q2, ... qn}, then the SQUAD-based interpolation from qi to +qi+1 can be calculated by using the time t ∈ [0, 1] as follows: +squad(qi, qi+1, ai, ai+1, t) += slerp(slerp(qi, qi+1, t), slerp(ai, ai+1, t), 2t(1−t)) +(5) +ai = qi exp +� +−logq∗ +i qi−1 +logq∗ +i qi+1 +4 +� +(6) +Here, the exponential of the quaternion exp(q) and the logarithm +of the quaternion logq for the quaternion q = a+bi+c j +dk are +defined as follows: +exp(q) = ea +� +cos∥bi+c j +dk∥+ bi+c j +dk +∥bi+c j +dk∥ sin∥bi+c j +dk∥ +� +(7) +logq = log∥q∥+ bi+c j +dk +∥bi+c j +dk∥ arctan ∥bi+c j +dk∥ +a +(8) +In addition, in the case of performing SQUAD-based interpolation +from q1 to q2, and from qn−1 to qn, we consider q0 = q1 and qn+1 = +qn. +Unit Sphere in Quaternion Space +: SLERP +: SQUAD +: Rotation Quaternion +Figure 5: Comparison of SLERP and SQUAD interpolation methods. + +7 +100 +80 +60 +40 +RdBu +20 +PiYG +PuOr3.3.3 +Implementation +We utilized the Kyoto Visualization System (KVS) [20] for imple- +menting the proposed viewpoint selection approach, based on depth +and lightness entropy, as well as the sequential camera path between +the selected viewpoints via SLERP- and SQUAD-based interpolation +methods. KVS is a cross-platform, open-source C++ visualization +library capable of running on a variety of hardware systems from +traditional x86/GPU systems to GPU-less HPC systems including +IBM Blue Gene L/P (PowerPC), K computer (SPARC VIIIfx), and +Fugaku (ARM A64FX). KVS supports hybrid MPI/OpenMP paral- +lelism and implements a sort-last parallel image composition method +based on Binary-Swap [13], with an extension to support non-power- +of-two number of nodes, which is named 234Compositor [17]. +The pseudocode of our implementation, using the SQUAD-based +interpolation, is described in Algorithm 1. In this pseudocode, I[], +V[], and Q[] respectively represent the queues for storing the out- +put image, simulation data, and the quaternion for the selected +viewpoint. In addition, is initial step(t) and is final step(t) are +functions that respectively return the true information in the first +and the final time step. The Vis(V, q) function renders the simu- +lation data V from the viewpoint represented by the quaternion of +q. Entropy Evaluation(V) is the function that calculates the en- +tropy for the simulation data V at all pre-arranged viewpoints on +the spherical surface, and returns the quaternion information of the +viewpoint with highest entropy value. Its pseudocode is described in +Algorithm 2. In this pseudocode, L represents the set of viewpoints +and the read back(V, l) is a function that renders the simulation data +V from the viewpoint l and returns its frame buffer. entropy( f) is +a function that calculates the entropy for the frame buffer f and re- +turns its value. quaternion() is a function that returns the quaternion +information from a given viewpoint. +4 +EXPERIMENTAL EVALUATIONS +We used the OpenFOAM CFD code and model for the experimental +evaluations. The simulation model used for the evaluations was +obtained from our collaborators [27], and refers to a sound propaga- +tion in the oral cavity by using irregular volume data composed of +3,197,279 hexahedral elements. We integrated the in situ KVS mod- +ule to the OpenFOAM code, and evaluated on two systems shown +in Tables 1 and 2. The irregular volume data was decomposed into 8 +blocks for the x86 Server, and up to 1,024 blocks for the Fugaku. +Table 1: x86/GPU-based Server System. +Nodes +1 +CPU +Intel Xeon Gold 6238R 2.20GHz 28Core×2 +Cores +28×2 = 56 +RAM +384 GB DRAM +GPU +NVIDIA Quadro RTX8000 +Compiler +GCC version 7.5.0 +MPI +OpenMPI 2.1.1 +Table 2: ARM-based Supercomputer Fugaku. +Nodes +158,976 +CPU +Fujitsu A64FX (Armv8.2-A SVE) +Cores +48 + 2 Assistant Cores +RAM +32GB HBM2 +Compiler +GCC-based Fujitsu Compiler Ver. 4.8.0 +MPI +OpenMPI with Fujitsu expensions for Tofu +Algorithm 1 In-situ visualization (using SQUAD interpolation). +1: function IN SITU VISUALIZATION(∆TV , ∆TE) +2: +I[], V[], Q[], t; +3: +while t ≤ tend do +4: +Vt = Sim(); +5: +if is initial step(t) then +6: +Qt = Entropy Evaluation(Vt); +7: +Q.push(Qt); Q.push(Qt); +8: +else if t%∆TV == 0 then +9: +if t%∆TE == 0 then +10: +Qt = Entropy Evaluation(Vt); +11: +Q.push(Qt); +12: +if Q.size() == 4 then +13: +q1 = Q.front(); Q.pop(); +14: +q2 = Q.front(); Q.pop(); +15: +q3 = Q.front(); Q.pop(); +16: +q4 = Q.front(); Q.pop(); +17: +for i = 0, 1, ··· , ∆TE −1 do +18: +s = i/TE; +19: +qs = squad(q1, q2, q3, q4, s); +20: +Vs = V. front(); V.pop(); +21: +Is = Vis(Vs, qs); +22: +I.push(Is); +23: +end for +24: +end if +25: +end if +26: +end if +27: +if is final step(t) then +28: +q1 = Q.front(); Q.pop(); +29: +q2 = Q.front(); Q.pop(); +30: +q3 = Q.front(); Q.pop(); +31: +q4 = q3; +32: +for i = 0, 1, ··· , ∆TE −1 do +33: +s = i/TE; +34: +qs = squad(q1, q2, q3, q4, s); +35: +Vs = V. front(); V.pop(); +36: +Is = Vis(Vs, qs); +37: +I.push(Is); +38: +end for +39: +while V.size() > 0 do +40: +Vs = V. front(); V.pop(); +41: +Is = Vis(Vs, q3); +42: +I.push(Is); +43: +end while +44: +end if +45: +t ++; +46: +end while +47: +return I; +48: end function +4.1 +Some Results +For the initial experimental evaluations, we selected the pressure +variable and used multi-isosurface rendering with three distinct iso- +values that are rendered as different colors. The total number of +simulation time steps for the utilized CFD model was 15,000, and +we used the parameters shown in Table 3 for the evaluations. We +evaluated the use of our proposed lightness entropy (Lightness) in ad- +dition to the depth entropy (Depth) proposed by Marsaglia et al. [14], +and also the use of the average of depth and lightness entropy (Depth +& Lightness). For the use of only lightness entropy, we experi- +mented with three diverging color maps (RdBu, PiYG, PuOr). We +selected three entropy evaluation intervals (10, 30, 50), which repre- +sent the visualization time step interval for performing the entropy +calculation. We also selected three sets of viewpoints with differ- + +Algorithm 2 Entropy Evaluation. +1: function ENTROPY EVALUATION(V) +2: +E = 0.0; +3: +Q = 1+0i+0j +0k; +4: +for l ∈ L do +5: +f = read back(V, l); +6: +e = entropy( f); +7: +if e > E then +8: +E = e; +9: +Q = l.quaternion(); +10: +end if +11: +end for +12: +return Q; +13: end function +ent numbers of viewpoints in the latitude and longitude directions +(latitude × longitude). Both SLERP- and SQUAD-based quaternion +interpolation methods were also evaluated for estimating the camera +path between the selected viewpoints. The x86/GPU Server was +used for the detailed evaluation using these different parameters, and +the supercomputer Fugaku was used for the scalability analysis by +using up to 1024 nodes, that is, 49,152 cores in hybrid MPI/OpenMP +parallelism. +Table 3: Parameters used for the experimental evaluations. +Entropy source +Depth; Lightness; Depth & Lightness +Color maps +RdBu; PiYG; PuOr +# of viewpoints +15×30; 25×50; 35×70 +Intervals (NE) +10; 30; 50 +Interpolation +SLERP; SQUAD +Fig. 6 shows some entropy heatmaps evaluated by using all three +entropy sources at different simulation time steps (2400, 6000, 9600, +and 13200). The set of viewpoints evenly distributed on the spherical +surface is mapped onto the 2D heatmap where the viewpoints on +the same latitude are placed on the same horizontal axis, and in the +same manner, the viewpoints on the same longitude are placed on +the same vertical axis. The blue-colored regions show the portions +where the evaluated entropy has low value, and on the other hand, +the red-colored regions show the portions where the evaluated en- +tropy has high value. Fig. 7 shows the multi-isosurface rendered +results from the selected viewpoints obtained in Fig. 6; Fig. 9 shows +the multi-isosurface rendered results from the selected viewpoints +obtained in Fig. 8; Fig. 11 shows the multi-isosurface rendered +results from the selected viewpoints obtained in Fig. 10. Table 4 +shows the average elapsed time of entropy calculation per image for +different entropy sources when using an image size of 512 × 512 +on the x86/GPU-based Server System. Compared to depth entropy, +we can observe that the computational costs when using lightness +become much higher. In addition, we can verify that the number +of viewpoints directly influences the computational cost. Code op- +timizations and the use of parallel processing for trying to reduce +this computational cost are planned as future works.Table 5 shows +a comparison of output images’ average entropy when varying the +number of viewpoints. Here, the utilized entropy source is Depth & +Lightness, the entropy evaluation interval is 30, and the interpolation +method is SQUAD. We can observe that the difference in the average +entropy when varying the number of viewpoints is small. +Fig. 12 shows a comparison of the accumulative distance from +the estimated camera path position to the viewpoint with the high- +est entropy, at each visualization time step, for different entropy +evaluation intervals; Fig. 13 shows a comparison of the accumula- +tive distance for different interpolation methods.Table 6 shows a +comparison of output images’ average entropy for different entropy +Table 4: Average elapsed time of entropy calculation for different +entropy sources (x86 System). +Entropy Sources +Average elapsed time [s] +Depth +2.24e-4 +Lightness +1.30e-3 +Depth & Lightness +1.53e-3 +Table 5: Average entropy when varying the number of viewpoints. +# of viewpoints +Average entropy +15×30 +3.09 +25×50 +3.10 +35×50 +3.09 +evaluation intervals. Here, the utilized entropy source is Depth & +Lightness, the number of viewpoints is 25×50, and the interpolation +method is SQUAD. In the case of NE = 1, rendered images of the +viewpoint with the highest entropy at each visualization time step +are output. From this figure and table, we can observe that as the +entropy evaluation interval increases, the accumulative distance also +increases, and the amount of average entropy decreases. Table 7 +shows the average elapsed time for path calculation between two +selected viewpoints for different interpolation methods. We can +observe that the computational cost is proportional to the number +of intervals, and the cost of SQUAD is much higher than that of +SLERP. However, it is worth noting that the influence on the total +computational cost compared to the entropy calculation cost is small +and almost neglectable. Table 8 shows a comparison of output im- +ages’ average entropy for different interpolation methods. Here, the +entropy source is Depth & Lightness, the number of viewpoints is +25×50, and the entropy evaluation interval is 30. We can observe +that when selecting SQUAD, the accumulative distance becomes +smaller and achieves a slight increase in average entropy. +Table 6: Average entropy for different entropy evaluation intervals. +Intervals (NE) +Average entropy +1 +3.17 +10 +3.12 +30 +3.10 +50 +3.08 +Table 7: Average elapsed time for path calculation between two se- +lected viewpoints using different interpolation methods (x86 System). +Interpolation +Intervals (NE) +method +10 +30 +50 +SLERP +2.80e-6 +5.16e-6 +6.93e-6 +SQUAD +9.80e-6 +3.04e-5 +3.54e-5 +Table 8: Average entropy for different interpolation methods. +Interpolation method +Average entropy +SLERP +3.09 +SQUAD +3.10 +4.2 +Discussions +Regarding the influence of different entropy sources (Fig. 6), we +observed that the lightness entropy has a higher influence than the + +Simulation step: +2400 +6000 +9600 +13200 +Depth: +Depth & Lightness: +Lightness: +Figure 6: Entropy heatmaps for different entropy sources. +Depth: +Simulation step: +2400 +6000 +9600 +13200 +Lightness: +Depth & Lightness: +Figure 7: Rendered images from the selected viewpoints. +depth entropy, and has an even higher influence when using both +depth and lightness entropy. Therefore, we opted to add both depth +and lightness entropies after normalization. In addition, due to a +large number of viewpoints with high entropy, the sequentially se- +lected viewpoints can be separated far apart from each other thus +resulting in an intense camera movement over the entire visualization +time steps. We also observed that when using the lightness entropy, +the entropy calculation took a little more time than using the depth +entropy. This was because of the necessary conversion from RGB +values to lightness values. It is worth noting that the selection of the +viewpoint evaluation metric will depend on the targeted simulation, +visualization method, and users’ analysis goals. Therefore, to satisfy +R1, it becomes important to implement a variety of viewpoint evalu- +ation metrics to handle different use case combinations. In addition, +depending on the use case, it may be helpful that different evaluation +metrics are interchangeable at run time in an adaptive manner. +Regarding the influence of the diverging color maps for the light- +ness entropy, we initially perceived almost no difference between +the heatmaps. However, there was a slight difference among them, +and at certain time steps, we observed that the selected viewpoints +were also different. Among the color maps, heatmaps for the PuOr +was especially different from the others. This may be because the +change in the lightness of the PuOr was also different from the other +diverging color maps. +Regarding the influence of the entropy evaluation intervals, as this +interval becomes smaller there will be fewer complementary images +between the selected viewpoints. As a result, changes in viewpoint +may become intense in a short period of time, this will lead to a non- +smooth video which affects the users’ post-hoc visual analysis tasks. +It is worth noting that when the simulation state is not expected +to change rapidly, there will be no necessity to frequently evaluate +the viewpoints. However, when utilizing larger entropy evaluation +intervals, a larger amount of memory will be required for temporarily +storing the simulation data. That is, there is a trade-off between the +entropy evaluation intervals and the memory consumption, and as a +result, depending on the simulation time step range and simulation +data size, large entropy evaluation intervals, such as the utilized +NE = 30 and NE = 50, may be sufficient to satisfy the R2. +Regarding the influence of the number of viewpoints on the spher- +ical surface, we verify that there was no significant difference for + +1.8 +2.6 +37.5 +2.4 +Colatitude +75.0 +2.2 +- 2.0 +112.5 +1.8 +150.0 +1.6 +1.4 +0.0 +72.0 +144.0 +216.0 +288.0 +Longitude1.8 +6.8 +37.5 +6.6 +latitude +75.0 +6.4 +Col +112.5 +6.2 +150.0 +6.0 +5.8 +0.0 +70.6 +141.2 +211.8 +282.4 +352.9 +Longitude1.8 +6.9 +37.5 +6.8 +Colatitude +6.7 +75.0 +6.6 +112.5 +6.5 +150.0 +6.4 +6.3 +0.0 +70.6 +141.2 +211.8 +282.4 +352.9 +Longitude1.8 +6.75 +37.5 +6.50 +6.25 +latitude +75.0 +6.00 +Col +112.5 +5.75 +5.50 +150.0 +5.25 +0.0 +70.6 +141.2 +211.8 +282.4 +352.9 +Longitude1.8 +3.00 +2.75 +37.5 +2.50 +Colatitude +75.0 +2.25 +2.00 +112.5 +- 1.75 +150.0 +1.50 +1.25 +0.0 +72.0 +144.0 +216.0 +288.0 +Longitude1.8 +2.8 +37.5 +2.6 +Colatitude +75.0 +2.4 +- 2.2 +112.5 +2.0 +150.0 +1.8 +0.0 +72.0 +144.0 +216.0 +288.0 +Longitude1.8 +2.8 +37.5 +2.6 +Colatitude +75.0 + 2.4 +112.5 +2.2 +2.0 +150.0 +1.8 +0.0 +72.0 +144.0 +216.0 +288.0 +Longitude1.8 +4.6 +37.5 +4.4 +latitude +4.2 +75.0 +4.0 +Col +112.5 +3.8 +150.0 + 3.6 +3.4 +0.0 +70.6 +141.2 +211.8 +282.4 +352.9 +Longitude1.8 +4.8 +37.5 +4.6 +latitude +4.4 +75.0 +4.2 +Col +112.5 +4.0 +150.0 +3.8 +3.6 +0.0 +70.6 +141.2 +211.8 +282.4 +352.9 +Longitude1.8 +4.8 +37.5 +4.7 +Colatitude +4.6 +75.0 +4.5 +112.5 +4.4 +150.0 +4.3 +4.2 +0.0 +70.6 +141.2 +211.8 +282.4 +352.9 +Longitude1.8 +4.6 +37.5 +latitude +- 4.4 +75.0 +4.2 +Col +112.5 +4.0 +150.0 +3.8 +0.0 +70.6 +141.2 +211.8 +282.4 +352.9 +Longitude1.8 +6.8 +6.6 +37.5 +6.4 +Colatitude +75.0 +6.2 +- 6.0 +112.5 +5.8 +150.0 +5.6 +5.4 +0.0 +70.6 +141.2 +211.8 +282.4 +352.9 +LongitudeCSimulation step: +2400 +6000 +9600 +13200 +25x50: +15x30: +35x70: +Figure 8: Lightness Entropy heatmaps for different diverging color maps. +varying number of viewpoints. However, it is worth noting that the +computational time required to select the viewpoints will increase +proportionately with the increase in the number of viewpoints. +Regarding the influence of the quaternion interpolation method +for estimating the camera path between selected viewpoints, we +observed that the camera path using SQUAD-based interpolation +passes closer to the viewpoint with highest entropy at the interme- +diate time steps. We also observed that jerky camera movements +tend to occur when using the SLERP-based interpolation. On the +other hand, smoother camera movement was observed when using +the SQUAD-based interpolation, and as a result, we can consider +that it will cause less discomfort to the user when seeing the ani- +mated rendering results since the camera movement will be more +natural. Therefore, we can consider that SQUAD-based quaternion +interpolation satisfies the R2. +Moreover, we carried out some evaluations with the domain sci- +entists who assisted in the development of previous work on in-situ +adaptive timestep selection [26]. We obtained technical feedback +from the generated visualization results in the form of animated +videos. According to them, the video generated by using the pro- +posed method seems to present more information than the video +generated by using fixed viewpoint camera settings, which has tradi- +tionally been used in their simulation analysis. However, they also +pointed out that the proposed video gives the impression of exces- +sive movement and sometimes tracking phenomena that do not need +much attention. As some suggestions, they mentioned that it would +be better to slightly reduce high viewpoint variations or suppress +unnecessary movement, and to improve evaluation methods for the +viewpoint selection. As an additional suggestion, they would prefer +to have the ability to zoom in on the target object to enable closer +observation. These suggestions will be taken into consideration for +further developments planned as future works. +In our current implementation, the set of volume data in the en- +tropy evaluation interval needs to be stored in the memory before +the processing, and this memory cost can become an impediment +for memory-hungry simulations. However, we consider that this +approach can be useful during test runs and model calibration runs, +before the main simulation run, when smaller models are usually +sufficient. In addition, the in-transit approach for flushing the simu- +lation data from the memory to another node or even system can be +considered helpful for minimizing this problem and is planned for +future work. Another planned future work is the application of the +adaptive timestep sampling [26] where larger time intervals will be +assigned to timestep regions with small variations between the simu- +lation results. This larger entropy evaluation time step by skipping +some simulation results may be helpful for accelerating the visu- +alization processing as well as reducing the excessive movements +pointed out by the domain scientists. +5 +CONCLUSIONS +In this work, we proposed an information entropy-based camera path +estimation method for in-situ visualization. Considering that most +of the images generated by traditional batch-based tightly coupled +in-situ visualization may have small or even no contribution for the +post-hoc visual analysis, we focused on generating a smooth video +that tries to provide as much information as possible to facilitate the +rapid understanding of the simulation or to narrow down the spatio- +temporal region of interest for posterior detailed analysis such as by +using traditional image-based visualization. The proposed method +focuses on selecting the most appropriate viewpoints, based on in- +formation entropy, at regular intervals. Intermediate images are +generated from the estimated camera path connecting these selected +viewpoints, and the produced smooth video that is produced is ex- +pected to be helpful for understanding the underlying simulation +phenomena. From the experimental evaluations and feedback from +domain scientists, we can confirm that the video generated by the +proposed approach provides more information compared to those +generated by using fixed viewpoint camera settings. However, there +is still need for improvements, and we can cite the following targets +for future works: implementation of better evaluation methods for +the viewpoint selection; implementation of zoom in and out func- +tionalities; integration with the adaptive timestep sampling (irregular +time intervals); improvement of computational performance such as +by applying parallel processing; and estimation of the focal point +for the camera. +ACKNOWLEDGMENTS +The authors are grateful to Tsukasa Yoshinaga (Toyohashi Univer- +sity of Technology) and Kazunori Nozaki (Osaka University) for +the simulation model and technical feedback. This work was par- +tially supported by JSPS KAKENHI (Grant Numbers: 20H04194, +21H04903, 22H03603), and the National Key R&D Program of +China under Grant No. 2021YFE0108400. This work used compu- +tational resources of supercomputer Fugaku provided by the RIKEN +Center for Computational Science. +REFERENCES +[1] J. Ahrens, S. Jourdain, P. O’Leary, J. Patchett, D. H. Rogers, and M. Pe- +tersen. An image-based approach to extreme scale in situ visualization +and analysis. In SC’14: Proceedings of the International Conference +for High Performance Computing, Networking, Storage and Analysis, +pp. 424–434. IEEE, 2014. +[2] U. Ayachit, A. Bauer, B. Geveci, P. O’Leary, K. Moreland, N. Fabian, +and J. Mauldin. Paraview Catalyst: Enabling in situ data analysis +and visualization. In Proceedings of the First Workshop on In Situ +Infrastructures for Enabling Extreme-Scale Analysis and Visualization, +pp. 25–29, 2015. + +1.8 +4.6 +37.5 +4.4 +latitude +4.2 +75.0 +4.0 +Col +112.5 +3.8 +150.0 + 3.6 +3.4 +0.0 +70.6 +141.2 +211.8 +282.4 +352.9 +Longitude1.8 +4.8 +37.1 +4.6 +latitude +74.1 +4.4 +4.2 +111.2 +4.0 +148.2 +3.8 +3.6 +0.0 +72.0 +144.0 +216.0 +288.0 +Longitude1.8 +4.8 +37.1 +4.7 +Colatitude +4.6 +74.1 +4.5 +111.2 +4.4 +148.2 +4.3 +4.2 +0.0 +72.0 +144.0 +216.0 +288.0 +Longitude1.8 +4.6 +37.1 +latitude +4.4 +74.1 +4.2 +8 +111.2 +4.0 +148.2 +3.8 +0.0 +72.0 +144.0 +216.0 +288.0 +Longitude1.8 +2.6 +37.5 +2.4 +Colatitude +75.0 +2.2 +- 2.0 +112.5 +1.8 +150.0 +1.6 +1.4 +0.0 +72.0 +144.0 +216.0 +288.0 +Longitude1.8 +4.8 +37.5 +4.6 +latitude +4.4 +75.0 +4.2 +Col +112.5 +4.0 +150.0 +3.8 +3.6 +0.0 +70.6 +141.2 +211.8 +282.4 +352.9 +Longitude1.8 +4.8 +37.5 +4.7 +Colatitude +4.6 +75.0 +4.5 +112.5 +4.4 +150.0 +4.3 +4.2 +0.0 +70.6 +141.2 +211.8 +282.4 +352.9 +Longitude1.8 +4.6 +37.5 +latitude +- 4.4 +75.0 +4.2 +Col +112.5 +4.0 +150.0 +3.8 +0.0 +70.6 +141.2 +211.8 +282.4 +352.9 +Longitude1.8 - +4.6 +38.6 +4.4 +Colatitude +4.2 +77.1 +4.0 +115.7 +3.8 +154.3 +3.6 +3.4 +0.0 +72.0 +144.0 +216.0 +288.0 +Longitude1.8 - +4.8 +38.6 +4.6 +Colatitude +4.4 +77.1 +4.2 +115.7 +4.0 +154.3 +3.8 +3.6 +0.0 +72.0 +144.0 +216.0 +288.0 +Longitude1.8 - +4.8 +38.6 +4.7 +Colatitude +4.6 +77.1 +4.5 +115.7 +4.4 +4.3 +154.3 +4.2 +0.0 +72.0 +144.0 +216.0 +288.0 +Longitude1.8 +4.6 +38.6 +Colatitude +4.4 +77.1 +4.2 +115.7 +4.0 +154.3 +3.8 +0.0 +72.0 +144.0 +216.0 +288.0 +Longitude1.8 +4.6 +37.1 +4.4 +latitude +4.2 +74.1 +4.0 +Col +111.2 +3.8 + 3.6 +148.2 +3.4 +0.0 +72.0 +144.0 +216.0 +288.0 +LongitudeFigure 9: Rendered images from the selected viewpoints for different diverging color maps. +[3] P. Barral, G. Dorme, and D. Plemenos. Scene understanding techniques +using a virtual camera. 2000. +[4] R. E. Blahut. Principles and practice of information theory. Addison- +Wesley Longman Publishing Co., Inc., 1987. +[5] U. Bordoloi and H.-W. Shen. View selection for volume rendering. +In VIS 05. IEEE Visualization, 2005., pp. 487–494, 2005. doi: 10. +1109/VISUAL.2005.1532833 +[6] H. Childs, S. D. Ahern, J. Ahrens, A. C. Bauer, J. Bennett, E. W. Bethel, +P.-T. Bremer, E. Brugger, J. Cottam, M. Dorier, et al. A terminology for +in situ visualization and analysis systems. The International Journal of +High Performance Computing Applications, 34(6):676–691, 2020. +[7] G. Ji and H.-W. Shen. Dynamic view selection for time-varying vol- +umes. IEEE Transactions on Visualization and Computer Graphics, +12(5):1109–1116, 2006. doi: 10.1109/TVCG.2006.137 +[8] A. Kageyama and N. Sakamoto. +4D street View: A video-based +visualization method. PeerJ Computer Science, (6:e305), 2020. +[9] A. Kageyama and T. Yamada. An approach to exascale visualiza- +tion: Interactive viewing of in-situ visualization. Computer Physics +Communications, 185(1):79–85, 2014. +[10] T. Kamada and S. Kawai. A simple method for computing general +position in displaying three-dimensional objects. Computer Vision, +Graphics, and Image Processing, 41(1):43–56, 1988. +[11] T. Kuhlen, R. Pajarola, and K. Zhou. Parallel in situ coupling of +simulation with a fully featured visualization system. In Proceedings +of the 11th Eurographics Conference on Parallel Graphics and Visu- +alization (EGPGV), vol. 10, pp. 101–109. Eurographics Association +Aire-la-Ville, Switzerland, 2011. +[12] J. Ling, W. P. Kegelmeyer, K. Aditya, H. Kolla, K. A. Reed, T. M. +Shead, and W. L. Davis. Using feature importance metrics to detect +events of interest in scientific computing applications. In 2017 IEEE +7th Symposium on Large Data Analysis and Visualization (LDAV), pp. +55–63. IEEE, 2017. +[13] K.-L. Ma, J. S. Painter, C. D. Hansen, and M. F. Krogh. Parallel volume +rendering using binary-swap compositing. IEEE Computer Graphics +and Applications, 14(4):59–68, 1994. +[14] N. Marsaglia, Y. Kawakami, S. D. Schwartz, S. Fields, and H. Childs. +An entropy-based approach for identifying user-preferred camera po- +sitions. In 2021 IEEE 11th Symposium on Large Data Analysis and +Visualization (LDAV), pp. 73–83. IEEE, 2021. +[15] N. Marsaglia, M. Majumder, and H. Childs. A trigger-based approach +for optimizing camera placement over time. 9 2022. +[16] K. Moreland. Diverging color maps for scientific visualization. In +International Symposium on Visual Computing, pp. 92–103. Springer, +2009. + +Simulation step: +2400 +6000 +9600 +13200 +25x50: +15x30: +35x70: +Figure 10: Entropy heatmaps when varying the number of viewpoints. +Simulation step: +2400 +6000 +9600 +13200 +25x50: +15x30: +35x70: +Figure 11: Rendered images from the selected viewpoints when varying the number of viewpoints. +Figure 12: Accumulative distance from the interpolated camera path +to the viewpoints, with the highest entropy, at each visualization time +step, for different entropy evaluation intervals. +[17] J. Nonaka, K. Ono, and M. Fujita. 234Compositor: A flexible parallel +image compositing framework for massively parallel visualization +environments. Future Generation Computer Systems, 82:647–655, +Figure 13: Accumulative distance to the viewpoints, with the highest +entropy, at each visualization time step, for different interpolation +methods. +2018. +[18] D. L. Page, A. F. Koschan, S. R. Sukumar, B. Roui-Abidi, and M. A. +Abidi. Shape analysis algorithm based on information theory. In +Proceedings 2003 International Conference on Image Processing (Cat. +No. 03CH37429), vol. 1, pp. I–229. IEEE, 2003. +[19] O. Polonsky, G. Patan´e, S. Biasotti, C. Gotsman, and M. Spagnuolo. + +1.8 +4.6 +37.5 +4.4 +latitude +4.2 +75.0 +4.0 +Col +112.5 +3.8 +150.0 + 3.6 +3.4 +0.0 +70.6 +141.2 +211.8 +282.4 +352.9 +Longitude1.8 +4.8 +37.1 +4.6 +latitude +74.1 +4.4 +4.2 +111.2 +4.0 +148.2 +3.8 +3.6 +0.0 +72.0 +144.0 +216.0 +288.0 +Longitude1.8 +4.8 +37.1 +4.7 +Colatitude +4.6 +74.1 +4.5 +111.2 +4.4 +148.2 +4.3 +4.2 +0.0 +72.0 +144.0 +216.0 +288.0 +Longitude1.8 +4.6 +37.1 +latitude +4.4 +74.1 +4.2 +8 +111.2 +4.0 +148.2 +3.8 +0.0 +72.0 +144.0 +216.0 +288.0 +Longitude1.8 +2.6 +37.5 +2.4 +Colatitude +75.0 +2.2 +- 2.0 +112.5 +1.8 +150.0 +1.6 +1.4 +0.0 +72.0 +144.0 +216.0 +288.0 +Longitude1.8 +4.8 +37.5 +4.6 +latitude +4.4 +75.0 +4.2 +Col +112.5 +4.0 +150.0 +3.8 +3.6 +0.0 +70.6 +141.2 +211.8 +282.4 +352.9 +Longitude1.8 +4.8 +37.5 +4.7 +Colatitude +4.6 +75.0 +4.5 +112.5 +4.4 +150.0 +4.3 +4.2 +0.0 +70.6 +141.2 +211.8 +282.4 +352.9 +Longitude1.8 +4.6 +37.5 +latitude +- 4.4 +75.0 +4.2 +Col +112.5 +4.0 +150.0 +3.8 +0.0 +70.6 +141.2 +211.8 +282.4 +352.9 +Longitude1.8 - +4.6 +38.6 +4.4 +Colatitude +4.2 +77.1 +4.0 +115.7 +3.8 +154.3 +3.6 +3.4 +0.0 +72.0 +144.0 +216.0 +288.0 +Longitude1.8 - +4.8 +38.6 +4.6 +Colatitude +4.4 +77.1 +4.2 +115.7 +4.0 +154.3 +3.8 +3.6 +0.0 +72.0 +144.0 +216.0 +288.0 +Longitude1.8 - +4.8 +38.6 +4.7 +Colatitude +4.6 +77.1 +4.5 +115.7 +4.4 +4.3 +154.3 +4.2 +0.0 +72.0 +144.0 +216.0 +288.0 +Longitude1.8 +4.6 +38.6 +Colatitude +4.4 +77.1 +4.2 +115.7 +4.0 +154.3 +3.8 +0.0 +72.0 +144.0 +216.0 +288.0 +Longitude1.8 +4.6 +37.1 +4.4 +latitude +4.2 +74.1 +4.0 +Col +111.2 +3.8 + 3.6 +148.2 +3.4 +0.0 +72.0 +144.0 +216.0 +288.0 +LongitudeAccumulativeDistance +7000 +interval: 10 +interval: 30 +6000 +interval: 50 +ance +Distal +5000 +Accumulative +4000 +3000 +2000 +1000 +0 +0 +2000 +4000 +6000 +8000100001200014000 +TimestepAccumulativeDistance +interval: 10 +6000 +interval: 30 +interval: 50 +4000 +2000 +0 +2500 +5000 +7500 +10000 +12500 +15000AccumulativeDistance +10000 +SLERP +SQUAD +Distance +8000 +Accumulative +6000 +4000 +2000 +0 +0 +2000 +4000 +6000 +8000 +10000 +1200014000 +Time stepAccumulativeDistance +10000 +SLERP +SQUAD +7500 +5000 +2500 +0 +0 +2500 +5000 +7500 +100001250015000What’s in an image? The Visual Computer, 21(8):840–847, 2005. +[20] N. Sakamoto and K. Koyamada. KVS: A simple and effective frame- +work for scientific visualization. Journal of Advanced Simulation in +Science and Engineering, 2(1):76–95, 2015. +[21] A. Secord, J. Lu, A. Finkelstein, M. Singh, and A. Nealen. Perceptual +models of viewpoint preference. +ACM Transactions on Graphics +(TOG), 30(5):1–12, 2011. +[22] S. Takahashi, I. Fujishiro, Y. Takeshima, and T. Nishita. A feature- +driven approach to locating optimal viewpoints for volume visualiza- +tion. In VIS 05. IEEE Visualization, 2005., pp. 495–502, 2005. doi: 10. +1109/VISUAL.2005.1532834 +[23] P.-P. V´azquez, M. Feixas, M. Sbert, and W. Heidrich. Viewpoint selec- +tion using viewpoint entropy. In VMV, vol. 1, pp. 273–280. Citeseer, +2001. +[24] P.-P. V´azquez, M. Feixas, M. Sbert, and W. Heidrich. Automatic view +selection using viewpoint entropy and its application to image-based +modeling. In Computer Graphics Forum, vol. 22, pp. 689–700. Wiley +Online Library, 2003. +[25] P.-P. V´azquez, M. Feixas, M. Sbert, and A. Llobet. Viewpoint en- +tropy: A new tool for obtaining good views of molecules. In ACM +International Conference Proceeding Series, vol. 22, pp. 183–188, +2002. +[26] Y. Yamaoka, K. Hayashi, N. Sakamoto, and J. Nonaka. In situ adap- +tive timestep control and visualization based on the spatio-temporal +variations of the simulation results. In Proceedings of the Workshop +on In Situ Infrastructures for Enabling Extreme-Scale Analysis and +Visualization, pp. 12–16, 2019. +[27] T. Yoshinaga, K. Nozaki, and S. Wada. Experimental and numerical +investigation of the sound generation mechanisms of sibilant fricatives +using a simplified vocal tract model. Physics of Fluids, 30(3):035104, +2018. +[28] Y. Zhang and B. Wang. Optimal viewpoint selection for volume ren- +dering based on shuffled frog leaping algorithm. In 2010 IEEE Inter- +national Conference on Progress in Informatics and Computing, vol. 2, +pp. 706–709. IEEE, 2010. + diff --git a/8NFJT4oBgHgl3EQfnSyO/content/tmp_files/load_file.txt b/8NFJT4oBgHgl3EQfnSyO/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..de761aebc5f14bbe0b09b8f7af3e74dd13755d7e --- /dev/null +++ b/8NFJT4oBgHgl3EQfnSyO/content/tmp_files/load_file.txt @@ -0,0 +1,1421 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf,len=1420 +page_content='Information Entropy-based Camera Path Estimation for In-Situ Visualization Ken Iwata* Kobe University Naohisa Sakamoto† Kobe University Jorji Nonaka‡ RIKEN R-CCS Chongke Bi§ Tianjin University Information Entropy (for viewpoint selection) Depth and Lightness Information 𝑑!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='𝑑"𝑑#⋯ 𝑑"$% ⋯ 𝐷𝑒𝑝𝑡ℎ 𝑙!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 𝑙" 𝑙# ⋯ 𝑙"$% ⋯ 𝐿𝑖𝑔ℎ𝑡𝑛𝑒𝑠𝑠 Selected Viewpoint “A” Camera Path Selected Viewpoint “A” Selected Viewpoint “B” Intermediate Images Selected Image “A” Selected Image “B” Selected Image “A” ︓Viewpoints Rendering Image (Information Source) Figure 1: Depth and lightness entropy-based viewpoint selection and camera path estimation for generating a smooth video, with as much information as possible, to assist the rapid understanding of the underlying simulation phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' ABSTRACT In-situ processing has widely been recognized as an effective ap- proach for the visualization and analysis of large-scale simulation outputs from modern HPC systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' One of the most common approaches for batch-based in-situ visualization is the image- or video-based approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' In this kind of approach, a large number of rendered images are generated from different viewpoints at each time step and has proven useful for detailed analysis of the main simulation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' However, during test runs and model calibration runs before the main simulation run, a quick overview might be sufficient and useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' In this work, we focused on selecting the viewpoints which provide as much information as possible by using information entropy to maximize the subsequent visual analysis task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' However, by simply following the selected viewpoints at each of the visualization time steps will probably lead to a rapidly changing video, which can impact the understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Therefore, we have also worked on an efficient camera path estimation approach for connecting selected viewpoints, at regular intervals, to generate a smooth video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' This resulting video is expected to assist in rapid understanding of the underlying simulation phenomena and can be helpful to narrow down the temporal region of interest to min- imize the turnaround time during detailed visual exploration via image- or video-based visual analysis of the main simulation run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' We implemented and evaluated the proposed approach using the OpenFOAM CFD application, on an x86-based Server and an ARM A64FX-based supercomputer (Fugaku), and we obtained positive evaluations from domain scientists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Index Terms: Human-centered computing—Visualization— e-mail: 228x202x@stu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='kobe-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='jp †e-mail: naohisa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='sakamoto@people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='kobe-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='jp ‡e-mail: jorji@riken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='jp §e-mail: bichongke@tju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='cn Visualization systems and tools—Visualization toolkits 1 INTRODUCTION High-end high performance computing (HPC) systems have contin- uously become more and more capable with higher computational capacity with every new system replacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' This was the case for the replacement of the K computer to the supercomputer Fugaku at the RIKEN R-CCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' The increased number of CPUs and computa- tional cores have been applied for Capability Computing to tackle even larger numerical simulations with higher spatio-temporal reso- lutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' In addition, this also has been used for Capacity Computing to handle an even larger number of parameters and members during parametric sweep and ensemble simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' On the other hand, this proportionately generates even larger simulation outputs, thus, making the visualization and analysis tasks even more challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' As a result, the importance of in-situ visualization and analysis has continuously become even more evident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' A variety of approaches have already been proposed and applied for the in-situ visualization and analysis as discussed in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' We can also verify that there are also a variety of existing applications and libraries for realizing in-situ visualization and analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' However, since in-situ processing is executed simultaneously with the simu- lation, it becomes highly important to collaborate with the domain scientists .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' We have been working with domain scientists working with computational fluid dynamics (CFD) simulation of the sound generation mechanisms [27], and we already worked on an adaptive in situ time-step sampling approach [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' In this work, we have used the same OpenFOAM CFD application and simulation model and obtained assistance from them for necessary technical feedback during the developments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Probably the most widely used image-based in-situ visualization approach is ParaView Cinema [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' In that approach, a large set of pre-computed images are generated in-situ on the HPC system side for the interactive post-hoc visual exploration on a local machine such as desktop PC and laptop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' There is also an image-based in-situ visualization approach that generates a set of images from omni- directional camera positions [9], and its extension for video-based arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='11591v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='GR] 27 Jan 2023 in-situ visualization [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' These image- or video-based in-situ visual- ization approaches have proven useful for detailed analysis of the main simulation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' In this work, we have focused on rapid un- derstanding of the underlying simulation during test runs and model calibration runs before the main simulation run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' For this purpose, we focused on selecting the most appropriate viewpoints, based on information entropy, at regular time intervals of the simulation in order to obtain as much information as possible trying to facilitate the rapid understanding of such kinds of simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 2 RELATED WORK There is an extensive work that culminated in the creation of a classi- fication and terminology for the in-situ visualization approaches [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Here, we will only focus on related works for realizing tightly cou- pled in-situ visualization, and techniques for selecting time steps and viewpoints that can be used for minimizing the amount of im- ages for the image- or video-based in-situ visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' VTK-based ParaView and VisIt are probably the most widely used visualization application for large data visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Both applications provide in- situ visualization APIs, ParaView Catalyst [2] and VisIt LibSim [11], for integrating to the simulation code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' In a batch-based in-situ visu- alization, a large amount of images can be generated for assisting the post-hoc visualization [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' To facilitate this post-hoc visual anal- ysis, Ahrens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' [1] proposed an image-based approach for the in-situ visualization and analysis, and was implemented as Paraview Cinema.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' In this approach a large set of images are generated in- situ, and a custom visualization application is used, on the local machine, to perform interactive visual analysis by automatically switching between the generated set of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Similar to this ap- proach, Kageyama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' [8] proposed a video-based approach by generating an omnidirectional animated video, from the set of in-situ generated images, which are explorable from a custom visualization application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Although these approaches have proven efficient, most of the generated images may have small or even no contribution to the visual analysis, thus it may be unnecessarily increasing the time spent on the post-hoc visual analysis task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' An approach to minimize the aforementioned amount of gen- erated images is the selection of the most valuable time steps for rendering the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' For this purpose, Ling et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' [12] proposed a method to estimate the probability density function of the simulation field, at each time step, by using the kernel density estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' They also applied machine learning for extracting feature quantity from the obtained estimation, and detected potentially valuable time steps where an important phenomena may occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' However, this method can cause false detections depending on the high correlation among the physical quantities on the simulation field as mentioned by the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Yamaoka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' [26] extend the aforementioned work, and proposed an adaptive time sampling method for in-situ visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' In this method, kernel density function and Kullback–Leibler diver- gence is applied to estimate the amount of change on the simulation field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' The sampling intervals are adaptively changed according to the estimated amount of change in the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Another approach for reducing the amount of images is the se- lection of viewpoints for generating the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' For this purpose, Kamada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' [10] considered the viewpoints capable of minimizing the number of degenerated face as being the optimal viewpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' However, they did not extend their work for the case when there exist multiple viewpoints with the same number of degenerated faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Barral et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' [3] solved this problem by adding the projected area as a weight to the number of degenerated faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' However, there still remains a problem on how to properly set these weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Vazquez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' [23] proposed a method to select the optimal view- point defined by the viewpoint entropy based on the information entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Since this method cannot handle the movement of view- points, the authors improved the viewpoint entropy and applied it to molecular objects [25] as well as to image-based modeling [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Page et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' [18] proposed a method to analyze the object shape by calculating the entropy for the silhouette and surface curvature of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Polonsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' [19] discussed evaluation indices for the viewpoint selection, and concluded that none of them could make the best choice in any situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' However, they also said that by improving each of these indices, it will become possible to make a better choice by using a combination of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Secord et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' [21] proposed some evaluation indices for the view- point selection, and showed that optimal viewpoints can be selected by combining these metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Takahashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' [22] proposed a method for estimating the optimal viewpoint for volume data by using in- formation entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Bordoloi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' [5] proposed an information entropy-based evaluation metric for the viewpoints during volume rendering by using the transfer function, data distribution, and voxel visibility information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' [28] also proposed an evaluation metric for volume rendering based on the opacity, brightness, and structural features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Ji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' [7] proposed a method to find the optimal time-varying views by using the viewpoint selection method to max- imize the amount of information for time-series volume data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' They showed that it is possible to create an animation with the largest amounts of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' This was realized by searching for a move- ment route with the largest amounts of information using dynamic programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Marsaglia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' [14] proposed a viewpoint quality evaluation metric based on information entropy involving the visible field data, depth, and shading values belonging to each of the pixels in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' In another work, they also utilized a trigger-based ap- proach in combination with information entropy to determine when to search for a new camera position as a simulation evolves [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Our work was inspired in their viewpoint quality evaluation metric, which we extended with the lightness information for evaluating the viewpoint quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' We will detail the methodology behind our proposed method in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 3 METHODOLOGY 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='1 Overview In this work, we focused on a viewpoint selection approach, based on information entropy, and on a camera path estimation approach, based on quaternion interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' The viewpoints selected at regular intervals are used as markers to estimate the smooth camera path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Following are the necessary requirements to meet this goal: R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Images from the selected viewpoints should capture important phenomenon from the underlying simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' The resulting video generated from the rendered outputs should be smooth for post-hoc analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Below is the adopted approach to satisfy the aforementioned requirements, and they are divided into the following three parts: A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Viewpoint evaluation The viewpoint quality will be evaluated using information entropy and will be used to select the most appropriate image for each evaluated time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Only images from the selected viewpoints will be output (R1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Camera path estimation The camera path connecting these selected viewpoints will be estimated, and the rendered images through this camera path will also be output as intermediate images (R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Video generation At the end of the simulation, these output images will be se- quentially concatenated to produce a video (R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Regarding part A, the simulation state usually does not often significantly change within a single simulation time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Therefore, Simulation time step Execute simulation Visualization time step Store simulation data Entropy evaluation time step Select optimal viewpoint Calculate camera path Times and Intervals Simulation time 𝑇!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' = 𝑖∆𝑇 Visualization time 𝑇" !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' = 𝑖∆𝑇" = 𝑖𝑁"∆𝑇 Entropy evaluation time 𝑇# !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' = 𝑖∆𝑇# = 𝑖𝑁"𝑁#∆𝑇 data1 time [ 𝑇" ] ∆𝑇" time [ 𝑇# ] ∆𝑇# time [ 𝑇 ] ∆𝑇 data2 data3 Rendering Selected Viewpoint “A” Selected Viewpoint “B” Intermediate Images Figure 2: Different time step intervals for the simulation, visualization and entropy evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' there is usually no need to visualize at every simulation time step, and the visualization can be performed at every set of simulation time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' In the same manner, the viewpoint evaluation for viewpoint selection will be performed at every set of visualization time steps to satisfy R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' In this paper, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 2, we use T to represent the simulation time step, TV to represent the visualization time step, and TE to represent the entropy evaluation time step, ∆T to represent the simulation time step interval, ∆TV to represent the visualization time step interval, and ∆TE to represent the entropy evaluation time step interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' The elapsed time for the ith simulation time step T i can be expressed as T i = i∆T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' In the same manner, the visualization time step and entropy evaluation time steps can be expressed respectively as T i V = i∆TV and T i E = i∆TE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Considering that the simulation is performed NV times for every visualization, and the visualization is performed NE times for every optimal viewpoint selection, then we can express these intervals as ∆TV = NV ∆T and ∆TE = NENV ∆T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Regarding part B, a camera path connecting viewpoints selected at every ∆TE entropy evaluation time step will be estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Although visualization is not performed during the visualization time steps in between the entropy evaluation time steps, the simulation data for each ∆TV visualization time step is stacked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' From the obtained camera path, the rendered images at the intermediate visualization time steps will be output as the intermediate images for generating a smooth video, and this satisfies R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' It is worth noting that it is also possible to generate the full set of images from the entire viewpoints for the detailed post-hoc analysis when necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Regarding part C, the set of output images generated at each ∆TV visualization time step will be joined sequentially to create a video file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' For this purpose, we can use existing tools such as the well- known FFmpeg available to a variety of platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' In the resulting video, the camera will automatically move and capture the important phenomenon, and this allows the R2 to be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Traditional approach requires the user to search for the best location, in the trial-and-error manner, to visually explore when searching for an important phenomenon during the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' However, by using the proposed method, this search for the best camera position may be alleviated and may facilitate narrowing down the spatio-temporal region of interest for the detailed visual analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='2 Viewpoint Selection In this section, we will detail the utilized viewpoint selection ap- proach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' The evaluation of the viewpoints is based on information entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' We used depth and lightness values from the rendered images for calculating the associated information entropy, that is, the depth entropy and lightness entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='1 Information Entropy Information entropy used in this work can be defined as the expected value for the amount of information obtained from a certain infor- mation source [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' The information entropy H(X) from a source X given by the set of probabilities P(x1),P(x2),··· ,P(xn) corre- sponding respectively to the set of information x1,x2,··· ,xn can be represented as follows: H(X) = − n ∑ i=1 P(xi)logP(xi) (1) Here, when P(xi) = 0, we will consider P(xi)logP(xi) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Re- garding the selection of the logarithm’s base, the base influences the multiplication factor and, thus, is arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Base 2 is commonly used in information theory, and was used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Taking into consideration the probability distribution of the information source, the larger the information bias, the smaller the value of information entropy, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='2 Depth Entropy Depth entropy used in this work is based on the viewpoint quality evaluation metric proposed by Marsaglia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' The information entropy is calculated by considering the image as the source of information, and by using the depth values belonging to each of the pixels in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' The depth values can vary in the range of 0 ∼ 1, and the closer the distance to the object, the smaller the value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' The background portion in the image where there is no object is set to infinity and will have their depth values corresponding to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' The depth values from all pixels of the rendered image are binned into 256 groups d0,d1,··· ,d255 for creating a discrete probability distribution D, which will be used to calculate the information en- tropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' At this time, the background portion in the image is considered to have no information, and only the pixels with some information will be used in the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' By using the discrete probability distribution, the depth entropy Hd can be calculated as follows: Hd = − 255 ∑ i=0 D(di)log2 D(di) (2) Here, D(di) corresponds to the probability for a given value, selected based on the probability distribution D, being di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' When evaluating a viewpoint using depth entropy, the resulting value will be larger for images with large dispersion in the distribution of depth values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Therefore, the viewpoints of images showing objects with high undulations will be highly evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='3 Lightness Entropy In this work, in addition to the depth entropy, we propose the use of lightness entropy to also take into consideration the color informa- tion in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Diverging color maps, proposed by Moreland [16], have become prevalent in scientific visualization as the substitute for the traditional but problematic rainbow color map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Although the change in color values, such as RGB values, in a color map may follow different behavior depending on the color map, diverging color maps usually show similar behavior in the lightness values in CIELAB color space as shown in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Therefore, lightness entropy is expected to work robustly for the diverging color maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' The proposed lightness entropy can be defined as an information en- tropy using the lightness values from the target image as the source of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' The lightness value is calculated from RGB values and can vary in the range of 0 ∼ 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' PiYG RdBu PuOr Lightness L* Figure 3: Lightness values for different diverging color maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' The lightness values, as well as the depth values, obtained from all pixels of the rendered image, are binned into 256 groups l0,l1,··· ,l255 for creating a discrete probability distribution L, which will be used to calculate the information entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' At this time, the background portion in the image is considered to have no infor- mation, and only the pixels with some information will be used in the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' By using the discrete probability distribution, the lightness entropy Hl can be calculated as follows: Hl = − 255 ∑ i=0 L(li)log2 L(li) (3) Here, L(li) corresponds to the probability for a given value, se- lected based on the probability distribution L, being li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' When a viewpoint is evaluated using the lightness entropy, the resulting value will be larger for images with large dispersion in the distribu- tion of lightness values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Therefore, the viewpoints of images with clear brightness and darkness will be highly evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='3 Path Estimation between Selected Viewpoints In this section, we will detail the utilized camera path estimation between the viewpoints selected by using the depth and/or lightness entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' In this work, we considered that the viewpoints are pre- arranged in a spherical surface as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 4, and the camera path from one viewpoint to another will move over this spherical surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' More specifically, the position and orientation of a given viewpoint will be represented as a quaternion, and the movement from one to another viewpoint will be obtained by using quaternion interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' In this work, we investigated two quaternion interpola- tion methods: spherical linear interpolation (SLERP) and spherical quadrangle interpolation (SQUAD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' In the following subsections, we will explain about spherical linear interpolation and spherical quadrangle interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='1 Spherical Linear Interpolation (SLERP) SLERP is an abbreviation for spherical linear interpolation, and is a quaternion interpolation method for connecting two points over a sphere in the straight line direction, or the shortest path, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' SLERP-based interpolation from a quaternion qA to the quaternion qB can be calculated by using time t ∈ [0, 1] as follows: : Viewpoints Figure 4: Viewpoint distribution over spherical surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' slerp(qA, qB, t) = sin(1−t)φ sinφ qA + sintφ sinφ qB (4) Here, φ = arccos⟨qA, qB⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' In addition, in the case of ⟨qA, qB⟩ < 0, the interpolation will be interpolated in the contrary direction over the sphere surface, that is by the longest path in the straight-line direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' To interpolate by the shortest path, then either qA or qB should be replaced with a quaternion with same rotation but in the opposite direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' For instance, by replacing qA with −qA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='2 Spherical Quadrangle Interpolation (SQUAD) SQUAD is an abbreviation for spherical quadrangle interpolation, and is a quaternion interpolation method to connect multiple points in a smoothness way so that the derivatives are continuous in the neighborhood of the points (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Considering a quaternion list {q1, q2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' qn},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' then the SQUAD-based interpolation from qi to qi+1 can be calculated by using the time t ∈ [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 1] as follows: squad(qi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' qi+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' ai+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' t) = slerp(slerp(qi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' qi+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' slerp(ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' ai+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 2t(1−t)) (5) ai = qi exp � −logq∗ i qi−1 +logq∗ i qi+1 4 � (6) Here,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' the exponential of the quaternion exp(q) and the logarithm of the quaternion logq for the quaternion q = a+bi+c j +dk are defined as follows: exp(q) = ea � cos∥bi+c j +dk∥+ bi+c j +dk ∥bi+c j +dk∥ sin∥bi+c j +dk∥ � (7) logq = log∥q∥+ bi+c j +dk ∥bi+c j +dk∥ arctan ∥bi+c j +dk∥ a (8) In addition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' in the case of performing SQUAD-based interpolation from q1 to q2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' and from qn−1 to qn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' we consider q0 = q1 and qn+1 = qn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Unit Sphere in Quaternion Space : SLERP : SQUAD : Rotation Quaternion Figure 5: Comparison of SLERP and SQUAD interpolation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 7 100 80 60 40 RdBu 20 PiYG PuOr3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='3 Implementation We utilized the Kyoto Visualization System (KVS) [20] for imple- menting the proposed viewpoint selection approach, based on depth and lightness entropy, as well as the sequential camera path between the selected viewpoints via SLERP- and SQUAD-based interpolation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' KVS is a cross-platform, open-source C++ visualization library capable of running on a variety of hardware systems from traditional x86/GPU systems to GPU-less HPC systems including IBM Blue Gene L/P (PowerPC), K computer (SPARC VIIIfx), and Fugaku (ARM A64FX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' KVS supports hybrid MPI/OpenMP paral- lelism and implements a sort-last parallel image composition method based on Binary-Swap [13], with an extension to support non-power- of-two number of nodes, which is named 234Compositor [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' The pseudocode of our implementation, using the SQUAD-based interpolation, is described in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' In this pseudocode, I[], V[], and Q[] respectively represent the queues for storing the out- put image, simulation data, and the quaternion for the selected viewpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' In addition, is initial step(t) and is final step(t) are functions that respectively return the true information in the first and the final time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' The Vis(V, q) function renders the simu- lation data V from the viewpoint represented by the quaternion of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Entropy Evaluation(V) is the function that calculates the en- tropy for the simulation data V at all pre-arranged viewpoints on the spherical surface, and returns the quaternion information of the viewpoint with highest entropy value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Its pseudocode is described in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' In this pseudocode, L represents the set of viewpoints and the read back(V, l) is a function that renders the simulation data V from the viewpoint l and returns its frame buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' entropy( f) is a function that calculates the entropy for the frame buffer f and re- turns its value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' quaternion() is a function that returns the quaternion information from a given viewpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 4 EXPERIMENTAL EVALUATIONS We used the OpenFOAM CFD code and model for the experimental evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' The simulation model used for the evaluations was obtained from our collaborators [27], and refers to a sound propaga- tion in the oral cavity by using irregular volume data composed of 3,197,279 hexahedral elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' We integrated the in situ KVS mod- ule to the OpenFOAM code, and evaluated on two systems shown in Tables 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' The irregular volume data was decomposed into 8 blocks for the x86 Server, and up to 1,024 blocks for the Fugaku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Table 1: x86/GPU-based Server System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Nodes 1 CPU Intel Xeon Gold 6238R 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='20GHz 28Core×2 Cores 28×2 = 56 RAM 384 GB DRAM GPU NVIDIA Quadro RTX8000 Compiler GCC version 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='0 MPI OpenMPI 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='1 Table 2: ARM-based Supercomputer Fugaku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Nodes 158,976 CPU Fujitsu A64FX (Armv8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='2-A SVE) Cores 48 + 2 Assistant Cores RAM 32GB HBM2 Compiler GCC-based Fujitsu Compiler Ver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='0 MPI OpenMPI with Fujitsu expensions for Tofu Algorithm 1 In-situ visualization (using SQUAD interpolation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 1: function IN SITU VISUALIZATION(∆TV , ∆TE) 2: I[], V[], Q[], t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 3: while t ≤ tend do 4: Vt = Sim();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 5: if is initial step(t) then 6: Qt = Entropy Evaluation(Vt);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 7: Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='push(Qt);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='push(Qt);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 8: else if t%∆TV == 0 then 9: if t%∆TE == 0 then 10: Qt = Entropy Evaluation(Vt);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 11: Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='push(Qt);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 12: if Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='size() == 4 then 13: q1 = Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='front();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='pop();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 14: q2 = Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='front();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='pop();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 15: q3 = Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='front();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='pop();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 16: q4 = Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='front();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='pop();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 17: for i = 0, 1, ··· , ∆TE −1 do 18: s = i/TE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 19: qs = squad(q1, q2, q3, q4, s);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 20: Vs = V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' front();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='pop();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 21: Is = Vis(Vs, qs);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 22: I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='push(Is);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 23: end for 24: end if 25: end if 26: end if 27: if is final step(t) then 28: q1 = Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='front();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='pop();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 29: q2 = Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='front();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='pop();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 30: q3 = Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='front();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='pop();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 31: q4 = q3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 32: for i = 0, 1, ··· , ∆TE −1 do 33: s = i/TE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 34: qs = squad(q1, q2, q3, q4, s);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 35: Vs = V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' front();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='pop();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 36: Is = Vis(Vs, qs);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 37: I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='push(Is);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 38: end for 39: while V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='size() > 0 do 40: Vs = V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' front();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='pop();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 41: Is = Vis(Vs, q3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 42: I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='push(Is);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 43: end while 44: end if 45: t ++;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 46: end while 47: return I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 48: end function 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='1 Some Results For the initial experimental evaluations, we selected the pressure variable and used multi-isosurface rendering with three distinct iso- values that are rendered as different colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' The total number of simulation time steps for the utilized CFD model was 15,000, and we used the parameters shown in Table 3 for the evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' We evaluated the use of our proposed lightness entropy (Lightness) in ad- dition to the depth entropy (Depth) proposed by Marsaglia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' [14], and also the use of the average of depth and lightness entropy (Depth & Lightness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' For the use of only lightness entropy, we experi- mented with three diverging color maps (RdBu, PiYG, PuOr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' We selected three entropy evaluation intervals (10, 30, 50), which repre- sent the visualization time step interval for performing the entropy calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' We also selected three sets of viewpoints with differ- Algorithm 2 Entropy Evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 1: function ENTROPY EVALUATION(V) 2: E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 3: Q = 1+0i+0j +0k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 4: for l ∈ L do 5: f = read back(V, l);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 6: e = entropy( f);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 7: if e > E then 8: E = e;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 9: Q = l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='quaternion();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 10: end if 11: end for 12: return Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 13: end function ent numbers of viewpoints in the latitude and longitude directions (latitude × longitude).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Both SLERP- and SQUAD-based quaternion interpolation methods were also evaluated for estimating the camera path between the selected viewpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' The x86/GPU Server was used for the detailed evaluation using these different parameters, and the supercomputer Fugaku was used for the scalability analysis by using up to 1024 nodes, that is, 49,152 cores in hybrid MPI/OpenMP parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Table 3: Parameters used for the experimental evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Entropy source Depth;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Lightness;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Depth & Lightness Color maps RdBu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' PiYG;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' PuOr # of viewpoints 15×30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 25×50;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 35×70 Intervals (NE) 10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 50 Interpolation SLERP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' SQUAD Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 6 shows some entropy heatmaps evaluated by using all three entropy sources at different simulation time steps (2400, 6000, 9600, and 13200).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' The set of viewpoints evenly distributed on the spherical surface is mapped onto the 2D heatmap where the viewpoints on the same latitude are placed on the same horizontal axis, and in the same manner, the viewpoints on the same longitude are placed on the same vertical axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' The blue-colored regions show the portions where the evaluated entropy has low value, and on the other hand, the red-colored regions show the portions where the evaluated en- tropy has high value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 7 shows the multi-isosurface rendered results from the selected viewpoints obtained in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 9 shows the multi-isosurface rendered results from the selected viewpoints obtained in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 11 shows the multi-isosurface rendered results from the selected viewpoints obtained in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Table 4 shows the average elapsed time of entropy calculation per image for different entropy sources when using an image size of 512 × 512 on the x86/GPU-based Server System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Compared to depth entropy, we can observe that the computational costs when using lightness become much higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' In addition, we can verify that the number of viewpoints directly influences the computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Code op- timizations and the use of parallel processing for trying to reduce this computational cost are planned as future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='Table 5 shows a comparison of output images’ average entropy when varying the number of viewpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Here, the utilized entropy source is Depth & Lightness, the entropy evaluation interval is 30, and the interpolation method is SQUAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' We can observe that the difference in the average entropy when varying the number of viewpoints is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 12 shows a comparison of the accumulative distance from the estimated camera path position to the viewpoint with the high- est entropy, at each visualization time step, for different entropy evaluation intervals;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 13 shows a comparison of the accumula- tive distance for different interpolation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='Table 6 shows a comparison of output images’ average entropy for different entropy Table 4: Average elapsed time of entropy calculation for different entropy sources (x86 System).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Entropy Sources Average elapsed time [s] Depth 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='24e-4 Lightness 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='30e-3 Depth & Lightness 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='53e-3 Table 5: Average entropy when varying the number of viewpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' # of viewpoints Average entropy 15×30 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='09 25×50 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='10 35×50 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='09 evaluation intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Here, the utilized entropy source is Depth & Lightness, the number of viewpoints is 25×50, and the interpolation method is SQUAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' In the case of NE = 1, rendered images of the viewpoint with the highest entropy at each visualization time step are output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' From this figure and table, we can observe that as the entropy evaluation interval increases, the accumulative distance also increases, and the amount of average entropy decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Table 7 shows the average elapsed time for path calculation between two selected viewpoints for different interpolation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' We can observe that the computational cost is proportional to the number of intervals, and the cost of SQUAD is much higher than that of SLERP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' However, it is worth noting that the influence on the total computational cost compared to the entropy calculation cost is small and almost neglectable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Table 8 shows a comparison of output im- ages’ average entropy for different interpolation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Here, the entropy source is Depth & Lightness, the number of viewpoints is 25×50, and the entropy evaluation interval is 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' We can observe that when selecting SQUAD, the accumulative distance becomes smaller and achieves a slight increase in average entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Table 6: Average entropy for different entropy evaluation intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Intervals (NE) Average entropy 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='17 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='12 30 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='10 50 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='08 Table 7: Average elapsed time for path calculation between two se- lected viewpoints using different interpolation methods (x86 System).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Interpolation Intervals (NE) method 10 30 50 SLERP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='80e-6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='16e-6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='93e-6 SQUAD 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='80e-6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='04e-5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='54e-5 Table 8: Average entropy for different interpolation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Interpolation method Average entropy SLERP 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='09 SQUAD 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='2 Discussions Regarding the influence of different entropy sources (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 6), we observed that the lightness entropy has a higher influence than the Simulation step: 2400 6000 9600 13200 Depth: Depth & Lightness: Lightness: Figure 6: Entropy heatmaps for different entropy sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Depth: Simulation step: 2400 6000 9600 13200 Lightness: Depth & Lightness: Figure 7: Rendered images from the selected viewpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' depth entropy, and has an even higher influence when using both depth and lightness entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Therefore, we opted to add both depth and lightness entropies after normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' In addition, due to a large number of viewpoints with high entropy, the sequentially se- lected viewpoints can be separated far apart from each other thus resulting in an intense camera movement over the entire visualization time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' We also observed that when using the lightness entropy, the entropy calculation took a little more time than using the depth entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' This was because of the necessary conversion from RGB values to lightness values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' It is worth noting that the selection of the viewpoint evaluation metric will depend on the targeted simulation, visualization method, and users’ analysis goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Therefore, to satisfy R1, it becomes important to implement a variety of viewpoint evalu- ation metrics to handle different use case combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' In addition, depending on the use case, it may be helpful that different evaluation metrics are interchangeable at run time in an adaptive manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Regarding the influence of the diverging color maps for the light- ness entropy, we initially perceived almost no difference between the heatmaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' However, there was a slight difference among them, and at certain time steps, we observed that the selected viewpoints were also different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Among the color maps, heatmaps for the PuOr was especially different from the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' This may be because the change in the lightness of the PuOr was also different from the other diverging color maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Regarding the influence of the entropy evaluation intervals, as this interval becomes smaller there will be fewer complementary images between the selected viewpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' As a result, changes in viewpoint may become intense in a short period of time, this will lead to a non- smooth video which affects the users’ post-hoc visual analysis tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' It is worth noting that when the simulation state is not expected to change rapidly, there will be no necessity to frequently evaluate the viewpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' However, when utilizing larger entropy evaluation intervals, a larger amount of memory will be required for temporarily storing the simulation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' That is, there is a trade-off between the entropy evaluation intervals and the memory consumption, and as a result, depending on the simulation time step range and simulation data size, large entropy evaluation intervals, such as the utilized NE = 30 and NE = 50, may be sufficient to satisfy the R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Regarding the influence of the number of viewpoints on the spher- ical surface, we verify that there was no significant difference for 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='6 37.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='8 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='0 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='6 141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='2 211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='8 282.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='4 352.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='9 LongitudeCSimulation step: 2400 6000 9600 13200 25x50: 15x30: 35x70: Figure 8: Lightness Entropy heatmaps for different diverging color maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' varying number of viewpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' However, it is worth noting that the computational time required to select the viewpoints will increase proportionately with the increase in the number of viewpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Regarding the influence of the quaternion interpolation method for estimating the camera path between selected viewpoints, we observed that the camera path using SQUAD-based interpolation passes closer to the viewpoint with highest entropy at the interme- diate time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' We also observed that jerky camera movements tend to occur when using the SLERP-based interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' On the other hand, smoother camera movement was observed when using the SQUAD-based interpolation, and as a result, we can consider that it will cause less discomfort to the user when seeing the ani- mated rendering results since the camera movement will be more natural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Therefore, we can consider that SQUAD-based quaternion interpolation satisfies the R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Moreover, we carried out some evaluations with the domain sci- entists who assisted in the development of previous work on in-situ adaptive timestep selection [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' We obtained technical feedback from the generated visualization results in the form of animated videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' According to them, the video generated by using the pro- posed method seems to present more information than the video generated by using fixed viewpoint camera settings, which has tradi- tionally been used in their simulation analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' However, they also pointed out that the proposed video gives the impression of exces- sive movement and sometimes tracking phenomena that do not need much attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' As some suggestions, they mentioned that it would be better to slightly reduce high viewpoint variations or suppress unnecessary movement, and to improve evaluation methods for the viewpoint selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' As an additional suggestion, they would prefer to have the ability to zoom in on the target object to enable closer observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' These suggestions will be taken into consideration for further developments planned as future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' In our current implementation, the set of volume data in the en- tropy evaluation interval needs to be stored in the memory before the processing, and this memory cost can become an impediment for memory-hungry simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' However, we consider that this approach can be useful during test runs and model calibration runs, before the main simulation run, when smaller models are usually sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' In addition, the in-transit approach for flushing the simu- lation data from the memory to another node or even system can be considered helpful for minimizing this problem and is planned for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Another planned future work is the application of the adaptive timestep sampling [26] where larger time intervals will be assigned to timestep regions with small variations between the simu- lation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' This larger entropy evaluation time step by skipping some simulation results may be helpful for accelerating the visu- alization processing as well as reducing the excessive movements pointed out by the domain scientists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 5 CONCLUSIONS In this work, we proposed an information entropy-based camera path estimation method for in-situ visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Considering that most of the images generated by traditional batch-based tightly coupled in-situ visualization may have small or even no contribution for the post-hoc visual analysis, we focused on generating a smooth video that tries to provide as much information as possible to facilitate the rapid understanding of the simulation or to narrow down the spatio- temporal region of interest for posterior detailed analysis such as by using traditional image-based visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' The proposed method focuses on selecting the most appropriate viewpoints, based on in- formation entropy, at regular intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Intermediate images are generated from the estimated camera path connecting these selected viewpoints, and the produced smooth video that is produced is ex- pected to be helpful for understanding the underlying simulation phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' From the experimental evaluations and feedback from domain scientists, we can confirm that the video generated by the proposed approach provides more information compared to those generated by using fixed viewpoint camera settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' However, there is still need for improvements, and we can cite the following targets for future works: implementation of better evaluation methods for the viewpoint selection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' implementation of zoom in and out func- tionalities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' integration with the adaptive timestep sampling (irregular time intervals);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' improvement of computational performance such as by applying parallel processing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' and estimation of the focal point for the camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' ACKNOWLEDGMENTS The authors are grateful to Tsukasa Yoshinaga (Toyohashi Univer- sity of Technology) and Kazunori Nozaki (Osaka University) for the simulation model and technical feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' This work was par- tially supported by JSPS KAKENHI (Grant Numbers: 20H04194, 21H04903, 22H03603), and the National Key R&D Program of China under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 2021YFE0108400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' This work used compu- tational resources of supercomputer Fugaku provided by the RIKEN Center for Computational Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' REFERENCES [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Ahrens, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Jourdain, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' O’Leary, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Patchett, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Rogers, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Pe- tersen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' An image-based approach to extreme scale in situ visualization and analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' In SC’14: Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 424–434.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' IEEE, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' [2] U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Ayachit, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Bauer, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Geveci, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' O’Leary, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Moreland, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Fabian, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Mauldin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Paraview Catalyst: Enabling in situ data analysis and visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' In Proceedings of the First Workshop on In Situ Infrastructures for Enabling Extreme-Scale Analysis and Visualization, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 25–29, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='6 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='4 latitude 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='2 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} 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+page_content=' Principles and practice of information theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Addison- Wesley Longman Publishing Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=', Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=', 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' [5] U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Bordoloi and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Shen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' View selection for volume rendering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' In VIS 05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' IEEE Visualization, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=', pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 487–494, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 1109/VISUAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='1532833 [6] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Childs, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Ahern, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Ahrens, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Bauer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Bennett, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Bethel, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Bremer, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Brugger, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Cottam, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Dorier, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' A terminology for in situ visualization and analysis systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' The International Journal of High Performance Computing Applications, 34(6):676–691, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' [7] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Ji and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Shen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Dynamic view selection for time-varying vol- umes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' IEEE Transactions on Visualization and Computer Graphics, 12(5):1109–1116, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='1109/TVCG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='137 [8] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Kageyama and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Sakamoto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 4D street View: A video-based visualization method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' PeerJ Computer Science, (6:e305), 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' [9] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Kageyama and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Yamada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' An approach to exascale visualiza- tion: Interactive viewing of in-situ visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Computer Physics Communications, 185(1):79–85, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' [10] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Kamada and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Kawai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' A simple method for computing general position in displaying three-dimensional objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Computer Vision, Graphics, and Image Processing, 41(1):43–56, 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' [11] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Kuhlen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Pajarola, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Zhou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Parallel in situ coupling of simulation with a fully featured visualization system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' In Proceedings of the 11th Eurographics Conference on Parallel Graphics and Visu- alization (EGPGV), vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 10, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 101–109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Eurographics Association Aire-la-Ville, Switzerland, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' [12] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Ling, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Kegelmeyer, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Aditya, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Kolla, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Reed, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Shead, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Davis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Using feature importance metrics to detect events of interest in scientific computing applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' In 2017 IEEE 7th Symposium on Large Data Analysis and Visualization (LDAV), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 55–63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' IEEE, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' [13] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Ma, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Painter, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Hansen, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Krogh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Parallel volume rendering using binary-swap compositing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' IEEE Computer Graphics and Applications, 14(4):59–68, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' [14] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Marsaglia, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Kawakami, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Schwartz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Fields, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Childs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' An entropy-based approach for identifying user-preferred camera po- sitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' In 2021 IEEE 11th Symposium on Large Data Analysis and Visualization (LDAV), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 73–83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' IEEE, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' [15] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Marsaglia, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Majumder, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Childs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' A trigger-based approach for optimizing camera placement over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 9 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' [16] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Moreland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Diverging color maps for scientific visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' In International Symposium on Visual Computing, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 92–103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Springer, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Simulation step: 2400 6000 9600 13200 25x50: 15x30: 35x70: Figure 10: Entropy heatmaps when varying the number of viewpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Simulation step: 2400 6000 9600 13200 25x50: 15x30: 35x70: Figure 11: Rendered images from the selected viewpoints when varying the number of viewpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Figure 12: Accumulative distance from the interpolated camera path to the viewpoints, with the highest entropy, at each visualization time step, for different entropy evaluation intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' [17] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Nonaka, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Ono, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Fujita.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 234Compositor: A flexible parallel image compositing framework for massively parallel visualization environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Future Generation Computer Systems, 82:647–655, Figure 13: Accumulative distance to the viewpoints, with the highest entropy, at each visualization time step, for different interpolation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' [18] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Page, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Koschan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Sukumar, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Roui-Abidi, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' A.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' [20] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Sakamoto and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Koyamada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' KVS: A simple and effective frame- work for scientific visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Journal of Advanced Simulation in Science and Engineering, 2(1):76–95, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' [21] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Secord, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Lu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Finkelstein, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Singh, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Nealen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Perceptual models of viewpoint preference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' ACM Transactions on Graphics (TOG), 30(5):1–12, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' [22] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Takahashi, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Fujishiro, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Takeshima, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Nishita.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' A feature- driven approach to locating optimal viewpoints for volume visualiza- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' In VIS 05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' IEEE Visualization, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=', pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 495–502, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 1109/VISUAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='1532834 [23] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' V´azquez, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Feixas, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Sbert, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Heidrich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Viewpoint selec- tion using viewpoint entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' In VMV, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 273–280.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Citeseer, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' [24] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' V´azquez, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Feixas, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Sbert, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Heidrich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Automatic view selection using viewpoint entropy and its application to image-based modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' In Computer Graphics Forum, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 22, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 689–700.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Wiley Online Library, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' [25] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' V´azquez, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Feixas, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Sbert, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Llobet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Viewpoint en- tropy: A new tool for obtaining good views of molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' In ACM International Conference Proceeding Series, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 22, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 183–188, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' [26] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Yamaoka, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Hayashi, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Sakamoto, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Nonaka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' In situ adap- tive timestep control and visualization based on the spatio-temporal variations of the simulation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' In Proceedings of the Workshop on In Situ Infrastructures for Enabling Extreme-Scale Analysis and Visualization, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 12–16, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' [27] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Yoshinaga, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Nozaki, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Wada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Experimental and numerical investigation of the sound generation mechanisms of sibilant fricatives using a simplified vocal tract model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Physics of Fluids, 30(3):035104, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' [28] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Zhang and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' Optimal viewpoint selection for volume ren- dering based on shuffled frog leaping algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' In 2010 IEEE Inter- national Conference on Progress in Informatics and Computing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' 706–709.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} +page_content=' IEEE, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFJT4oBgHgl3EQfnSyO/content/2301.11591v1.pdf'} diff --git a/8tE4T4oBgHgl3EQfCwu_/content/tmp_files/2301.04864v1.pdf.txt b/8tE4T4oBgHgl3EQfCwu_/content/tmp_files/2301.04864v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..47c5a1ab79a858cd1a42335586ca3266fefc5385 --- /dev/null +++ b/8tE4T4oBgHgl3EQfCwu_/content/tmp_files/2301.04864v1.pdf.txt @@ -0,0 +1,3154 @@ +arXiv:2301.04864v1 [math.AG] 12 Jan 2023 +TOPOLOGICAL STRUCTURES OF MODULI SPACES OF CURVES AND ANABELIAN GEOMETRY +IN POSITIVE CHARACTERISTIC +ZHI HU, YU YANG, AND RUNHONG ZONG +ABSTRACT. In the present paper, we study a new kind of anabelian phenomenon concerning the smooth pointed stable +curves in positive characteristic. It shows that the topological structures of moduli spaces of curves can be understood from +the viewpoint of anabelian geometry. We formulate some new anabelian-geometric conjectures relating the tame fundamental +groups of curves over algebraically closed fields of characteristic p > 0 to the moduli spaces of curves. These conjectures +are generalized versions of the weak Isom-version of the Grothendieck conjecture for curves over algebraically closed fields +of characteristic p > 0 which was formulated by Tamagawa. Moreover, we prove that the conjectures hold for certain points +lying in the moduli space of curves of genus 0. +CONTENTS +1. +Introduction +1 +1.1. +The mystery of fundamental groups in positive characteristic +1 +1.2. +Topology structures of moduli spaces of curves and anabelian geometry +2 +1.3. +Main results +3 +1.4. +Some further developments +4 +1.5. +Structure of the present paper +6 +1.6. +Acknowledgements +6 +2. +Conjectures +6 +2.1. +The weak Hom-version conjecture +6 +2.2. +The pointed collection conjecture +7 +3. +Reconstructions of marked points +8 +3.1. +Anabelian reconstructions +9 +3.2. +The set of marked points +9 +3.3. +Reconstructions of inertia subgroups +12 +3.4. +Reconstructions of inertia subgroups via surjections +14 +3.5. +Reconstructions of additive structures via surjections +21 +4. +Main theorems +23 +4.1. +The first main theorem +23 +4.2. +The second main theorem +29 +References +31 +1. INTRODUCTION +1.1. The mystery of fundamental groups in positive characteristic. +sec111 +1.1.1. +Let k be an algebraically closed field of characteristic p ≥ 0, and let (X, DX) be a smooth pointed stable curve +of type (gX, nX) over k (i.e. 2gX + nX − 2 > 0, see +K[K, Definition 1.1 (iv)]), where X denotes the underlying curve, +DX denotes the (ordered) finite set of marked points, gX denotes the genus of X, and nX denotes the cardinality +1 + +2 +ZHI HU, YU YANG, AND RUNHONG ZONG +#(DX) of DX. We put UX := X \ DX. By choosing a base point of UX, we have the tame fundamental group +πt +1(UX) of UX. +If p = 0, it is well-known that πt +1(UX) is isomorphic to the profinite completion of the topological fundamental +group of a Riemann surface of type (gX, nX). Hence, almost no geometric information about UX can be carried out +from πt +1(UX). By contrast, if p > 0, the situation is quite different from that in characteristic 0. The tame fundamental +group πt +1(UX) is very mysterious and its structure is no longer known, in particular, there exist anabelian phenomena +for curves over algebraically closed fields of characteristic p > 0. +1.1.2. +Firstly, let us explain some general background about anabelian geometry. In the 1980s, A. Grothendieck +suggested a theory of arithmetic geometry called anabelian geometry ( +G[G]). The central question of the theory is as +follows: Can we reconstruct the geometric information of a variety group-theoretically from various versions of its +algebraic fundamental group? The original anabelian geometry suggested by Grothendieck focused on varieties over +arithmetic fields, in particular, the fields finitely generated over Q. In the case of curves in characteristic 0, anabelian +geometry has been deeply studied (e.g. +N[N], +T1 +[T1]) and, in particular, the most important case (i.e., the fields finitely +generated over Q, or more general, sub-p-adic fields) has been established completely( +M[M]). Note that the actions of +the Galois groups of the base fields on the geometric fundamental groups play a crucial role for recovering geometric +information of curves over arithmetic fields. +Next, we return to the case where k is an algebraically closed field of characteristic p > 0. In +T2 +[T2], A. Tamagawa +discovered that there also exist anabelian phenomena for curves over algebraically closed fields of characteristic p. +This came rather surprisingly since it means that, in positive characteristic, the geometry of curves can be determined +by their geometric fundamental groups without Galois actions. Since the late 1990s, this kind of anabelian phenom- +enon has been studied further by M. Raynaud ( +R2 +[R2]), F. Pop-M. Saïdi ( +PS +[PS]), Tamagawa ( +T2 +[T2], +T4 +[T4], +T5 +[T5]), and the +second author of the present paper ( +Y1 +[Y1], +Y2 +[Y2], +Y4 +[Y4]). More precisely, they focused on the so-called weak Isom- +version of Grothendieck’s anabelian conjecture for curves over algebraically closed fields of characteristic p > 0 (or +the “weak Isom-version conjecture” for short) formulated by Tamagawa ( +T3 +[T3, Conjecture 2.2]), which says that curves +are isomorphic if and only if their tame (or étale) fundamental groups are isomorphic. At the present, this conjecture +is still wide-open. +1.2. Topology structures of moduli spaces of curves and anabelian geometry. In the present paper, we study a +new kind of anabelian phenomenon concerning curves over algebraically closed fields of characteristic p > 0 which +shows that the topological structures of moduli spaces of curves can be understood by their fundamental groups. +1.2.1. +Let Fp be the prime field of characteristic p > 0, and let Mord +g,n,Z be the moduli stack over Z parameterizing +smooth n-pointed stable curves of type (g, n) (in the sense of +K[K]). We put Mord +g,n,Fp := Mord +g,n,Z ×Z Fp. Note that +the set of marked points of an n-smooth pointed stable curve admits a natural action of the n-symmetric group Sn. +Moreover, we denote by Mg,n,Fp := [Mord +g,n,Fp/Sn] the quotient stack, and denote by Mg,n,Fp the coarse moduli space +of Mg,n,Fp. +Let q ∈ Mg,n,Fp be an arbitrary point, k(q) the residue field of q, kq an algebraically closed field containing k(q), +and Vq := {q} the topological closure of {q} in Mg,n,Fp. Write (Xkq, DXkq ) for the smooth pointed stable curve +of type (g, n) over kq determined by the natural morphism Speckq → Mg,n,Fp and put UXkq := Xkq \ DXkq . In +particular, we put (Xkq, DXkq ) := (Xq, DXq) and UXq := Xq \ DXq if kq is an algebraic closure of k(q). Since +the isomorphism class of the tame fundamental group πt +1(UXkq ) depends only on q, we shall write πt +1(q) for the tame +fundamental group πt +1(UXkq ). +sec122 +1.2.2. +We maintain the notation introduced above. The weak Isom-version conjecture of Tamagawa can be reformu- +lated as follows: + +TOPOLOGICAL STRUCTURES OF MODULI SPACES OF CURVES AND ANABELIAN GEOMETRY IN POSITIVE CHARACTERISTIC +3 +Weak Isom-version Conjecture . Let qi ∈ Mg,n,Fp, i ∈ {1, 2}, be an arbitrary point of Mg,n,Fp. The set of +continuous isomorphisms of profinite groups +Isompg(πt +1(q1), πt +1(q2)) +is non-empty if and only if Vq1 = Vq2 (namely, UXq1 ∼= UXq2 as schemes). +The weak Isom-version conjecture means that moduli spaces of curves can be reconstructed “as sets” from the iso- +morphism classes of the tame fundamental groups of curves. This conjecture has been only confirmed by Tamagawa +( +T4 +[T4, Theorem 0.2]) in the following case: +Suppose that q1 is a closed point of M0,n,Fp. Then the weak Isom-version conjecture holds true. +Next, we propose a new conjecture as follows, that is the weak Hom-version of the Grothendieck conjecture for +curves over algebraically closed fields of characteristic p > 0 (or is called weak Hom-version conjecture for simplic- +ity), as a generalization of the weak Isom-version conjecture. +Weak Hom-version Conjecture . Let qi ∈ Mg,n,Fp, i ∈ {1, 2}, be an arbitrary point of Mg,n,Fp. The set of open +continuous homomorphisms of profinite groups +Homop +pg(πt +1(q1), πt +1(q2)) +is non-empty if and only if Vq1 ⊇ Vq2. +The weak Hom-version conjecture means that the sets of deformations of a smooth pointed stable curve can be re- +constructed group-theoretically from the sets of open continuous homomorphisms of their tame fundamental groups. +Therefore, it provides a new kind of anabelian phenomenon: +The moduli spaces of curves in positive characteristic can be understood not only as sets but also “as +topological spaces” from the sets of open continuous homomorphisms of tame fundamental groups +of curves in positive characteristic. +Roughly speaking, this means that a smooth pointed stable curve corresponding to a geometric point over q2 can be +deformed to a smooth pointed stable curve corresponding to a geometric point over q1 if and only if the set of open +continuous homomorphisms of tame fundamental groups Homop +pg (πt +1(q1), πt +1(q2)) is not empty. +1.3. Main results. +1.3.1. +The main result of the present paper is the following (see Theorem +them-4 +4.6 (iv) for a more general statement): +maintheorem +Theorem 1.1. The Weak Hom-version Conjecture holds when q1 is a closed point of M0,n,Fp. +Theorem +maintheorem +1.1 follows from the following “Hom-type" anabelian result (see Theorem +them-3 +4.4 for a more precise statement) +which is a generalization of Tamagawa’s result (i.e. +T4 +[T4, Theorem 0.2]): +them-0-1 +Theorem 1.2. Let qi ∈ Mg,n,Fp, i ∈ {1, 2}, be an arbitrary point of Mg,n,Fp. Suppose that q1 is a closed point of +Mg,n,Fp. Then the set of open continuous homomorphisms +Homop +pg(πt +1(q1), πt +1(q2)) +is non-empty if and only if UXq1 ∼= UXq2 as schemes. +Note that Theorem +them-0-1 +1.2 is essentially different from +T4 +[T4, Theorem 0.2]. The reason is the following: We a priori do +not know whether or not +Isompg(πt +1(q1), πt +1(q2)) + +4 +ZHI HU, YU YANG, AND RUNHONG ZONG +is non-empty even through Homop +pg(πt +1(q1), πt +1(q2)) is non-empty. In fact, for arbitrary qi ∈ Mg,n,Fp, i ∈ {1, 2}, we +have +Isompg(πt +1(q1), πt +1(q2)) = ∅, Homop +pg(πt +1(q1), πt +1(q2)) ̸= ∅ +in general ( +T5 +[T5, Theorem 0.3]). +On the other hand, to verify Theorem +them-0-1 +1.2, we need to establish various anabelian reconstructions from open contin- +uous homomorphisms of tame fundamental groups which are much harder than the case of isomorphisms in general. +We explain in more detail about this point in the following. +1.3.2. +Let us explain the main differences between the proofs of Tamagawa’s result (i.e. +T4 +[T4, Theorem 0.2]) and +our result (i.e. Theorem +them-0-1 +1.2), and the new ingredient in our proof. First, we recall the key points of the proof of +Tamagawa’s result. Roughly speaking, Tamagawa’s proof consists of two parts: +(1) He proved that the sets of inertia subgroups of marked points and the field structures associated to inertia +subgroups of marked points of smooth pointed stable curves can be reconstructed group-theoretically from +tame fundamental groups. This is the most difficult part of Tamagawa’s proof. +(2) By using the inertia subgroups and their associated field structures, if g = 0, he proved that the coordinates of +marked points can be calculated group-theoretically. +The group-theoretical reconstructions in Tamagawa’s proofs (1) and (2) are isomorphic version reconstructions. +This means that the reconstructions should fix an isomorphism class of a tame fundamental group. To explain this, +let us show an example. Let UXi, i ∈ {1, 2}, be a curve of type (gX, nX) over an algebraically closed field k of +characteristic p > 0 introduced above, πt +1(UXi) the tame fundamental group of UXi, φ : πt +1(UX1) → πt +1(UX2) an +open continuous homomorphism, H2 ⊆ πt +1(UX2) an open subgroup, and H1 := φ−1(H2). In Tamagawa’s proof, +since φ is an isomorphism, we have H1 ≃ H2. Then the group-theoretical reconstruction for types implies that the +type (gXH1 , nXH1 ) and the type (gXH2 , nXH2 ) of the curves corresponding to H1 and H2, respectively, are equal. This +is a key point in the proof of Tamagawa’s group-theoretical reconstruction of the inertia subgroups of marked points. +Unfortunately, his method cannot be applied to the present paper. The reason is that we need to treat the case where +φ is an arbitrary open continuous homomorphism. Since H1 is not isomorphic to H2 in general (e.g. specialization +homomorphism), we do not know whether or not (gXH1 , nXH1 ) = (gXH2 , nXH2 ). This is one of the main difficulties +of “Hom-type” problems appeared in anabelian geometry. Similar difficulties for generalized Hasse-Witt invariants +will appear if we try to reconstruct the field structure associated to inertia subgroups of marked points. +To overcome the difficulties mentioned above, we have the following key observation: +The inequalities of Avrp(Hi) (i.e., the p-averages of generalized Hasse-Witt invariants (see +paverage +3.4.3)) in- +duced by φ play roles of the comparability of (outer) Galois representations in the theory of anabelian +geometry of curves over algebraically closed fields of characteristic p > 0. +In the present paper, our method for reconstructing inertia subgroups of marked points is completely different from +Tamagawa’s reconstruction. We develop a new group-theoretical algorithm for reconstructing the inertia subgroups of +marked points whose input datum is a profinite group which is isomorphic to πt +1(UXi), i ∈ {1, 2}, and whose output +data are inertia subgroups of marked points (Theorem +them-2 +3.18). Moreover, we prove that the group-theoretical algorithm +and the reconstructions for field structures are compatible with arbitrary surjection φ (Proposition +pro-4 +3.19). By using +Theorem +them-2 +3.18 and Proposition +pro-4 +3.19, we may prove that Tamagawa’s calculation of coordinates is compatible with our +reconstructions. This implies Theorem +them-0-1 +1.2. +1.4. Some further developments. +1.4.1. Moduli spaces of fundamental groups. Let us explain some further developments for the anabelian phenomenon +concerning the weak Hom-verson conjecture. In +Y6 +[Y6], the second author of the present paper introduced a topological +space Πg,n (or more general, Πg,n) determined group-theoretically by the tame fundamental groups of smooth pointed + +TOPOLOGICAL STRUCTURES OF MODULI SPACES OF CURVES AND ANABELIAN GEOMETRY IN POSITIVE CHARACTERISTIC +5 +stable curves (or more general, the geometric log étale fundamental groups of arbitrary pointed stable curves) of +type (g, n) which is called the moduli spaces of fundamental groups of curves, whose underlying set is the sets of +isomorphism classes of fundamental groups, and whose topology is determined by the sets of finite quotients of +fundamental groups. Moreover, he posed the so-called homeomorphism conjecture, roughly speaking, which says +that (by quotiening a certain equivalence relation induced by Frobenius actions) the moduli spaces of curves are +homeomorphic to the moduli spaces of fundamental groups. +In the present literatures, the term “anabelian” means that a geometric object can be determined by its fundamental +group. Furthermore, the homeomorphism conjecture concerning moduli spaces of fundamental groups supplies a new +point of view to understand anabelian phenomena as follows: +The term “anabelian” means that not only a geometric object can be determined by its fundamental +groups, but also a certain moduli space of geometric objects can be determined by the fundamental +groups of geometric objects. +Under this point of view, the homeomorphism conjecture is regarded as the analogue of a famous theorem in the theory +of classic Teichmüller spaces which states that the Teichmüller spaces of complex hyperbolic curves are homeomorphic +to the spaces of discrete and faithful representations of topological fundamental groups of underlying surfaces into the +group PSL2(R). +Now Theorem +maintheorem +1.1 implies that M0,4,Fp is homeomorphic to Π0,4 as topological spaces (note that Tamagawa’s +result (i.e. +T4 +[T4, Theorem 0.2]) only says that the natural map M0,4,Fp → Π0,4 is a bijection as sets). Moreover, based +on +Y1 +[Y1], +Y3 +[Y3], +Y4 +[Y4], +Y5 +[Y5], and the main results of the present paper, the homeomorphism conjecture is confirmed +for 1-dimensional moduli spaces of pointed stable curves in +Y6 +[Y6] and +Y7 +[Y7]. For the homeomorphism conjecture in +the case of higher dimensional moduli spaces of curves, the weak Hom-version conjecture and the pointed collection +conjecture (see Section +pcc +2.2 of the present paper) are also the main steps toward understanding it (see +Y8 +[Y8, Section +1.2.3]). +1.4.2. The sets of finite quotients of tame fundamental groups. We maintain the notation introduced in +sec111 +1.1.1. The +techniques developed in § +mpanabelian +3 of the present paper have important applications for understanding the set of finite quotients +πt +A(UX) of the tame fundamental groups πt +1(UX) of UX. +Note that, if UX is affine, the set πét +A(UX) of finite quotients of the étale fundamental groups πét +1 (UX) of UX can +be completely determined by its type (gX, nX) (i.e. Abhyankar’s conjecture proved by Raynaud and D. Harbater). +However, the structure of πét +1 (UX) cannot be carried out from πét +A(UX) since πét +1 (UX) is not topologically finitely +generated when UX is affine. +By contrast, the isomorphism class of πt +1(UX) can be completely determined by πt +A(UX) since πt +1(UX) is topolog- +ically finitely generated, and one cannot expect that there exists an explicit description for the entire set πt +A(UX) since +there exists anabelian phenomenon mentioned above (i.e. πt +A(UX) depends on the isomorphism class of UX). On the +other hand, for understanding more precisely the relationship between the structures of tame fundamental groups and +the anabelian phenomena in positive characteristic world, it is naturally to ask the following interesting problem: +How does the scheme structure of UX affect explicitly the set of finite quotients πt +A(UX)? +In +Y9 +[Y9], by applying the techniques developed in § +mpanabelian +3 of the present paper and +Y5 +[Y5, Theorem 1.2], we obtain the +following result: +Let q1 ∈ Mg1,n1,Fp and q2 ∈ M0,n2,Fp be arbitrary points and πt +A(qi) the set of finite quotients of +the tame fundamental group πt +1(qi). Suppose that q2 is a closed point of M0,n2,Fp, and that πt +1(q1) ̸∼= +πt +1(q2). Then we can construct explicitly a finite group Gq2 depending on q2 such that Gq2 ∈ πt +A(q1) +and Gq2 ̸∈ πt +A(q2). + +6 +ZHI HU, YU YANG, AND RUNHONG ZONG +1.5. Structure of the present paper. The present paper is organized as follows. In Section +sec-1 +2, we formulate the +the weak Hom-version conjecture and the pointed collection conjecture. In Section +mpanabelian +3, we give a group-theoretical +algorithm for reconstructions of inertia subgroups associated to marked points, and prove that the group-theoretical +algorithm is compatible with arbitrary open surjective homomorphisms of tame fundamental groups. In Section +sec-5 +4, we +prove our main results. +1.6. Acknowledgements. The second author was supported by JSPS Grant-in-Aid for Young Scientists Grant Num- +bers 16J08847 and 20K14283. +2. CONJECTURES +sec-1 +In this section, we formulate two new conjectures concerning anabelian geometry of curves over algebraically +closed fields of characteristic p > 0. +2.1. The weak Hom-version conjecture. In this subsection, we formulate the first conjecture of the present paper +which we call “the weak Hom-version conjecture”. +curves +2.1.1. +Let k be an algebraically closed field of characteristic p > 0, and let +(X, DX) +be a smooth pointed stable curve of type (gX, nX) over k, where X denotes the (smooth) underlying curve of genus +gX and DX denotes the (ordered) finite set of marked points with cardinality nX := #(DX) satisfying +K[K, Definition +1.1 (iv)] (i.e. 2gX + nX − 2 > 0). Note that UX := X \ DX is a hyperbolic curve over k. +Let (Y, DY ) and (X, DX) be smooth pointed stable curves over k, and let f : (Y, DY ) → (X, DX) be a morphism +of smooth pointed stable curves over k. We shall say that f is étale (resp. tame, Galois étale, Galois tame) if f is étale +over X (resp. f is étale over UX and is at most tamely ramified over DX, f is a Galois covering and is étale, f is a +Galois covering and is tame). +By choosing a base point of x ∈ UX, we have the tame fundamental group πt +1(UX, x) of UX and the étale funda- +mental group π1(X, x) of X. Since we only focus on the isomorphism classes of fundamental groups in the present +paper, for simplicity of notation, we omit the base point and denote by πt +1(UX) and π1(X) the tame fundamental +group πt +1(UX, x) of UX and the étale fundamental group π1(X, x) of X, respectively. Note that there is a natural +continuous surjective homomorphism πt +1(UX) ։ π1(X). +moduli212 +2.1.2. +Let Fp be an algebraic closure of Fp, and let Mord +g,n,Fp be the moduli stack over Z parameterizing smooth +pointed stable curves of type (g, n) in the sense of +K[K, Definition 1.1]. The set of marked points of a smooth pointed +stable curve admits a natural action of the n-symmetric group Sn, we put Mg,n,Z := [Mord +g,n,Z/Sn] the quotient stack. +Moreover, we denote by Mord +g,n := Mg,n,Z ×Z Fp, Mg,n,Fp := Mg,n,Z ×Z Fp, and Mg,n := Mg,n,Z ×Z Fp, and +denote by M ord +g,n , Mg,n,Fp, and Mg,n the coarse moduli spaces of Mord +g,n, Mg,n,Fp, and Mg,n, respectively. +Let q ∈ M ord +g,n be an arbitrary point and k(q) the residue field of q, and kq an algebraically closed field containing +k(q). Write (Xkq, DXkq ) for the smooth pointed stable curve of type (g, n) over kq determined by the natural mor- +phism Speckq → Speck(q) → M ord +g,n and UXkq for Xkq \ DXkq . In particular, if kq is an algebraic closure of k(q), we +shall write (Xq, DXq) for (Xkq, DXkq ). +Since the isomorphism class of the tame fundamental group πt +1(UXkq ) depends only on q (i.e., the isomorphism +class does not depend on the choices of kq), we shall write πt +1(q) and πt +A(q) for πt +1(UXkq ) and the set of finite quotients +of πt +1(UXkq ), respectively. +FJ +[FJ, Proposition 16.10.7] implies that for any points q1, q2 ∈ M ord +g,n , πt +1(q1) ∼= πt +1(q2) as +profinite groups if and only if πt +A(q1) = πt +A(q2) as sets. + +TOPOLOGICAL STRUCTURES OF MODULI SPACES OF CURVES AND ANABELIAN GEOMETRY IN POSITIVE CHARACTERISTIC +7 +On the other hand, Let q ∈ M ord +g,n and q′ ∈ Mg,n,Fp be arbitrary points. We denote by Vq ⊆ M ord +g,n and Vq′ ⊆ +Mg,n,Fp the topological closures of q and q′ in M ord +g,n and Mg,n,Fp, respectively. +2.1.3. +We have the following definition. +Definition 2.1. +def-2 +(1) Let c1, c2 ∈ M ord,cl +g,n +be closed points, where (−)cl denotes the set of closed points of (−). Then c1 ∼fe c2 if +there exists m ∈ Z such that ν(c2) = ν(c(m) +1 +), where c(m) +1 +denotes the closed point corresponding to the curve +obtained by mth Frobenius twist of the curve corresponding to c1. Here “fe" means “Frobenius equivalence". +(2) Let q1, q2 ∈ M ord +g,n be arbitrary points. We denote by Vq1 ⊇fe Vq2 if, for each closed point c2 ∈ V cl +q2 , there +exists a closed point c1 ∈ V cl +q1 such that c1 ∼fe c2. Moreover, we denote by Vq1 =fe Vq2 if Vq1 ⊇fe Vq2 and +Vq1 ⊆fe Vq2. Moreover, we also denote by q1 ∼fe q2 if Vq1 =fe Vq2. +We have the following proposition. +pro-5 +Proposition 2.2. Let ω : M ord +g,n → Mg,n,Fp be the morphism induced by the natural morphism Mord +g,n → Mg,n,Fp. +Let i ∈ {1, 2}, and let qi ∈ M ord +g,n and q′ +i := ω(qi) ∈ Mg,n,Fp. Then we have Vq1 ⊇fe Vq2 if and only if Vq′ +1 ⊇ Vq′ +2. In +particular, we have Vq1 =fe Vq2 if and only if Vq′ +1 = Vq′ +2. Namely, we have Vq1 =fe Vq2 if and only if UXq1 ∼= UXq2 +as schemes. +Proof. Suppose that qi, i ∈ {1, 2}, is a closed point of M ord +g,n . If Vq1 ⊇fe Vq2, we see immeidately q1 ∼ q2. Thus, we +obtain UXq1 ∼= UXq2 as schemes. This means q′ +1 = q′ +2. Conversely, if Vq′ +1 ⊇ Vq′ +2, then we have q′ +1 = q′ +2. Thus, we +obtain q1 ∼ q2. +Suppose that qi, i ∈ {1, 2}, is an aribtrary point of M ord +g,n . If Vq1 ⊇fe Vq2, then the case of closed points implies +V cl +q′ +1 ⊇ V cl +q′ +2 . Since Vq′ +1 and Vq′ +2 are irreducible, we obtain Vq′ +1 ⊇ Vq′ +2. Conversely, if Vq′ +1 ⊇ Vq′ +2, we note that Vqi is an +irreducible component of (ω)−1(Vq′ +i). Then the case of closed points implies Vq1 ⊇fe Vq2. +□ +2.1.4. +Denote by Homop +pg(−, −) the set of open continuous homomorphisms of profinite groups, and by +Isompg(−, −) the set of isomorphisms of profinite groups. We have the following conjecture. +Weak Hom-version Conjecture . Let qi ∈ Mg,n (resp. qi ∈ Mg,n,Fp), i ∈ {1, 2}, be an arbitrary point. Then we +have +Homop +pg(πt +1(q1), πt +1(q2)) +is non-empty if and only if Vq1 ⊇fe Vq2 (resp. Vq1 ⊇ Vq2). +The weak Hom-version conjecture means that the topological structures of the moduli spaces of smooth pointed stable +curves can be understood by the tame fundamental groups of curves. In particular, the weak Hom-version conjecture +implies the following conjecture which was essentially formulated by Tamagawa ( +T3 +[T3]). +Weak Isom-version Conjecture . Let qi ∈ Mg,n (resp. qi ∈ Mg,n,Fp), i ∈ {1, 2}, be an arbitrary point. Then we +have +Isompg(πt +1(q1), πt +1(q2)) +is non-empty if and only if Vq1 =fe Vq2 (resp. Vq1 = Vq2). +The weak Isom-version conjecture means that the set structures of the moduli spaces of smooth pointed stable curves +can be understood by the tame fundamental groups of curves. +pcc +2.2. The pointed collection conjecture. In this subsection, we formulate the second conjecture of the present paper +which we call “the pointed collection conjecture”. We maintain the notation introduced in +moduli212 +2.1.2. + +8 +ZHI HU, YU YANG, AND RUNHONG ZONG +2.2.1. +Let q be an arbitrary point of M ord +g,n and G ∈ πt +A(q) an arbitrary finite group. We put +UG := {q′ ∈ M ord +g,n | G ∈ πt +A(q′)} ⊆ M ord +g,n . +Then we have the following result. +pro-6 +Proposition 2.3. Let q be an arbitrary point of M ord +g,n and G ∈ πt +A(q) an arbitrary finite group. Then the set UG +contains an open neighborhood of q in M ord +g,n . +Proof. Proposition +pro-6 +2.3 was proved by K. Stevenson when n = 0 and q is a closed point of Mg,0 (cf. +Ste +[Ste, Proposition +4.2]). Moreover, by similar arguments to the arguments given in the proof of +Ste +[Ste, Proposition 4.2], Proposition +pro-6 +2.3 +also holds for n ≥ 0. +□ +def-3 +Definition 2.4. We denote by qgen the generic point of M ord +g,n , and let +C ⊆ πt +A(qgen) = +� +q∈Mord,cl +g,n +πt +A(q) +be a subset of πt +A(qgen). We shall say that C is a pointed collection if the following conditions are satisfied: (i) +0 < #((� +G∈C UG) ∩ M ord,cl +g,n +) < ∞; (ii) UG′ ∩ (� +G∈C UG) ∩ M ord,cl +g,n += ∅ for each G′ ∈ πt +A(qgen) such that G′ ̸∈ C. +On the other hand, for each closed point t ∈ M ord,cl +g,n +, we may define a set associated to t as follows: +Ct := {G ∈ πt +A(qgen) | t ∈ UG}. +Note that, if t ∈ V cl +q , then Ct ⊆ πt +A(q). Moreover, we denote by +Cq := {C is a pointed collection | C ⊆ πt +A(q)}. +2.2.2. +At present, no published results are known concerning the weak Hom-version conjecture (or the weak Isom- +version conjecture) for non-closed points. The main difficulty of proving the weak Hom-version conjecture (or the +weak Isom-version conjecture) for non-closed points of M ord +g,n is the following: For each q ∈ M ord +g,n , we do not know +how to reconstruct the tame fundamental groups of closed points of Vq group-theoretically from πt +1(q). +Once the tame fundamental groups of the closed points of Vq can be reconstructed group-theoretically from πt +1(q), +then the weak Hom-version conjecture for closed points of M ord +g,n implies that the set of closed points of Vq can be +reconstructed group-theoretically from πt +1(q). Thus, the weak Hom-version conjecture for non-closed points of M ord +g,n +can be deduced from the weak Hom-version conjecture for closed points of M ord +g,n . +Let q ∈ M ord +g,n . Since the isomorphism class of πt +1(q) as a profinite group can be determined by the set πt +A(q), the +following conjecture tell us how to reconstruct group-theoretically the set of finite quotients of a closed point of Vq +from πt +A(q) (or πt +1(q)). +Pointed Collection Conjecture . For each t ∈ M ord,cl +g,n +, the set Ct associated to t is a pointed collection. Moreover, +let q ∈ M ord +g,n . Then the natural map +colleq : V cl +q +→ Cq, [t] �→ Ct, +is a bijection, where [t] denotes the image of t in V cl +q +:= V cl +q / ∼fe. +Write q′ ∈ Mg,n,Fp for the image ω(q). Then we have V cl +q += V cl +q′ . This means that the pointed collection conjecture +holds if and only if the weak Hom-version conjecture holds. +3. RECONSTRUCTIONS OF MARKED POINTS +mpanabelian +The main purposes of the present section are as follows: We will give a new mono-anabelian reconstruction of +Ine(πt +1(UX)), and prove that the mono-anabelian reconstruction (i.e., the group-theoretical algorithm) is compatible +with any open continuous homomorphisms of tame fundamental groups of smooth pointed stable curves with a fixed +type. + +TOPOLOGICAL STRUCTURES OF MODULI SPACES OF CURVES AND ANABELIAN GEOMETRY IN POSITIVE CHARACTERISTIC +9 +3.1. Anabelian reconstructions. We maintain the notation introduced in +curves +2.1.1. +3.1.1. +Let us recall the definitions concerning “anabelian reconstructions". +definition 1 +Definition 3.1. Let F be a geometric object and ΠF a profinite group associated to the object F. Suppose that we +are given an invariant InvF depending on the isomorphism class of F (in a certain category), and that we are given +an additional structure AddF (e.g. a family of subgroups, a family of quotient groups) on the profinite group ΠF +depending functorially on F. +We shall say that InvF can be mono-anabelian reconstructed from ΠF if there exists a group-theoretical algorithm +whose input datum is ΠF, and whose output datum is InvF. We shall say that AddF can be mono-anabelian recon- +structed from ΠF if there exists a group-theoretical algorithm whose input datum is ΠF, and whose output datum is +AddF. +Let Fi, i ∈ {1, 2}, be a geometric object and ΠFi a profinite group associated to Fi. Suppose that we are given +an additional structure AddFi on the profinite group ΠFi depending functorially on Fi. We shall say that a map +(or a morphism) AddF1 → AddF2 can be mono-anabelian reconstructed from an open continuous homomorphism +ΠF1 → ΠF2 if there exists a group-theoretical algorithm whose input datum is ΠF1 → ΠF2, and whose output datum +is AddF1 → AddF2. +unicov313 +3.1.2. +Let K be the function field of X, and let �K be the maximal Galois extension of K in a fixed separable closure +of K, unramified over UX and at most tamely ramified over DX. Then we may identify πt +1(UX) with Gal( �K/K). +We define the universal tame covering of (X, DX) associated to πt +1(UX) to be ( � +X, D � +X), where � +X denotes the nor- +malization of X in �K, and D � +X denotes the inverse image of DX in � +X. Then there is a natural action of πt +1(UX) on +( � +X, D � +X). For each �e ∈ D � +X, we denote by I�e the inertia subgroup of πt +1(UX) associated to �e (i.e., the stabilizer of �e +in πt +1(UX)). Then we have I�e ∼= �Z(1)p′, where �Z(1)p′ denotes the prime-to-p part of �Z(1). The following result was +proved by Tamagawa ( +T4 +[T4, Lemma 5.1 and Theorem 5.2]). +Proposition 3.2. +proposition 1 +(1) The type (gX, nX) can be mono-anabelian reconstructed from πt +1(UX). +(2) Let �e and �e′ be two points of D � +X distinct from each other. Then the intersection of I�e and I�e′ is trivial in +πt +1(UX). Moreover, the map +D � +X → Sub(πt +1(UX)), �e �→ I�e, +is an injection, where Sub(−) denotes the set of closed subgroups of (−). +(3) Write Ine(πt +1(UX)) for the set of inertia subgroups in πt +1(UX), namely the image of the map D � +X → +Sub(πt +1(UX)). Then Ine(πt +1(UX)) can be mono-anabelian reconstructed from πt +1(UX). In particular, the +set of marked points DX and π1(X) can be mono-anabelian reconstructed from πt +1(UX). +sec-2 +3.2. The set of marked points. We maintain the notation introduced in +curves +2.1.1. Moreover, we suppose that gX ≥ 2 +and nX > 0. +3.2.1. +We will prove that the set of marked points can be regarded as a quotient set of a set of cohomological classes +of a suitable covering of curves (i.e. Proposition +pro-2 +3.3). The main idea is the following: By taking a suitable étale +covering with a prime degree f : (Y, DY ) → (X, DX), for every marked point x ∈ DX, there exists a set of tame +coverings with a prime degree which is totally ramified over the inverse image f −1(x). Then x can be regarded as the +set of cohomological classes corresponding to such coverings. +triple +3.2.2. +Let h : (W, DW ) → (X, DX) be a connected Galois tame covering over k. We put +Ramh := {e ∈ DX | h is ramified over e}. + +10 +ZHI HU, YU YANG, AND RUNHONG ZONG +Let (Y, DY ) be a smooth pointed stable curve over k. We shall say that +(ℓ, d, f : (Y, DY ) → (X, DX)) +is an mp-triple associated to (X, DX) if the following conditions hold: (i) ℓ and d are prime numbers distinct from +each other such that (ℓ, p) = (d, p) = 1 and ℓ ≡ 1 (mod d); then all dth roots of unity are contained in Fℓ; (ii) f is +a Galois étale covering over k whose Galois group is isomorphic to µd, where µd ⊆ F× +ℓ denotes the subgroup of dth +roots of unity. Here, “mp” means “marked points”. +Then we have a natural injection H1 +ét(Y, Fℓ) ֒→ H1 +ét(UY , Fℓ) induced by the natural surjection πt +1(UY ) ։ π1(Y ). +Note that every non-zero element of H1 +ét(UY , Fℓ) induces a connected Galois tame covering of (Y, DY ) of degree ℓ. +We obtain an exact sequence +0 → H1 +ét(Y, Fℓ) → H1 +ét(UY , Fℓ) → Div0 +DY (Y ) ⊗ Fℓ → 0 +with a natural action of µd. +sec31aaa +3.2.3. +Let (Div0 +DY (Y ) ⊗ Fℓ)µd ⊆ Div0 +DY (Y ) ⊗ Fℓ be the subset of elements on which µd acts via the character +µd ֒→ F× +ℓ and M ∗ +Y ⊆ H1 +ét(UY , Fℓ) the subset of elements whose images are non-zero elements of (Div0 +DY (Y )⊗Fℓ)µd. +For each α ∈ M ∗ +Y , write gα : (Yα, DYα) → (Y, DY ) for the tame covering induced by α. We define ǫ : M ∗ +Y → Z, +where ǫ(α) := #DYα. Denote by +MY := {α ∈ M ∗ +Y | #Ramgα = d} = {α ∈ M ∗ +Y | ǫ(α) = ℓ(dnX − d) + d}. +Note that MY is non-empty. +For each α ∈ MY , since the image of α is contained in (Div0 +DY (Y ) ⊗ Fℓ)µd, we obtain that the action of µd +on Ramgα ⊆ DY is transitive. Thus, there exists a unique marked point eα ∈ DX such that f(y) = eα for each +y ∈ Ramgα. +For each e ∈ DX, we put +MY,e := {α ∈ MY | gα is ramified over f −1(e)}. +Then, for any marked points e, e′ ∈ DX distinct from each other, we have MY,e ∩ MY,e′ = ∅ and the disjoint union +MY = +� +e∈DX +MY,e. +315 +3.2.4. +Next, we define a pre-equivalence relation ∼ on MY as follows: Let α, β ∈ MY . Then α ∼ β if λα + µβ ∈ +MY for each λ, µ ∈ F× +ℓ for which λα + µβ ∈ M ∗ +Y . Then we have the following proposition. +pro-2 +Proposition 3.3. The pre-equivalence relation ∼ on MY is an equivalence relation. Moreover, the map +ϑX : MY / ∼→ DX, [α] �→ eα, +is a bijection, where [α] denotes the image of α in MY / ∼. +Proof. Let β, γ ∈ MY . If Ramgβ = Ramgγ, then, for each λ, µ ∈ F× +ℓ for which λβ + µγ ̸= 0, we have Ramgλβ+µγ = +Ramgβ = Ramgγ. Thus we obtain that β ∼ γ. On the other hand, if β ∼ γ, we have Ramgβ = Ramgγ. Otherwise, we +have #Ramgβ+γ = 2d. This means that β ∼ γ if and only if Ramgβ = Ramgγ. Then ∼ is an equivalence relation on +MY . +Let us prove that ϑX is a bijection. It is easy to see that ϑX is an injection. On the other hand, for each e ∈ DX, +the structure of the maximal pro-ℓ tame fundamental groups implies that we may construct a connected tame Galois +covering of h : (Z, DZ) → (Y, DY ) such that the element of H1 +ét(UY , Fℓ) induced by h is contained in MY . Then ϑX +is a surjection. This completes the proof of Proposition +pro-2 +3.3. +□ + +TOPOLOGICAL STRUCTURES OF MODULI SPACES OF CURVES AND ANABELIAN GEOMETRY IN POSITIVE CHARACTERISTIC +11 +rem-2-1 +Remark 3.4. We claim that the set MY / ∼ does not depend on the choices of mp-triples associated to (X, DX). Let +(ℓ∗, d∗, f ∗ : (Y ∗, DY ∗) → (X, DX)) +be an arbitrary mp-triple associated to (X, DX). Hence we obtain a resulting set MY ∗/ ∼ and a natural bijection ϑ∗ +X : +MY ∗/ ∼→ DX. We will prove that there exists a natural bijection δ : MY ∗/ ∼ +≃ +−→ MY / ∼ such that ϑ∗ +X = ϑX ◦ δ. +First, suppose that ℓ ̸= ℓ∗ and d ̸= d∗. Then we may construct a natural bijection δ : MY ∗/ ∼ +≃ +−→ MY / ∼ as +follows. Let α ∈ MY and α∗ ∈ MY ∗. Write (Yα, DYα) → (Y, DY ) and (Yα∗, DYα∗) → (Y ∗, DY ∗) for the Galois +tame coverings induced by α and α∗, respectively. We consider the following fiber product in the category of smooth +pointed stable curves +(Yα, DYα) ×(X,DX) (Yα∗, DYα∗) +which is a smooth pointed stable curve over k. Thus, we obtain a connected tame covering (Yα, DYα) ×(X,DX) +(Yα∗, DYα∗) → (X, DX) of degree dd∗ℓℓ∗. Then it is easy to check that ϑX([α]) = ϑ∗ +X([α∗]) if and only if the +cardinality of the set of marked points of (Yα, DYα) ×(X,DX) (Yα∗, DYα∗) is equal to dd∗(ℓℓ∗(nX − 1) + 1). We put +[α] := δ([α∗]) if ϑX([α]) = ϑ∗ +X([α∗]). Moreover, by the construction above, we obtain that ϑ∗ +X = ϑX ◦ δ. In the +general case, we may choose an mp-triple +(ℓ∗∗, d∗∗, f ∗∗ : (Y ∗∗, DY ∗∗) → (X, DX)) +associated to (X, DX) such that ℓ∗∗ ̸= ℓ, ℓ∗∗ ̸= ℓ∗, d∗∗ ̸= d, and d∗∗ ̸= d∗. Hence we obtain a resulting set MY ∗∗/ ∼ +and a natural bijection ϑ∗∗ +X : MY ∗∗/ ∼→ DX. Then the proof given above implies that there are natural bijections +δ1 : MY ∗∗/ ∼ +≃ +−→ MY / ∼ and δ2 : MY ∗∗/ ∼ +≃ +−→ MY ∗/ ∼. Thus, we may put +δ := δ1 ◦ δ−1 +2 +: MY ∗/ ∼ +≃ +−→ MY / ∼ . +rem-2-2 +Remark 3.5. Let H ⊆ πt +1(UX) be an arbitrary open normal subgroup and fH : (XH, DXH) → (X, DX) the Galois +tame covering over k induced by the natural inclusion H ֒→ πt +1(UX). Let +(ℓ, d, f : (Y, DY ) → (X, DX)) +be an mp-triple associated to (X, DX) such that (#(πt +1(UX)/H), ℓ) = (#(πt +1(UX)/H), d) = 1. Then we obtain an +mp-triple +(ℓ, d, g : (Z, DZ) := (Y, DY ) ×(X,DX) (XH, DXH) → (XH, DXH)) +associated to (XH, DXH) induced by (ℓ, d, f : (Y, DY ) → (X, DX)), where (Y, DY ) ×(X,DX) (XH, DXH) denotes +the fiber product in the category of smooth pointed stable curves. The mp-triple associated to (XH, DXH) induces a +set MZ/ ∼ which can be identified with the set of marked points DXH of (XH, DXH) by applying Proposition +pro-2 +3.3. +Moreover, for each eX ∈ DX and each αY,eX ∈ MY,eX, αY,eX induces an element +αZ = +� +eXH ∈f −1 +H (eX) +αZ,eXH +over (Z, DZ) via the natural morphism (Z, DZ) → (Y, DY ), where αZ,eXH ∈ MZ,eXH . On the other hand, for each +e′ +XH ∈ DXH and each e′ +X ∈ DX, we have that fH(e′ +XH) = e′ +X if and only if there exists an element αY,e′ +X ∈ MY,e′ +X +such that the following two conditions hold: +• the element α′ +Z, induced by αY,e′ +X via the natural morphism (Z, DZ) → (Y, DY ), can be represented by a +linear combination +α′ +Z = +� +eXH ∈SXH +α′ +Z,eXH , +where SXH is a subset of DXH, and αZ,eXH ∈ MZ,eXH ; +• e′ +XH ∈ SXH. + +12 +ZHI HU, YU YANG, AND RUNHONG ZONG +lem-1 +Lemma 3.6. Let (ℓ, d, f : (Y, DY ) → (X, DX)) be a triple associated to (X, DX) and gY the genus of Y . Then we +have #(MY,e) = ℓ2gY +1 − ℓ2gY , e ∈ DX. Moreover, we have #(MY ) = nX(ℓ2gY +1 − ℓ2gY ). +Proof. Let e ∈ DX. Write De ⊆ DY for the set f −1(e). Then MY,e can be naturally regarded as a subset of +H1 +ét(Y \ De, Fℓ) via the natural open immersion Y \ De ֒→ Y. Write Le for the Fℓ-vector space generated by MY,e in +H1 +ét(Y \ De, Fℓ). Then we have MY,e = Le \ H1 +ét(Y, Fℓ). Write He for the quotient Le/H1 +ét(Y, Fℓ). We have an exact +sequence as follows: +0 → H1 +ét(Y, Fℓ) → Le → He → 0. +Since the action of µd on f −1(e) is transitive, we obtain dimFℓ(He) = 1. On the other hand, since dimFℓ(H1 +ét(Y, Fℓ)) = +2gY , we obtain #(MY,e) = ℓ2gY +1 − ℓ2gY . Thus, we have #(MY ) = nX(ℓ2gY +1 − ℓ2gY ). This completes the proof +of the lemma. +□ +sec-3 +3.3. Reconstructions of inertia subgroups. We maintain the notation introduced in +curves +2.1.1. +3.3.1. +We will prove that the inertia subgroups of marked points can be mono-anabelian reconstructed from πt +1(UX) +(i.e. Proposition +them-1 +3.10). The main idea is as follows: Let H ⊆ πt +1(UX) be an arbitrary normal open subgroup +and (XH, DXH) → (X, DX) the tame covering corresponding to H. Firstly, by using some numerical conditions +induced by the Riemann-Hurwitz formula, the étale fundamental group π1(X) can be mono-anabelian reconstructed +from πt +1(UX). Then the results obtained in Section +sec-2 +3.2 implies that DX can be mono-anabelian reconstructed from +πt +1(UX). Moreover, DXH can also be mono-anabelian reconstructed from H. Secondly, since the natural injection +H ֒→ πt +1(UX) induces a map of sets of cohomological classes obtained in Section +sec-2 +3.2, we obtain that the natural map +DXH → DX can be mono-anabelian reconstructed from H ֒→ πt +1(UX). Thus, by taking a cofinal system of open +normal subgroups of πt +1(UX), we obtain a new mono-anabelian reconstruction of Ine(πt +1(UX)). +3.3.2. +First, we have the following lemma. +lem-2 +Lemma 3.7. +(1) The prime number p (i.e., the characteristic of k) can be mono-anabelian reconstructed from πt +1(UX). +(2) The étale fundamental group π1(X) can be mono-anabelian reconstructed from πt +1(UX). +Proof. (1) Let P be the set of prime numbers, and let Q be an arbitrary open subgroup of πt +1(UX) and rQ an integer +such that +#{l ∈ P | rQ = dimFl(Qab ⊗ Fl)} = ∞. +Then we see immediately that the characteristic of k is the unique prime number p such that there exists an open +subgroup T ⊆ πt +1(UX) and rT ̸= dimFp(T ab ⊗ Fp). +(2) Let H be an arbitrary open normal subgroup of πt +1(UX). We denote by (XH, DXH) the smooth pointed stable +curve of type (gXH, nXH) over k induced by H, and denote by fH : (XH, DXH) → (X, DX) the morphism of +smooth pointed stable curves over k induced by the natural inclusion H ֒→ πt +1(UX). We note that fH is étale if and +only if gXH − 1 = #(πt +1(UX)/H)(gX − 1). We put +Et(πt +1(UX)) := {H ⊆ πt +1(UX) is an open normal subgroup : gXH − 1 = #(πt +1(UX)/H)(gX − 1)}. +Moreover, Proposition +proposition 1 +3.2 (1) implies that gXH and gX can be mono-anabelian reconstructed from H and πt +1(UX), +respectively. Then the set Et(πt +1(UX)) can be mono-anabelian reconstructed from πt +1(UX). We obtain that +π1(X) = πt +1(UX)/ +� +H∈Et(πt +1(UX)) +H. +This completes the proof of the lemma. +□ + +TOPOLOGICAL STRUCTURES OF MODULI SPACES OF CURVES AND ANABELIAN GEOMETRY IN POSITIVE CHARACTERISTIC +13 +gpmptriple +3.3.3. +Suppose gX ≥ 2. Let us define a group-theoretical object corresponding to an mp-triple which was introduced +in +triple +3.2.2. We shall say that (ℓ, d, y) is an mp-triple associated to πt +1(UX) if the following two conditions hold: +• ℓ and d are prime numbers distinct from each other such that (ℓ, p) = (d, p) = 1 and ℓ ≡ 1 (mod d); then all +d-th roots of unity are contained in Fℓ; +• y ∈ Hom(π1(X), µd) such that y ̸= 0, where µd ⊆ F× +ℓ denotes the subgroup of d-th roots of unity. +3.3.4. +Moreover, by applying Lemma +lem-2 +3.7, there is a triple (ℓ, d, y) associated to πt +1(UX) which can be mono- +anabelian reconstructed from πt +1(UX). Let f : (Y, DY ) → (X, DX) be a Galois étale covering induced by y. +Then we see immediately that (ℓ, d, f : (Y, DY ) → (X, DX)) is an mp-triple associated to (X, DX) defined in +triple +3.2.2. We denote by πt +1(UY ) the kernel of the composition of the surjections πt +1(UX) ։ π1(X) +y։ µd. Since +H1 +ét(Y, Fℓ) ∼= Hom(π1(Y ), Fℓ) and H1 +ét(UY , Fℓ) ∼= Hom(πt +1(UY ), Fℓ), Lemma +lem-2 +3.7 implies immediately that the fol- +lowing exact sequence +0 → H1 +ét(Y, Fℓ) → H1 +ét(UY , Fℓ) → Div0 +DY (Y ) ⊗ Fℓ → 0 +can be mono-anabelian reconstructed from πt +1(UY ). Thus, Proposition +proposition 1 +3.2 (1) implies that the set MY / ∼ defined in +315 +3.2.4 can be mono-anabelian reconstructed from πt +1(UY ). Note that, by Remark +rem-2-1 +3.4, the set MY / ∼ does not depend +on the choices of mp-triples. Then we put +Dgp +X := MY / ∼, +where “gp" means “group-theoretical". By Proposition +pro-2 +3.3, we may identify Dgp +X with the set of marked points DX +of (X, DX) via the bijection ϑX : Dgp +X +≃ +−→ DX defined in Proposition +pro-2 +3.3. +pro-3 +Proposition 3.8. Let H ⊆ πt +1(UX) be an arbitrary open normal subgroup and +fH : (XH, DXH) → (X, DX) +the morphism of smooth pointed stable curves over k induced by the natural inclusion H ֒→ πt +1(UX). Suppose +gX ≥ 2. Then the sets Dgp +X and Dgp +XH can be mono-anabelian reconstructed from πt +1(UX) and H, respectively. +Moreover, the inclusion H ֒→ πt +1(UX) induces a map γH,πt +1(UX) : Dgp +XH → Dgp +X such that the following commutative +diagram holds: +Dgp +XH +ϑXH +−−−−→ DXH +γH,πt +1(UX )� +�γfH +Dgp +X +ϑX +−−−−→ DX, +where γfH denotes the map of the sets of marked points induced by fH. +Proof. We only need to prove the “moreover" part of Proposition +pro-3 +3.8. We maintain the notation introduced in Remark +rem-2-2 +3.5. Note that, for each eX ∈ DX and each eXH ∈ DXH, the sets MY,eX and MZ,eXH can be mono-anabelian +reconstructed from πt +1(UX) and H, respectively. Then the “moreover" part follows from Remark +rem-2-2 +3.5. +□ +rem-pro-3-1 +Remark 3.9. We maintain the notation introduced in Proposition +pro-3 +3.8. Let π1(XH) be the étale fundamental group of +XH. Then we have a natural surjection H ։ π1(XH). Note that π1(XH) admits an action of πt +1(UX)/H induced by +the outer action of πt +1(UX)/H on H which is induced by the exact sequence +1 → H → πt +1(UX) → πt +1(UX)/H → 1. +Moreover, the action of πt +1(UX)/H on π1(XH) induces an action of πt +1(UX)/H on Dgp +XH. On the other hand, it is +easy to check that the action of πt +1(UX)/H on Dgp +XH coincides with the natural action of πt +1(UX)/H on DXH when +we identify Dgp +X with DX. + +14 +ZHI HU, YU YANG, AND RUNHONG ZONG +3.3.5. +We have the following result. +them-1 +Proposition 3.10. Write Ine(πt +1(UX)) for the set of inertia subgroups in πt +1(UX). Then Ine(πt +1(UX)) can be mono- +anabelian reconstructed from πt +1(UX). +Proof. Let CX := {Hi}i∈Z>0 be a set of open normal subgroups of πt +1(UX) such that lim +←−i πt +1(UX)/Hi ∼= πt +1(UX) +(i.e., a cofinal system of open normal subgroups). +Let �e ∈ D � +X. For each i ∈ Z>0, we write (XHi, DXHi) for the smooth pointed stable curve of type (gXHi , nXHi ) +induced by Hi and eXHi ∈ DXHi for the image of �e. Then we obtain a sequence of marked points +ICX +�e +: · · · �→ eXH2 �→ eXH1 +induced by CX. Note that the sequence ICX +�e +admits a natural action of πt +1(UX). We may identify the inertia subgroup +I�e associated to �e with the stabilizer of ICX +�e +. +Moreover, since Proposition +proposition 1 +3.2 (1) implies that (gXHi , nXHi ) can be mono-anabelian reconstructed from Hi, by +choosing a suitable set of open normal subgroups CX, we may assume that gXH1 ≥ 2. If nXH1 = 0, Proposition +them-1 +3.10 +is trivial. Then we may assume that nXH1 > 0. +On the other hand, Proposition +pro-3 +3.8 implies that, for each Hi, i ∈ Z>0, the set Dgp +XHi can be mono-anabelian +reconstructed from Hi. For each eXHi ∈ DXHi , we denote by +egp +XHi := ϑ−1 +XHi (eXHi ). +Then the sequence of marked points ICX +�e +induces a sequence +ICX +�egp : · · · �→ egp +XH2 �→ egp +XH1 . +By applying the “moreover” part of Proposition +pro-3 +3.8, we see that ICX +�egp can be mono-anabelian reconstructed from CX. +Then Remark +rem-pro-3-1 +3.9 implies that the stabilizer of ICX +�egp is equal to the stabilizer of ICX +�e +. This completes the proof of the +proposition. +□ +sec-4 +3.4. Reconstructions of inertia subgroups via surjections. In this subsection, we will prove that the mono- +anabelian reconstructions obtained in Proposition +them-1 +3.10 are compatible with any open continuous homomorphisms +(i.e. Theorem +them-2 +3.18). +sett331 +3.4.1. Settings. Let (Xi, DXi), i ∈ {1, 2}, be a smooth pointed stable curve of type (gX, nX) over an algebraically +closed field ki of characteristic p > 0, UXi := Xi \ DXi, πt +1(UXi) the tame fundamental group of UXi, and π1(Xi) +the étale fundamental group of Xi. Then Lemma +lem-2 +3.7 implies that π1(Xi) can be mono-anabelian reconstructed from +πt +1(UXi). Moreover, in this subsection, we suppose that nX > 0, and that φ : πt +1(UX1) ։ πt +1(UX2) is an arbitrary +open continuous surjective homomorphism of profinite groups. +Note that, since (Xi, DXi), i ∈ {1, 2}, is a smooth pointed stable curve of type (gX, nX), φ induces a natural +surjection φp′ : πt +1(UX1)p′ ։ πt +1(UX2)p′, where (−)p′ denotes the maximal prime-to-p quotient of (−). Since +πt +1(UXi)p′, i ∈ {1, 2}, is topologically finitely generated, and πt +1(UX1)p′ is isomorphic to πt +1(UX2)p′ as abstract +profinite groups, we obtain that φp′ : πt +1(UX1)p′ +≃ +−→ πt +1(UX2)p′ is an isomorphism ( +FJ +[FJ, Proposition 16.10.6]). +3.4.2. +We explain the main idea in the proof of Theorem +them-2 +3.18. Let H2 ⊆ πt +1(UX2) be an arbitrary open normal +subgroup and H1 := φ−1(H2) ⊆ πt +1(UX1). We write (XHi, DXHi), i ∈ {1, 2}, for the smooth pointed smooth curve +of type (gXHi , nXHi ) over ki induced by Hi. To prove the compatibility, we need to prove that, for any prime number +ℓ ̸= p, the weight-monodromy filtration of Hab +2 +⊗ Fℓ induces the weight-monodromy filtration of Hab +1 +⊗ Fℓ via the +natural surjection φ|H1 : H1 ։ H2. Note that the weight 1 part of Hab +i +⊗ Fℓ corresponds to π1(XHi)ab ⊗ Fℓ, and the +weight 2 part of Hab +i +⊗ Fℓ corresponds to the image of the subgroup of Hi generated by the inertia subgroups of the +marked points of DXHi. The key observation is as follows: + +TOPOLOGICAL STRUCTURES OF MODULI SPACES OF CURVES AND ANABELIAN GEOMETRY IN POSITIVE CHARACTERISTIC +15 +The inequality of the limit of p-averages (see Proposition +coro-p-average +3.11 (1) below) +Avrp(H1) ≥ Avrp(H2) +of H1 and H2 induced by the surjection φ|H1 : H1 ։ H2 plays a role of the comparability of +“Galois actions” in the theory of the anabelian geometry of curves over algebraically closed fields of +characteristic p > 0. +paverage +3.4.3. +Firstly, we have the following proposition. +Proposition 3.11. +coro-p-average +(1) Let (X, DX) be a pointed stable curve of type (gX, nX) over an algebraically closed field k of characteristic +p > 0, UX := X \ DX, and πt +1(UX) the tame fundamental group of UX. Let r ∈ N be a natural number, and +let Kpr−1 be the kernel of the natural surjection πt +1(UX) ։ πt +1(UX)ab ⊗ Z/(pr − 1)Z, where (−)ab denotes +the abelianization of (−). Then we have +Avrp(πt +1(UX)) := lim +r→∞ +dimFp(Kab +pr−1 ⊗ Fp) +#(πt +1(UX)ab ⊗ Z/(pr − 1)Z) = +� +gX − 1, +if nX ≤ 1, +gX, +if nX > 1. +(2) We maintain the setting introduced in +sett331 +3.4.1. Let H2 ⊆ πt +1(UX2) be an open normal subgroup such that +([πt +1(UX2) : H2], p) = 1 and H1 := φ−1(H2). Write gHi, i ∈ {1, 2}, for the genus of the smooth pointed +stable curve over ki corresponding to Hi ⊆ πt +1(UXi). Then we have gH1 ≥ gH2. +Proof. (1) is the Tamagawa’s result concerning the limit of p-averages of πt +1(UX) ( +T4 +[T4, Theorem 0.5]). Let us prove +(2). The surjection φ induces a surjection φp′ : πt +1(UX1)p′ ։ πt +1(UX2)p′, where (−)p′ denotes the maximal prime- +to-p quotient of (−). Moreover, since πt +1(UXi)p′, i ∈ {1, 2}, is topologically finitely generated, and πt +1(UX1)p′ +is isomorphic to πt +1(UX2)p′ as abstract profinite groups (since the types of (X1, DX1) and (X2, DX2) are equal to +(gX, nX)), we obtain that φp′ is an isomorphism (cf. +FJ +[FJ, Proposition 16.10.6]). +On the other hand, since [πt +1(UX1) : H1] = [πt +1(UX2) : H2] and ([πt +1(UX2) : H2], p) = 1, we obtain that the +natural homomorphism φp′ +H : Hp′ +1 ։ Hp′ +2 induced by φH := φ|H1 : H1 ։ H2 is also an isomorphism. This implies +#(Hab +1 +⊗ Z/(pr − 1)Z) = #(Hab +2 +⊗ Z/(pr − 1)Z) +for all r ∈ N. Let KHi,pr−1, i ∈ {1, 2}, be the kernel of the natural surjection Hi ։ Hab +i +⊗ Z/(pr − 1)Z. Then the +surjection φH implies +Avrp(H1) := lim +r→∞ +dimFp(Kab +H1,pr−1 ⊗ Fp) +#(Hab +1 +⊗ Z/(pr − 1)Z) ≥ Avrp(H2) := lim +r→∞ +dimFp(Kab +H2,pr−1 ⊗ Fp) +#(Hab +2 ⊗ Z/(pr − 1)Z). +Thus, the corollary follows from (2). +□ +3.4.4. +We have the following lemmas. +lem-3 +Lemma 3.12. Let ℓ be a prime number distinct from p. Then the isomorphism (φp′)−1 : πt +1(UX2)p′ +≃ +−→ πt +1(UX1)p′ +induces an isomorphism +ψℓ +X : H1 +ét(X1, Fℓ) ≃ Hom(π1(X1), Fℓ) +≃ +−→ Hom(π1(X2), Fℓ) ≃ H1 +ét(X2, Fℓ). +Proof. Let f1 : (Y1, DY1) → (X1, DX1) be an étale covering of degree ℓ over k1. Write f2 : (Y2, DY2) → (X2, DX2) +for the connected Galois tame covering of degree ℓ over k2 induced by φp′. Then we will prove that f2 is also an étale +covering over k2. +Write gY1 and gY2 for the genus of Y1 and Y2, respectively. Since f1 is an étale covering of degree ℓ, the Riemann- +Hurwitz formula implies gY1 = ℓ(gX1 − 1) + 1. On the other hand, the Riemann-Hurwitz formula implies gY2 = + +16 +ZHI HU, YU YANG, AND RUNHONG ZONG +ℓ(gX2 − 1) + 1 + 1 +2(ℓ − 1)#(Ramf2). By applying Proposition +coro-p-average +3.11 (2), the surjection φ implies gY1 ≥ gY2. This +means #(Ramf2) = 0. So f2 is an étale covering over k2. Then the morphism (φp′)−1 induces an injection +ψℓ +X : Hom(π1(X1), Fℓ) ֒→ Hom(π1(X2), Fℓ). +Furthermore, since dimFℓ(Hom(π1(X1), Fℓ)) = dimFℓ(Hom(π1(X2), Fℓ)) = 2gX, we obtain that ψℓ +X is a bijection. +This completes the proof of the lemma. +□ +lem-4 +Lemma 3.13. Suppose gX ≥ 2. Then the surjection φ : πt +1(UX1) ։ πt +1(UX2) induces a bijection +ρφ : Dgp +X1 +≃ +−→ Dgp +X2, +and the bijection ρφ can be mono-anabelian reconstructed from φ. +Proof. Let (ℓ, d, y2) be an mp-triple associated to πt +1(UX2) (see +gpmptriple +3.3.3). Then Lemma +lem-3 +3.12 implies that φ induces an +mp-triple (ℓ, d, y1) associated to πt +1(UX1), where y1 := (ψd +X)−1(y2) ∈ Hom(π1(X1), µd). +Let fi : (Yi, DYi) → (Xi, DXi), i ∈ {1, 2}, be the étale covering of degree d over ki induced by yi. Then the +mp-triple (ℓ, d, yi) associated to πt +1(UXi) determines an mp-triple +(ℓ, d, fi : (Yi, DYi) → (Xi, DXi)) +associated to (Xi, DXi) over ki. Note that the types of (Y1, DY1) and (Y2, DY2) are equal. +Write πt +1(UYi), i ∈ {1, 2}, for the kernel of πt +1(UXi) ։ π1(Xi) +yi +։ µd. By replacing (Xi, DXi) by (Yi, DYi), +Lemma +lem-3 +3.12 implies that (φ|p′ +πt +1(UY1 ))−1 induces a commutative diagram as follows: +0 −−−−→ H1 +ét(Y1, Fℓ) −−−−→ H1 +ét(UY1, Fℓ) −−−−→ Div0 +DY1 (Y1) ⊗ Fℓ −−−−→ 0 +ψℓ +Y +� +ψt,ℓ +Y +� +� +0 −−−−→ H1 +ét(Y2, Fℓ) −−−−→ H1 +ét(UY2, Fℓ) −−−−→ Div0 +DY2 (Y2) ⊗ Fℓ −−−−→ 0, +where all the vertical arrows are isomorphisms. We note that H1 +ét(Yi, Fℓ), H1 +ét(UYi, Fℓ), and Div0 +DYi (Yi) ⊗ Fℓ, i, ∈ +{1, 2}, are naturally isomorphic to Hom(π1(Yi), Fℓ), Hom(πt +1(UYi), Fℓ), and Hom(πt +1(UYi), Fℓ)/Hom(π1(Yi), Fℓ), +respectively. Then Lemma +lem-2 +3.7 implies that the commutative diagram above can be mono-anabelian reconstructed from +φ|πt +1(UY1 ) : πt +1(UY1) ։ πt +1(UY2). +Write MYi ⊆ M ∗ +Yi for the subsets of H1 +ét(UYi, Fℓ) defined in +sec31aaa +3.2.3. Since the actions of µd on the exact sequences +are compatible with the isomorphisms appearing in the commutative diagram above, we have ψt,ℓ +Y (M ∗ +Y1) = M ∗ +Y2. +Next, we prove ψt,ℓ +Y (MY1) = MY2. +Let α1 ∈ MY1 and gα1 : (Yα1, DYα1) → (Y1, DY1) the Galois tame covering of degree ℓ over k1 induced by α1. +Write gα2 : (Yα2, DYα2) → (Y2, DY2) for the Galois tame covering of degree ℓ over k2 induced by α2 := ψt,ℓ +Y (α1). +Write gYα1 and gYα2 for the genus of Yα1 and Yα2, respectively. Then Proposition +coro-p-average +3.11 (2) and the Riemann-Hurwitz +formula imply that gYα1 −gYα2 = 1 +2(d−#(Ramgα2 ))(ℓ−1) ≥ 0. This means d−#(Ramgα2 ) ≥ 0. Since α2 ∈ M ∗ +Y2, +we have d | #(Ramgα2 ). Thus, either #(Ramgα2 ) = 0 or #(Ramgα2 ) = d holds. +If #(Ramgα2 ) = 0, then gα2 is an étale covering over k2. Then Lemma +lem-3 +3.12 implies that gα1 is an étale covering +over k1. This provides a contradiction to the fact that α1 ∈ MY1. Then we have #(Ramgα2 ) = d. This means +α2 ∈ MY2. Thus, we obtain ψt,ℓ +Y (MY1) ⊆ MY2. On the other hand, Lemma +lem-1 +3.6 implies #(MY1) = #(MY2). We +have ψt,ℓ +Y : MY1 +≃ +−→ MY2. Then Proposition +pro-2 +3.3 implies that ψt,ℓ +Y induces a bijection +ρφ : Dgp +X1 +≃ +−→ Dgp +X2. +Moreover, since MYi and M ∗ +Yi can be mono-anabelian reconstructed from πt +1(UYi), the bijection ρφ can be mono- +anabelian reconstructed from φ. This completes the proof of the lemma. +□ + +TOPOLOGICAL STRUCTURES OF MODULI SPACES OF CURVES AND ANABELIAN GEOMETRY IN POSITIVE CHARACTERISTIC +17 +sec334 +3.4.5. +Let H2 ⊆ πt +1(UX2) be an arbitrary open normal subgroup and H1 := φ−1(H2). We write (XHi, DXHi ), +i ∈ {1, 2}, for the smooth pointed stable curve of type (gXHi , nXHi ) over ki induced by Hi and fHi : (XHi, DXHi ) → +(Xi, DXi) for the Galois tame coverings over ki induced by the inclusion Hi ֒→ πt +1(UXi). Moreover, Proposition +pro-3 +3.8 +implies that the inclusion Hi ֒→ πt +1(UXi) induces a map γHi,πt +1(UXi ) : Dgp +XHi → Dgp +Xi which fits into the following +commutative diagram: +Dgp +XHi +ϑXHi +−−−−→ DXHi +γHi,πt +1(UXi )� +�γfHi +Dgp +Xi +ϑXi +−−−−→ DXi, +where γfHi denotes the map of the sets of marked points induced by fHi. We may identify πt +1(UX1)/H1 with +πt +1(UX2)/H2 via the isomorphism πt +1(UX1)/H1 +≃ +−→ πt +1(UX2)/H2 induced by φ, and denote by G := πt +1(UX1)/H1 ∼= +πt +1(UX2)/H2. Then we have the following lemma. +lem-5 +Lemma 3.14. Suppose that gX ≥ 2, and that (gXH1 , nXH1 ) = (gXH2 , nXH2 ). Then the commutative diagram of +profinite groups +H1 +φ|H1 +−−−−→ +H2 +� +� +πt +1(UX1) +φ +−−−−→ πt +1(UX2) +(3.1) +induces a commutative diagram +Dgp +XH1 +ρφ|H1 +−−−−→ Dgp +XH2 +γH1,πt +1(UX1 )� +�γH2,πt +1(UX2 ) +Dgp +X1 +ρφ +−−−−→ Dgp +X2. +(3.2) +Moreover, the commutative diagram (2) can be mono-anabelian reconstructed from (1). +Proof. Proposition +pro-3 +3.8 and Lemma +lem-4 +3.13 imply the diagram +Dgp +XH1 +ρφ|H1 +−−−−→ Dgp +XH2 +γH1,πt +1(UX1 )� +�γH2,πt +1(UX2 ) +Dgp +X1 +ρφ +−−−−→ Dgp +X2 +can be mono-anabelian reconstructed from the commutative diagram of profinite groups +H1 +φ|H1 +−−−−→ +H2 +� +� +πt +1(UX1) +φ +−−−−→ πt +1(UX2). +To verify Lemma +lem-5 +3.14, it is sufficient to check that the diagram is commutative. +Let egp +XH1 ∈ Dgp +XH1 , egp +XH2 := ρφ|H1 (egp +XH1 ) ∈ Dgp +XH2 , egp +X1 := γH1,πt +1(UX1 )(egp +XH1 ) ∈ Dgp +X1, egp +X2 := (γH2,πt +1(UX2 ) ◦ +ρφ|H1)(egp +XH1 ) ∈ Dgp +X2, and egp,∗ +X1 +:= ρ−1 +φ (egp +X2) ∈ Dgp +X1. Let us prove +egp +X1 = egp,∗ +X1 . +We put Sgp +XH1 := γ−1 +H1,πt +1(UX1)(egp,∗ +X1 ) and Sgp +XH2 := γ−1 +H2,πt +1(UX2 )(egp +X2), respectively. Note that egp +XH2 ∈ Sgp +XH2 . To +verify egp +X1 = egp,∗ +X1 , it is sufficient to prove that egp +XH1 ∈ Sgp +XH1 . Moreover, for each i ∈ {1, 2}, we put +eXi := ϑXi(egp +Xi), eXHi := ϑXHi (egp +Xi), e∗ +X1 := ϑX1(egp,∗ +X1 ), SXi := Sgp +Xi, SXHi := Sgp +XHi . + +18 +ZHI HU, YU YANG, AND RUNHONG ZONG +Then to verify the lemma, we only need to prove that eXH1 ∈ ϑXH1 (SXH1 ). +Let (ℓ, d, y2) be an mp-triple associated to πt +1(UX2). Then Lemma +lem-3 +3.12 implies that φ induces an mp-triple (ℓ, d, y1) +associated to πt +1(UX1), where y1 := (ψd +X)−1(y2) ∈ Hom(π1(X1), µd). Let fi : (Yi, DYi) → (Xi, DXi), i ∈ {1, 2}, +be the tame covering of degree d over ki induced by yi. Then the mp-triple (ℓ, d, yi) associated to πt +1(UXi) induces an +mp-triple +(ℓ, d, fi : (Yi, DYi) → (Xi, DXi)) +associated to (Xi, DXi) over ki. Note that since f1 and f2 are étale, the types of (Y1, DY1) and (Y2, DY2) are equal. +On the other hand, we have an mp-triple +(ℓ, d, g2 : (Z2, DZ2) := (Y2, DY2) ×(X2,DX2 ) (XH2, DXH2 ) → (XH2, DXH2 )) +associated to (XH2, DXH2 ) induced by the natural inclusion H2 ֒→ πt +1(UX2) and the mp-triple (ℓ, d, f2 : (Y2, DY2) → +(X2, DX2)). By Lemma +lem-3 +3.12 again, we obtain an mp-triple +(ℓ, d, g1 : (Z1, DZ1) := (Y1, DY1) ×(X1,DX1 ) (XH1, DXH1 ) → (XH1, DXH1 )) +associated to (XH1, DXH1 ) induced by φ|H1 and the triple (ℓ, d, g2 : (Z2, DZ2) → (XH2, DXH2 )). +Let α2 ∈ MY2,eX2 . The final paragraph of the proof of Lemma +lem-4 +3.13 implies that we have a bijection MY1 = +� +e∈DX1 MY1,e +≃ +−→ MY2 = � +e∈DX2 MY2,e induced by φ. Then α2 induces an element α1 ∈ MY1,e∗ +X1 . Write +(Yα1, DYα1) and (Yα2, DYα2) for the smooth pointed stable curves over k1 and k2 induced by α1 and α2, respectively. +Consider the connected Galois tame covering +(Yα2, DYα2) ×(X2,DX2 ) (XH2, DXH2 ) → (Z2, DZ2) +of degree ℓ over k2, and write β2 for an element of M ∗ +Z2 corresponding to this connected Galois tame covering. Then +we have +β2 = +� +c2∈SXH2 +tc2βc2, +where tc2 ∈ (Z/ℓZ)× and βc2 ∈ MZ2,c2. On the other hand, the proof of Lemma +lem-4 +3.13 implies that β2 induces an +element +β1 := +� +c2∈SXH2 \{eXH2 } +tc2βρ−1 +φ|H1 +(c2) + teXH2 βρ−1 +φ|H1 +(eXH2 ) += +� +c2∈SXH2 \{eXH2 } +tc2βρ−1 +φ|H1 +(c2) + teXH2 βeXH1 ∈ M ∗ +Z1. +Then we have that the coefficient teXH2 of βeXH1 is not equal to 0. Thus, the composition +(Yα1, DYα1) ×(X1,DX1 ) (XH1, DXH1 ) → (Z1, DZ1) +g1 +→ (XH1, DXH1) +is tamely ramified over eXH1 . This means that eXH1 is contained in SXH1 . This completes the proof of the lemma. +□ +rem-lem-5-1 +Remark 3.15. Remark +rem-pro-3-1 +3.9 implies that Dgp +XHi , i ∈ {1, 2}, admits a natural action of G. Moreover, the commutative +diagram +Dgp +XH1 +ρφ|H1 +−−−−→ Dgp +XH2 +γH1,πt +1(UX1 )� +�γH2,πt +1(UX2 ) +Dgp +X1 +ρφ +−−−−→ Dgp +X2 +is compatible with the actions of G. + +TOPOLOGICAL STRUCTURES OF MODULI SPACES OF CURVES AND ANABELIAN GEOMETRY IN POSITIVE CHARACTERISTIC +19 +3.4.6. +Next, we prove that the condition (gXH1 , nXH1 ) = (gXH2 , nXH2 ) mentioned in Lemma +lem-5 +3.14 can be omitted. +Firstly, we treat the case of abelian groups. +lem-6 +Lemma 3.16. We maintain the notation introduced in +sec334 +3.4.5. Suppose that gX ≥ 2, and that G is an abelian group. +Then we have (gXH1 , nXH1 ) = (gXH2 , nXH2). +Proof. We write m for #G and put K2 := ker(πt +1(UX2) ։ πt +1(UX2)ab ⊗ Z/mZ). Then we see immediately that K2 +is contained in H2. Let K1 := φ−1(K2) ⊆ H1. Write (XKi, DXKi ) for the smooth pointed stable curves of type +(gXKi , nXKi ) over ki induced by Ki and fKi : (XKi, DXKi ) → (Xi, DXi) for the tame covering over ki induced by +the inclusion Ki ֒→ πt +1(UXi). We identify πt +1(UX1)/K1 with πt +1(UX2)/K2 via the isomorphism induced by φ, and +denote by A := πt +1(UX1)/K1 ≃ πt +1(UX2)/K2. +Since each p-Galois tame covering is étale (i.e., Galois tame coverings whose Galois group is a p-group), we see +immediately that (gXK1 , nXK1 ) = (gXK2 , nXK2 ). Then Lemma +lem-5 +3.14 implies that the commutative diagram +K1 +φ|K1 +−−−−→ +K2 +� +� +πt +1(UX1) +φ +−−−−→ πt +1(UX2) +of profinite groups induces a commutative diagram +Dgp +XK1 +ρφ|K1 +−−−−→ Dgp +XK2 +γK1,πt +1(UX1 )� +�γK2,πt +1(UX2 ) +Dgp +X1 +ρφ +−−−−→ Dgp +X2. +Moreover, Remark +rem-lem-5-1 +3.15 implies that the commutative diagram above admits a natural action of A. Then, for each +egp +XK1 ∈ Dgp +XK1 , the inertia subgroup Iegp +XK1 in A associated to egp +XK1 (i.e., the stabilizer of egp +XK1 under the action +of A) is equal to the inertia subgroup Iegp +XK2 in A associated to egp +XK2 := ρφ|K1 (egp +XK1 ) ∈ Dgp +XK2 . On the other +hand, write F for the kernel of the natural morphism A ։ G induced by the inclusion Ki ֒→ Hi, i ∈ {1, 2}. +Since (XHi, DXHi ) ≃ (XKi, DXKi )/F, the set of ramification indices of the Galois tame covering (XKi, DXKi ) → +(XHi, DXHi ) with Galois group F are equal to {#(F ∩ Iegp +XKi )}egp +XKi ∈Dgp +XKi . Then by the Riemann-Hurwitz formula, +we have (gXH1 , nXH1 ) = (gXH2 , nXH2 ). This completes the proof of the lemma. +□ +Next, we treat the general case. +lem-7 +Lemma 3.17. We maintain the notation introduced in +sec334 +3.4.5. Suppose that gX ≥ 2 and nX ≥ 2. Then there exists an +open normal subgroup P2 ⊆ πt +1(UX2) which is contained in H2 such that the following holds: +Write (XPi, DXPi ), i ∈ {1, 2}, for the smooth pointed stable curve of type (gXPi , nXPi ) over ki +induced by Pi, where P1 = φ−1(P2). We have (gXP1 , nXP1 ) = (gXP2 , nXP2 ). +Proof. First, suppose that G is a simple finite group. By applying Lemma +lem-6 +3.16, we may assume that G is non-abelian. +Moreover, we claim that we may assume that nX is a positive even number. Let us prove this claim. Suppose p ̸= 2. +Let R2 ⊆ πt +1(UX2) be an open subgroup such that #(πt +1(UX2)/R2) = 2, and that R2 ⊇ ker(πt +1(UX2) ։ π1(X2)) +(i.e., the cyclic Galois tame covering corresponding to R2 is étale). Let R1 := φ−1(R2) ⊆ πt +1(UX1). Then we have +that #(πt +1(UX1)/R1) = 2, and that Lemma +lem-3 +3.12 implies R1 ⊇ ker(πt +1(UX1) ։ π1(X1)). By replacing Hi and +πt +1(UXi), i ∈ {1, 2}, by Hi ∩ Ri and Ri, respectively, we may assume that nX is a even positive number. Suppose +that p = 2. Let ℓ be a prime number such that (ℓ, 2) = (ℓ, #G) = 1. By +R1 +[R1, Théorème 4.3.1], there exists an open +subgroup R∗ +2 ⊆ πt +1(UX2) such that #(πt +1(UX2)/R∗ +2) = ℓ, that R∗ +2 ⊇ ker(πt +1(UX2) ։ π1(X2)), and that +dimFp(R∗,ab +2 +⊗ Fp) > 0. + +20 +ZHI HU, YU YANG, AND RUNHONG ZONG +Let R∗ +1 := φ−1(R∗ +2) ⊆ πt +1(UX1). Then we have that #(πt +1(UX1)/R∗ +1) = ℓ, that dimFp(R∗,ab +1 +⊗ Fp) > 0, and that +Lemma +lem-3 +3.12 implies R∗ +1 ⊇ ker(πt +1(UX1) ։ π1(X1)). Thus, we may take an open subgroup R′ +2 ⊆ R∗ +2 such that +πt +1(UX2)/R′ +2 ≃ Z/2Z ⋊ Z/ℓZ, +and that R′ +2 ⊇ ker(πt +1(UX2) ։ π1(X2)). +We put R′ +1 := φ−1(R′ +2). +Then the construction of R′ +1 implies +πt +1(UX1)/R′ +1 ≃ Z/2Z ⋊ Z/ℓZ and R′ +1 ⊇ ker(πt +1(UX1) ։ π1(X1)). By replacing Hi and πt +1(UXi), i ∈ {1, 2}, +by Hi ∩ R′ +i and R′ +i, respectively, we may assume that nX is a even positive number. This completes the proof of the +claim. +Let #G := ptm′ such that (m′, p) = 1. Since nX is a positive even number, we may choose a Galois tame covering +f2 : (Y2, DY2) → (X2, DX2) +over k2 with Galois group Z/m′Z such that f2 is totally ramified over every marked point of DX2. Write (gY2, nY2) +for the type of (Y2, DY2), Q2 ⊆ πt +1(UX2) for the open normal subgroup induced by f2, Q1 := φ−1(Q2) ⊆ πt +1(UX1), +f1 : (Y1, DY1) → (X1, DX1) +for the Galois tame covering over k1 with Galois group Z/m′Z induced by the natural inclusion Q1 ֒→ πt +1(UX1), and +(gY1, nY1) for the type of (Y1, DY1). Then Lemma +lem-6 +3.16 implies that (gY1, nY1) = (gY2, nY2) and f1 is also totally +ramified over every marked point of DX1. +We consider the Galois tame covering +(Zi, DZi) := (XHi, DXHi ) ×(Xi,DXi) (Yi, DYi) → (Xi, DXi), i ∈ {1, 2}, +over ki with Galois group G × Z/m′Z which is the composition of (Zi, DZi) → (Yi, DYi) and (Yi, DYi) → +(Xi, DXi). Note that since G is a non-abelian simple finite group, (Zi, DZi) is connected. Moreover, by Abhyankar’s +lemma, we obtain that (Zi, DZi) → (Yi, DYi) is an étale covering over ki. Since (gY1, nY1) = (gY2, nY2) and +(Zi, DZi) → (Yi, DYi) is unramified, the Riemann-Hurwitz formula implies (gZ1, nZ1) = (gZ2, nZ2). +Next, let us prove the lemma in the case where G is an arbitrary finite group. Let G1 ⊆ G2 ⊆ · · · ⊆ Gn := G +be a sequence of subgroups of G such that Gi/Gi−1 is a simple group for all i ∈ {2, . . .n}. In order to verify the +lemma, we see that it is sufficient to prove the lemma when n = 2. Let N2 be the kernel of the natural homomorphism +πt +1(UX2) ։ G ։ G1 and N1 := φ−1(N2). Then by replacing G by G1 and by applying the lemma for the simple +group G1, we obtain an open normal subgroup M2 ⊆ πt +1(UX2) which is contained in N2 such that (gXM1 , nXM1 ) = +(gXM2 , nXM2 ), where M1 := φ−1(M2), and (gXMi , nXMi ), i ∈ {1, 2}, denotes the type of the smooth pointed stable +curve corresponding to Mi. +If Mi ⊆ Hi, i ∈ {1, 2}, then we may put Pi := Mi. If Hi, i ∈ {1, 2}, does not contain Mi, we put Oi := Mi ∩ Hi. +Then we have Mi/Oi ≃ G/G1. Note that G/G1 is a simple group. Then the lemma follows from the lemma when +we replace (Xi, DXi) and G by (XMi, DXMi ) and the simple group G/G1, respectively. This completes the proof of +the lemma. +□ +3.4.7. +Now, we prove the main result of the present section. +them-2 +Theorem 3.18. Let ( � +Xi, D � +Xi), i ∈ {1, 2}, be the universal tame covering of (Xi, DXi) defined in +unicov313 +3.1.2. +Let +φ : πt +1(UX1) ։ πt +1(UX2) be an arbitrary open continuous surjective homomorphism. Then the group-theoretical +algorithm of the mono-anabelian reconstruction concerning Ine(πt +1(UXi)) obtained in Proposition +them-1 +3.10 is compatible +with the surjection φ : πt +1(UX1) ։ πt +1(UX2). Namely, the following holds: Let �e2 ∈ D � +X2 and I�e2 ∈ Ine(πt +1(UX2)) +the inertia subgroup associated to �e2. Then there exists an inertia subgroup I�e1 ∈ Ine(πt +1(UX1)) associated to a point +�e1 ∈ D � +X1 such that +φ(I�e1) = I�e2, +and that the restriction homomorphism φ|I�e1 : I�e1 ։ I�e2 is an isomorphism. + +TOPOLOGICAL STRUCTURES OF MODULI SPACES OF CURVES AND ANABELIAN GEOMETRY IN POSITIVE CHARACTERISTIC +21 +Proof. If nX = 0, then the theorem is trivial. We suppose nX > 0. Let m >> 0 be an integer such that (m, p) = 1. +We put Ki := ker(πt +1(UXi) ։ πt +1(UXi)ab ⊗ Z/mZ), i ∈ {1, 2}. Write (XKi, DKi) for the smooth pointed stable +curve of type (gXKi , nXKi ) over ki induced by Ki. Moreover, the condition m >> 0 implies gXK1 = gXK2 ≥ +2, nXK1 = nXK2 ≥ 2. +By applying Lemma +lem-7 +3.17, we may choose a set of open subgroups CX2 := {H2,j}j∈Z>0 of πt +1(UX2) such that the +following three conditions hold: +• H2,1 = K2; +• lim +←−j πt +1(UX2)/H2,j ≃ πt +1(UX2) (i.e. CX2 is a cofinal system); +• write {H1,j := φ−1(H2,j)}j∈Z>0 for the set of open subgroups of πt +1(UX1) induced by φ, and for each +j ∈ Z>0, write (XHi,j, DXHi,j ), i ∈ {1, 2}, for the smooth pointed stable curve of type (gXHi,j , nXHi,j ) over +ki induced by Hi,j, then we have (gXH1,j , nXH1,j ) = (gXH2,j , nXH2,j ). +For each j ∈ Z>0, we write eXH2,j ∈ DXH2,j for the image of �e2. Then we obtain a sequence of marked points +I +CX2 +�e2 +: · · · �→ eH2,2 �→ eH2,1. +Proposition +pro-3 +3.8 implies that, for each H2,j, j ∈ Z>0, the set Dgp +XH2,j can be mono-anabelian reconstructed from H2,j. +For each eXH2,j ∈ DXH2,j , we denote by +egp +XH2,j := ϑ−1 +XH2,j (eXH2,j ). +Then the sequence of marked points ICX +�e2 +induces a sequence +ICX +�egp +2 +: · · · �→ egp +XH2,2 �→ egp +XH2,1 . +Then Remark +rem-pro-3-1 +3.9 implies that the inertia subgroup associated to �e2 is equal to the stabilizer of ICX +�egp +2 . +By Lemma +lem-5 +3.14 and Lemma +lem-7 +3.17, I +CX2 +�egp +2 +induces a sequence as follows: +· · · �→ egp +XH1,2 := ρ−1 +φ|H1,2 (egp +XH2,2 ) ∈ Dgp +XH1,2 �→ egp +XH1,1 := ρ−1 +φ|H1,1(egp +XH2,1 ) ∈ Dgp +XH1,1 +with an action of I�e2. Then Proposition +them-1 +3.10 implies that we have a sequence +· · · �→ eXH1,2 := ϑXH1,2 (egp +XH1,2 ) ∈ DXH1,2 �→ eXH1,1 := ϑXH1,1 (egp +XH1,1 ) ∈ DXH1,1 +with an action of I�e2 +Let Kker(φ) be the subfield of �K induced by the closed subgroup ker(φ) of πt +1(UX1), � +X1,ker(φ) the normalization of +X1 in Kker(φ), and D � +X1,ker(φ) the inverse image of DX1 in � +X1,ker(φ). Then the sequence +· · · �→ eXH1,2 �→ eXH1,1 . +determines a point �e1,ker(φ) ∈ D � +X1,ker(φ). We choose a point of �e1 ∈ D � +X1 such that the image of �e1 in D � +X1,ker(φ) +is �e1,ker(φ). Then we have φ(I�e1) = I�e2. Moreover, since I�e1 and I�e2 are isomorphic to �Z(1)p′, the restriction +homomorphism φ|I�e1 is an isomorphism. This completes the proof of the theorem. +□ +sec-new6 +3.5. Reconstructions of additive structures via surjections. We maintain the settings introduced in +sett331 +3.4.1. +3.5.1. +Let �e2 be an arbitrary point of D � +X2. By applying Theorem +them-2 +3.18, there exists a point �e1 ∈ D � +X1 such that +φ|I�e1 : I�e1 +≃ +−→ I�e2 is an isomorphism. Write Fp,i, i ∈ {1, 2}, for the algebraic closure of Fp in ki. We put +F�ei := (I�ei ⊗Z (Q/Z)p′ +i ) ⊔ {∗�ei}, i ∈ {1, 2}, +where {∗�ei} is an one-point set, and (Q/Z)p′ +i denotes the prime-to-p part of Q/Z which can be canonically identified +with � +(p,m)=1 µm(ki). Moreover, let a�ei be a generator of I�ei. Then we have a natural bijection +I�ei ⊗Z (Q/Z)p′ +i +≃ +−→ Z ⊗Z (Q/Z)p′ +i , a�ei ⊗ 1 �→ 1 ⊗ 1. + +22 +ZHI HU, YU YANG, AND RUNHONG ZONG +Thus, we obtain the following bijections +I�ei ⊗Z (Q/Z)p′ +i +≃ +−→ Z ⊗Z (Q/Z)p′ +i +≃ +−→ +� +(p,m)=1 +µm(ki) +≃ +−→ F +× +p,i. +This means that F�ei can be identified with Fp,i as sets, and hence admits a structure of field whose multiplicative group +is I�ei ⊗Z (Q/Z)p′ +i , and whose zero element is ∗�ei. +3.5.2. +We will prove that φ|I�e1 : I�e1 +≃ +−→ I�e2 induces an isomorphism F�e1 +≃ +−→ F�e2 as fields (i.e. Proposition +pro-4 +3.19). +The main idea is as follows: First, we reduce the problem to the case where nX = 3 by applying Theorem +them-2 +3.18. +Second, the field structure of F�ei (i.e., the set of isomorphisms of F�ei and Fp,i as fields) can be translated to certain +problem concerning generalized Hasse-Witt invariants (e.g. γχi(Mχi) in the proof of Proposition +pro-4 +3.19). Then by +applying Theorem +them-2 +3.18 again, we obtained the result by comparing γχ1(Mχ1) with γχ2(Mχ2). +3.5.3. +We have the following proposition. +pro-4 +Proposition 3.19. The field structure of F�ei, i ∈ {1, 2}, can be mono-anabelian reconstructed from πt +1(UXi). More- +over, the isomorphism φ|I�e1 : I�e1 +≃ +−→ I�e2 induces an isomorphism +θφ,�e1,�e2 : F�e1 +≃ +−→ F�e2 +as fields. +Proof. First, we claim that we may assume nX = 3. If gX = 0, then nX ≥ 3. Suppose that gX ≥ 1. Theorem +them-2 +3.18 implies that φ : πt +1(UX1) ։ πt +1(UX2) induces an open continuous surjection φét : π1(X1) ։ π1(X2). Let +H′ +2 ⊆ π1(X2) be an open normal subgroup such that #(π1(X2)/H′ +2) ≥ 3 and H′ +1 := (φét)−1(H′ +2). Write Hi ⊆ +πt +1(UXi), i ∈ {1, 2}, for the inverse image of H′ +i of the natural surjection πt +1(UXi) ։ π1(Xi), and (XHi, DXHi ) +for the smooth pointed stable curve of type (gXHi , nXHi ) over ki induced by Hi. Note that gXH1 = gXH2 ≥ 2 and +nXH1 = nXH2 ≥ 3. By replacing (Xi, DXi) by (XHi, DXHi), we may assume gX ≥ 2 and nX ≥ 3. The surjection +φ induces a bijection +DX1 +ϑ−1 +X1 +−−−→ Dgp +X1 +ρφ +−→ Dgp +X2 +ϑX2 +−−−→ DX2. +Let D′ +X1 := {e1,1, e1,2, e1,3} ⊆ DX1 and D′ +X2 := {e2,1 := ϑX2 ◦ρφ◦ϑ−1 +X1(e1,1), e2,2 := ϑX2 ◦ρφ◦ϑ−1 +X1(e1,2), e2,3 := +ϑX2 ◦ ρφ ◦ ϑ−1 +X1(e1,3)} ⊆ DX2. Then (Xi, D′ +Xi), i ∈ {1, 2}, is a smooth pointed stable curve of type (gX, 3) over ki. +Write Ii, i ∈ {1, 2}, for the closed subgroup of πt +1(UXi) generated by the inertia subgroups associated to the elements +of D � +Xi whose images in DXi are contained in DXi \ D′ +Xi. Then we have an isomorphism +πt +1(Xi \ D′ +Xi) ∼= πt +1(UXi)/Ii, i ∈ {1, 2}. +Moreover, Theorem +them-2 +3.18 implies that φ induces an open continuous surjective homomorphism +φ′ : πt +1(X1 \ D′ +X1) ։ πt +1(X2 \ D′ +X2). +Thus, by replacing (Xi, DXi), πt +1(UXi), and φ by (Xi, D′ +Xi), πt +1(Xi \ D′ +Xi), and φ′, respectively, we may assume +nX = 3. +Let r ∈ N. We denote by Fpr,�ei, i ∈ {1, 2}, the unique subfield of F�ei whose cardinality is equal to pr. On the +other hand, we fix any finite field Fpr of cardinality pr and an algebraic closure Fp of Fp. By Proposition +them-1 +3.10, we +have that F× +pr,�ei = I�ei/(pr − 1) can be mono-anabelian reconstructed from πt +1(UXi). Then reconstructing the field +structure of Fpr,�ei is equivalent to reconstructing Homfields(Fpr,�ei, Fpr) as a subset of Homgroup(F× +pr,�ei, F× +pr). Note +that, in order to reconstruct the field structure of F�ei, it is sufficient to reconstruct the subset Homfields(Fpr,�ei, Fpr) for +r in a cofinal subset of N with respect to division. + +TOPOLOGICAL STRUCTURES OF MODULI SPACES OF CURVES AND ANABELIAN GEOMETRY IN POSITIVE CHARACTERISTIC +23 +Let χi ∈ Homgroups(πt +1(UXi)ab ⊗ Z/(pr − 1)Z, F× +pr). Write Hχi for the kernel of πt +1(UXi) ։ πt +1(UXi)ab ⊗ +Z/(pr − 1)Z +χi +→ F× +pr, Mχi for Hab +χi ⊗ Fp, and (XHχi , DXHχi ) for the smooth pointed stable curve over ki induced by +Hχi. We define +Mχi[χi] := {a ∈ Mχi ⊗Fp Fp | σ(a) = χi(σ)a for all σ ∈ πt +1(UXi)ab ⊗ Z/(pr − 1)Z} +and γχi(Mχi) := dimFp(Mχi[χi]) (i.e. a generalized Hasse-Witt invariant (see +Y5 +[Y5, Section 2.2])). Then +T4 +[T4, Remark +3.7] implies γχi(Mχi) ≤ gX + 1. Moreover, we define two maps +Resi,r : Homgroups(πt +1(UXi)ab ⊗ Z/(pr − 1)Z, F× +pr) → Homgroups(F× +pr,�ei, F× +pr), +Γi,r : Homgroups(πt +1(UXi)ab ⊗ Z/(pr − 1)Z, F× +pr) → Z≥0, χi �→ γχi(Mχi), +where the map Resi,r is the restriction with respect to the natural inclusion F× +pr,�ei ֒→ πt +1(UXi)ab ⊗ Z/(pr − 1)Z. +Let m0 be the product of all prime numbers ≤ p − 2 if p ̸= 2, 3 and m0 = 1 if p = 2, 3. Let r0 be the order of p in +the multiplicative group (Z/m0Z)×. Then +T4 +[T4, Claim 5.4] implies the following result: +there exists a constant C(gX) which depends only on gX such that, for each r > logp(C(gX) + 1) +divisible by r0, we have +Homfields(Fpr,�ei, Fpr) = Homsurj +groups(F× +pr,�ei, F× +pr) \ Resi,r(Γ−1 +i,r ({gX + 1})), i ∈ {1, 2}, +where Homsurj +groups(−, −) denotes the set of surjections whose elements are contained in +Homgroups(−, −). +Let κ2 ∈ Homgroups(πt +1(UX2)ab ⊗ Z/(pr − 1)Z, F× +pr). Then φ induces a character +κ1 ∈ Homgroups(πt +1(UX1)ab ⊗ Z/(pr − 1)Z, F× +pr). +Moreover, the surjection φ|Hκ1 induces a surjection Mκ1[κ1] ։ Mκ2[κ2]. Suppose that κ2 ∈ Γ−1 +2,r({gX + 1}). The +surjection Mκ1[κ1] ։ Mκ2[κ2] implies γκ1(Mκ1) = gX + 1. This means κ1 ∈ Γ−1 +1,r({gX + 1}). On the other hand, +the isomorphism φ|I�e1 : I�e1 +≃ +−→ I�e2 induces an injection +Res2,r(Γ−1 +2,r({gX + 1})) ֒→ Res1,r(Γ−1 +1,r({gX + 1})). +Since #(Homfields(Fpr,�e1, Fpr)) += +#(Homfields(Fpr,�e2, Fpr)), +we obtain that φ|I�e1 +induces a bijection +Homfields(Fpr,�e2, Fpr) +≃ +−→ Homfields(Fpr,�e1, Fpr). Thus, φ|I�e1 induces a bijection +Homfields(F�e2, Fp) +≃ +−→ Homfields(F�e1, Fp). +If we choose Fp = F�e2, then the image of idF�e2 via the bijection above induces an isomorphism θφ,�e1,�e2 : F�e1 +≃ +−→ F�e2 +as fields. This completes the proof of the proposition. +□ +4. MAIN THEOREMS +sec-5 +4.1. The first main theorem. In this subsection, we apply the results obtained in the previous sections to prove that +the curves of type (0, n) over Fp can be reconstructed group-theoretically from open continuous homomorphism (i.e. +Theorem +them-3 +4.4). + +24 +ZHI HU, YU YANG, AND RUNHONG ZONG +4.1.1. Settings. We fix some notation. Let (Xi, DXi), i ∈ {1, 2}, be a smooth pointed stable curve of type (gX, nX) +over an algebraically closed field ki of characteristic p > 0, UXi := Xi \ DXi, πt +1(UXi) the tame fundamental +group of UXi, π1(Xi) the étale fundamental group of Xi, and ( � +Xi, D � +Xi) the universal tame covering of (Xi, DXi) +associated to πt +1(UXi) ( +unicov313 +3.1.2). Let km +i , i ∈ {1, 2}, be the minimal algebraically closed subfield of ki over which UXi +can be defined. Thus, by considering the function field of Xi, we obtain a smooth pointed stable curve (Xm +i , DXm +i ) +(i.e., a minimal model of (Xi, DXi) (cf. +T3 +[T3, Definition 1.30 and Lemma 1.31])) such that UXi ∼= UXm +i ×km +i ki as +ki-schemes, where UXm +i := Xm +i \ DXm +i . Note that πt +1(UXm +i ) is naturally isomorphic to πt +1(UXi). We shall denote by +Fp,i the algebraic closure of Fp in ki. Moreover, we put +d(Xi,DXi ) := +� +0, +if km +i ∼= Fp,i, +1, +if km +i ̸∼= Fp,i. +4.1.2. +Firstly, we have the following lemma. +lemsurj +Lemma 4.1. Let φ : πt +1(UX1) → πt +1(UX2) be an arbitrary open continuous homomorphism. Then φ is a surjection. +Proof. We denote by Πφ the image of φ which is an open subgroup of πt +1(UX2). Let (Xφ, DXφ) be the smooth +pointed stable curve of type (gXφ, nXφ) over k2 induced by Πφ and fφ : (Xφ, DXφ) → (X2, DX2) the tame covering +of smooth pointed stable curves over k2 induced by the inclusion Πφ ֒→ πt +1(UX2). Since fφ is a tame covering, +we have that nXφ ≥ nX. On the other hand, if gX = 0, we have gφ ≥ 0. If gX > 0, the Riemann-Hurwitz +formula implies gXφ ≥ [πt +1(UX2) : Πφ](gX − 1) + 1 ≥ gX. Then we have gφ ≥ gX and nXφ ≥ nX. Note that +πt +1(UX1) ։ Πφ ֒→ πt +1(UX2) implies +2gX + nX − 1 ≥ 2gXφ + nXφ − 1 ≥ 2gX + nX − 1. +Then we obtain that 2gX + nX − 1 = 2gXφ + nXφ − 1. Moreover, Proposition +coro-p-average +3.11 (ii) and the natural surjection +πt +1(UX1) ։ Πφ induced by φ imply that gX ≥ gXφ. Then we obtain that gX = gXφ. Thus, we have (gX, nX) = +(gXφ, nXφ). This means that the tame covering fφ : (Xφ, DXφ) → (X2, DX2) is totally ramified over every marked +point of DX2. +Let us prove [πt +1(UX2) : Πφ] = 1. Suppose [πt +1(UX2) : Πφ] ̸= 1. Since fφ is totally ramified, the Riemann- +Hurwitz formula implies gXφ > gX if nX ̸= 0 and gX ̸= 0. This is a contradiction. If nX = 0, the Riemann-Hurwitz +formula implies gX = 1 if gX ̸= 0. This contradicts the assumption that (Xi, DXi) is a pointed stable curve. Then +we obtain gX = gXφ = 0. Moreover, by applying the Riemann-Hurwitz formula again, since nX = nXφ, we +obtain nX = nXφ = 2. This contradicts the assumption that (Xi, DXi) is pointed stable curve. Then we have +[πt +1(UX2) : Πφ] = 1. This means that φ is a surjection. +□ +4.1.3. Further settings. In the remainder of this subsection, we suppose (gX, nX) = (0, n). We fix two marked +points e1,∞, e1,0 ∈ DX1 distinct from each other. Moreover, we choose any field k′ +1 ∼= k1, and choose any isomor- +phism ϕ1 : X1 +≃ +−→ P1 +k′ +1 as schemes such that ϕ1(e1,∞) = ∞ and ϕ1(e1,0) = 0. Then the set of k1-rational points +X1(k1) \ {e1,∞} is equipped with a structure of Fp-module via the bijection ϕ1. Note that since any k′ +1-isomorphism +of P1 +k′ +1 fixing ∞ and 0 is a scalar multiplication, the Fp-module structure of X1(k1) \ {e1,∞} does not depend on the +choices of k′ +1 and ϕ1 but depends only on the choices of e1,∞ and e1,0. Then we shall say that X1(k1) \ {e1,∞} is +equipped with a structure of Fp-module with respect to e1,∞ and e1,0. +By applying Theorem +them-2 +3.18, in the next lemma, we will prove that Tamagawa’s group-theoretical criterion (i.e., +T2 +[T2, +Lemma 3.3]) for linear conditions is compatible with arbitrary open continuous surjective homomorphism. +lem-8 +Lemma 4.2. Let φ : πt +1(UX1) ։ πt +1(UX2) be an open continuous surjective homomorphism. By Lemma +lem-4 +3.13, φ +induces a bijection ρφ : Dgp +X1 +≃ +−→ Dgp +X2. We may identify Dgp +Xi, i ∈ {1, 2}, with DXi via the bijection ϑXi : Dgp +Xi +≃ +−→ + +TOPOLOGICAL STRUCTURES OF MODULI SPACES OF CURVES AND ANABELIAN GEOMETRY IN POSITIVE CHARACTERISTIC +25 +DXi. Write e2,∞ and e2,0 for ρφ(e1,∞) and ρφ(e1,0), respectively. Let +� +e1∈DX1 \{e1,∞,e1,0} +be1e1 = e1,0 +be a linear condition with respect to e1,∞ and e1,0 on (X1, DX1), where be1 ∈ Fp for each e1 ∈ DX1 \ {e1,∞, e1,0}. +Then the linear condition +� +e1∈DX1 \{e1,∞,e1,0} +be1ρφ(e1) = ρφ(e1,0) = e2,0 +with respect to e2,∞ and e2,0 on (X2, DX2) also holds. +Proof. Let �e2,∞ ∈ D � +X2 be a point over e2,∞. The set F�e2,∞ := (I�e2,∞ ⊗Z (Q/Z)p′ +2 ) ⊔ {∗�e2,∞} admits a structure +of field, and Proposition +pro-4 +3.19 implies that the field structure can be mono-anabelian reconstructed from πt +1(UX2). +Theorem +them-2 +3.18 implies that there exists a point �e1,∞ ∈ D � +X1 over e1,∞ such that φ(I�e1,∞) = �e2,∞. By Proposition +pro-4 +3.19 again, the set F�e1,∞ := (I�e1,∞ ⊗Z (Q/Z)p′ +1 ) ⊔ {∗�e1,∞} admits a structure of field which can be mono-anabelian +reconstructed from πt +1(UX1), and φ induces an isomorphism θφ,�e1,∞,�e2,∞ : F�e1,∞ +≃ +−→ F�e2,∞ as fields. +For each e1 ∈ DX1, we take b′ +e1 ∈ Z≥0 such that b′ +e1 ≡ be1 (mod p) and +� +e1∈DX1 \{e1,∞,e1,0} +b′ +e1 ≥ 2. +Let r ≥ 1 such that pr − 2 ≥ � +e1∈DX1 \{e1,∞,e1,0} b′ +e1. For each �e1 ∈ D � +X1 over e1, write I�e1,ab for the image of the +natural morphism I�e1 ֒→ πt +1(UX1) ։ πt +1(UX1)ab. Moreover, since the image of I�e1,ab does not depend on the choices +of �e1, we may write Ie1 for I�e1,ab. The structure of maximal prime-to-p quotient of πt +1(UX1) implies that πt +1(UX1)ab +is generated by {Ie1}e1∈DX1 , and that there exists a generator ae1, e1 ∈ DX1, of Ie1 such that � +e1∈DX1 ae1 = 1. We +define +Ie1,∞ → Z/(pr − 1)Z, ae1,∞ �→ 1, +Ie1,0 → Z/(pr − 1)Z, ae1,0 �→ ( +� +e1∈DX1 \{e1,∞,e1,0} +b′ +e1) − 1, +and +Ie1 → Z/(pr − 1)Z, ae1 �→ −b′ +e1, e1 ∈ DX1 \ {e1,∞, e1,0}. +Then the homomorphisms of inertia subgroups defined above induces a surjection δ1 : πt +1(UX1) ։ πt +1(UX1)ab ։ +Z/(pr − 1)Z. Note that ker(δ1) does not depend on the choices of the generators {ae1}e1∈DX1 . +Let I�e2 := φ(I�e1), �e1 ∈ D � +X1, and Ie2, e2 ∈ DX2 be the image of the natural homomorphism I�e2 ֒→ πt +1(UX2) ։ +πt +1(UX2)ab. Since (p, pr − 1) = 1, by Theorem +them-2 +3.18, δ1 and the isomorphism φp′ : πt +1(UX1)p′ +≃ +−→ πt +1(UX2)p′ imply +the following homomorphisms of inertia subgroups: +Ie2,∞ → Z/(pr − 1)Z, ae2,∞ �→ 1, +Ie2,0 → Z/(pr − 1)Z, ae2,0 �→ ( +� +e1∈DX1 \{e1,∞,e1,0} +b′ +e1) − 1, +and +Ie2 → Z/(pr − 1)Z, ae2 �→ −b′ +e1, e2 ∈ DX2 \ {e2,∞, e2,0}, +where ae2, e2 ∈ DX2, denotes the element induced by ae1, e1 ∈ DX1, via φ. Then the homomorphisms of inertia +subgroups defined above induces a sujection δ2 : πt +1(UX2) ։ πt +1(UX2)ab ։ Z/(pr − 1)Z. +We put Hδi := ker(δi), Mδi := Hab +δi ⊗ Fp, i ∈ {1, 2}. Write (XHδi , DXHδi ) for the smooth pointed stable curve +over ki induced by Hδi, where Hδ1 = φ−1(Hδ2). The Fp-vector space Mδi admits a natural action of I�ei,∞ via +conjugation which coincides with the action via the following character +χI�ei,∞,r : I�ei,∞ ֒→ πt +1(UXi) +δi։ Z/(pr − 1)Z = I�ei,∞/(pr − 1) ֒→ F× +�ei,∞, i ∈ {1, 2}. + +26 +ZHI HU, YU YANG, AND RUNHONG ZONG +We put Mδi[χI�ei,∞ ,r] +:= +{a +∈ +Mδi ⊗Fp F�ei,∞ | σ(a) += +χI�ei,∞,r(σ)a for all σ +∈ +I�ei,∞} (in fact, +dimF�ei,∞ (Mδi[χI�ei,∞ ,r]) is the first generalized Hasse-Witt invariant associated to the tame covering of UXi corre- +sponding to Hδi ⊆ πt +1(UXi) (see +Y5 +[Y5, Section 2.2])). Since the action of I�ei,∞ on Mδi is semi-simple, we obtain a +surjection Mδ1[χI�e1,∞,r] ։ Mδ2[χI�e2,∞,r] induced by φ|Hδ1 and θφ,�e1,∞,�e2,∞. On the other hand, the third and the +final paragraphs of the proof of +T2 +[T2, Lemma 3.3] imply that the linear condition +� +e1∈DX1 \{e1,∞,e1,0} +be1e1 = e1,0 +with respect to e1,∞ and e1,0 on (X1, DX1) holds if and only if Mδ1[χI�e1,∞ ,r] = 0. Thus, we obtain Mδ2[χI�e2,∞ ,r] = +0. Then the third and the final paragraphs of the proof of +T2 +[T2, Lemma 3.3] imply that the linear condition +� +e1∈DX1\{e1,∞,e1,0} +be1ρφ(e1) = e2,0 +with respect to e2,∞ and e2,0 on (X2, DX2) holds. This completes the proof of the lemma. +□ +rem-lem-8-1 +Remark 4.3. Note that, if X1 = P1 +k, then the linear condition is as follows: +� +e1∈DX1 \{∞,0} +be1e1 = 0 +with respect to ∞ and 0. +4.1.4. +Now, we prove the first main theorem of the present paper. +them-3 +Theorem 4.4. We maintain the notation and settings introduced above. Then we have the following claims. +(1) d(Xi,DXi ), i ∈ {1, 2}, can be mono-anabelian reconstructed from πt +1(UXi). +(2) Suppose km +1 ∼= Fp,1. Then the set of open continuous homomorphisms +Homop +pg(πt +1(UX1), πt +1(UX2)) +is non-empty if and only if UXm +1 ∼= UXm +2 as schemes. In particular, if this is the case, we have km +2 ∼= Fp,2 and +Homop +pg(πt +1(UX1), πt +1(UX2)) = Isompg(πt +1(UX1), πt +1(UX2)). +Proof. Firstly, let us prove (2). The “if" part of (2) is trivial. We treat the “only if" part of (2). Suppose that +Homop +pg(πt +1(UX1), πt +1(UX2)) is a non-empty set, and let φ ∈ Homop +pg(πt +1(UX1), πt +1(UX2)). Then Lemma +lemsurj +4.1 implies +that φ is a surjection. +We identify Dgp +Xi, i ∈ {1, 2}, with DXi via the bijection ϑXi : Dgp +Xi +≃ +−→ DXi. Since φ is a surjection, Lemma +lem-4 +3.13 +implies that φ induces a bijection ρφ : DX1 +≃ +−→ DX2. We put e2,0 := ρφ(e1,0) and e2,∞ := ρφ(e1,∞). Let �e2,0 ∈ D � +X2 +be a point over e2,0. Theorem +them-2 +3.18 implies that there exists a point �e1,0 ∈ D � +X1 over e1,0 such that φ(I�e1,0) = I�e2,0. +Then F�ei,0 := (I�ei,0 ⊗Z (Q/Z)p′ +i )⊔{∗�ei,0}, i ∈ {1, 2}, admits a structure of field. Moreover, Proposition +pro-4 +3.19 implies +that the field structure can be mono-anabelian reconstructed from πt +1(UXi), and that φ induces a field isomorphism +θφ,�e1,0,�e2,0 : F�e1,0 +≃ +−→ F�e2,0. +Proposition +proposition 1 +3.2 (1) implies that n can be mono-anabelian reconstructed from πt +1(UXi), i ∈ {1, 2}. If n = 3, +(ii) is trivial, so we may assume n ≥ 4. Moreover, since km +1 +∼= Fp,1, without loss of generality, we may assume +k1 = Fp,1 = F�e1,0, X1 = P1 +Fp,1, and +DX1 = {e1,∞ = ∞, e1,0 = 0, e1,1 = 1, e1,2, . . . , e1,n−2}. +Here, e1,2, . . . , e1,n−2 ∈ Fp,1 \ {e1,0, e1,1} are distinct from each other. +Step 1: In this step, we will construct a linear condition on a certain tame covering of (X1, DX1). + +TOPOLOGICAL STRUCTURES OF MODULI SPACES OF CURVES AND ANABELIAN GEOMETRY IN POSITIVE CHARACTERISTIC +27 +We see that there exists a natural number r prime to p such that Fp(ζr) contains rth roots of e1,2, . . . , e1,n−2, where +ζr denotes a fixed primitive rth root of unity in Fp,1. Let s := [Fp(ζr) : Fp]. For each e1,u ∈ {e1,2, . . . , e1,n−2}, we +fix an rth root e1/r +1,u in Fp,1. Then we have +e1/r +1,u = +s−1 +� +v=0 +b1,uvζv +r , u ∈ {2, . . ., n − 2}, +where b1,uv ∈ Fp for each u ∈ {2, . . . , n − 2} and each v ∈ {0, . . . , s − 1}. +Let X1 \ {e1,∞} = SpecFp,1[x1], fH1 : (XH1, DXH1) → (X1, DX1) the Galois tame covering over Fp,1 with +Galois group Z/rZ determined by the equation yr +1 = x1, and H1 the open normal subgroup of πt +1(UX1) induced by +the tame covering fH1. Then fH1 is totally ramified over {e1,∞ = ∞, e1,0 = 0} and is étale over DX1 \ {∞, 0}. Note +that XH1 = P1 +Fp,1, and the points of DXH1 over {e1,∞, e1,0} are {eH1,∞ := ∞, eH1,0 := 0}. We put +eH1,u := e1/r +1,u ∈ DXH1 , u ∈ {2, . . ., n − 2}, ev +H1,1 := ζv +r ∈ DXH1 , v ∈ {0, . . . , s − 1}. +Thus, we obtain a linear condition +eH1,u = +s−1 +� +v=0 +b1,uvev +H1,1 +with respect to eH1,∞ and eH1,0 on (XH1, DXH1 ) for each u ∈ {2, . . . , n − 2}. +Step 2: In this step, we will prove that the linear condition on a certain tame covering of (X1, DX1) constructed in +Step 1 induces a linear condition on a certain tame covering of (X2, DX2) via the surjection φ. +Write H2 for φ(H1). Since (r, p) = 1, we have the following commutative diagram of profinite groups: +H1 +φ|H1 +−−−−→ +H2 +� +� +πt +1(UX1) +φ +−−−−→ πt +1(UX2) +� +� +Z/rZ +Z/rZ. +We denote by fH2 : (XH2, DXH2) → (X2, DX2) the Galois tame covering over Fp,2 with Galois group Z/rZ induced +by H2. Note that Lemma +lem-6 +3.16 implies that (XH1, DXH1) and (XH2, DXH2 ) are equal types. Moreover, Lemma +lem-5 +3.14 +implies that the following commutative diagram can be mono-anabelian reconstructed from the commutative diagram +of profinite groups above: +DXH1 +ρφ|H1 +−−−−→ DXH2 +� +� +DX1 +ρφ +−−−−→ DX2. +We put +e2,∞ := ρφ(e1,∞), e2,u := ρφ(e1,u), u ∈ {0, . . ., n − 2}, +eH2,∞ := ρφ|H1(eH1,∞), eH2,0 := ρφ|H1(eH1,0), eH2,u := ρφ|H1 (eH1,u), u ∈ {2, . . ., n − 2}, +and +ev +H2,1 := ρφ|H1(ev +H1,1), v ∈ {0, . . . , s − 1}. +Remark +rem-lem-5-1 +3.15 implies that fH2 is totally ramified over {e2,∞, e2,0} and is étale over X2 \ {e2,∞, e2,0}. Then we +may assume that X2 = P1 +k2, and that e2,∞ = ∞, e2,0 = 0, e2,1 = 1. We regard e2,u, u ∈ {2, . . . , n − 2}, as an +element of k2 \ {e2,0, e2,1}. Moreover, we have eH2,∞ = ∞ and eH2,0 = 0. + +28 +ZHI HU, YU YANG, AND RUNHONG ZONG +We put ξr := θφ,�e1,0,�e2,0(ζr) which is an rth root of unity in F�e2,0. Since ζr(ev +H1,1) = ev+1 +H1,1, we obtain ξr(ev +H2,1) = +ev+1 +H2,1, v ∈ {0, . . ., s − 2}. By applying Lemma +lem-8 +4.2 for φ|H1 : H1 ։ H2, the following linear condition +eH2,u = +s−1 +� +v=0 +b1,uvξv +r(e0 +H2,1) +with respect to eH2,∞ and eH2,0 on (XH2, DXH2 ) holds for each u ∈ {2, . . . , n − 2}. Since (eH2,u)r = e2,u, +u ∈ {2, . . . , n − 2}, we obtain +e2,u = ( +s−1 +� +v=0 +b1,uvξv +r (e0 +H2,1))r. +Moreover, if we put e0 +H2,1 = 1, then we obtain that +e2,u = ( +s−1 +� +v=0 +b1,uvξv +r )r +for each u ∈ {2, . . . , n − 2}. Since θφ,�e1,0,�e2,0(ζr) = ξr, we have +UX1 = UXm +1 = P1 +Fp,1 \ {e1,∞ = ∞, e1,0 = 0, e1,1 = 1, e1,2, . . . , e1,n−2} +≃ +−→ P1 +F�e2,0 \ {e2,∞ = ∞, e2,0 = 0, e2,1 = 1, θφ,�e1,0,�e2,0(e1,2), . . . , θφ,�e1,0,�e2,0(e1,n−2)} +∼= P1 +Fp,2 \ {e2,∞ = ∞, e2,0 = 0, e2,1 = 1, e2,2, . . . , e2,n−2} +and +P1 +Fp,2 \ {e2,∞ = ∞, e2,0 = 0, e2,1 = 1, e2,2, . . . , e2,n−2} ×Fp,2 k2 ∼= UX2. +This means UXm +1 ∼= UXm +2 as schemes. In particular, we have km +2 ∼= Fp,2. +Finally, we prove that Homop +pg(πt +1(UX1), πt +1(UX2)) = Isompg(πt +1(UX1), πt +1(UX2)). The “⊇" part is trivial. We only +need to prove the “⊆" part. We may assume Homop +pg(πt +1(UX1), πt +1(UX2)) ̸= ∅. Let φ′ ∈ Homop +pg(πt +1(UX1), πt +1(UX2)). +Then πt +1(UX1) is isomorphic to πt +1(UX2) as abstract profinite groups. By Lemma +lemsurj +4.1, φ′ is a surjection. Then +FJ +[FJ, +Proposition 16.10.6] implies that φ′ is an isomorphism. Thus, we obtain φ′ ∈ Isompro-gps(πt +1(UX1), πt +1(UX2)). This +completes the proof of (2). +Next, let us prove (1). Without loss of generality, we only treat the case where i = 1. Moreover, let (X, DX) := +(X1, DX1), +DX = {e∞ = ∞, e0 = 0, e1 = 1, e2, . . . , en−2}, +k := k1, and Fp := F�e0. Let (r, Q) be a pair such that the following two conditions hold: +• (r, p) = 1; +• Q is an open normal subgroup of πt +1(UX) such that πt +1(UX)/Q ∼= Z/rZ, and that the Galois tame covering +fQ : (XQ, DXQ) → (X, DX) over k induced by Q is totally ramified over {e∞, e0} and is étale over +DX \ {e∞, e0}. +By applying Theorem +them-2 +3.18, we see immediately that the set of pairs defined above can be mono-anabelian recon- +structed from πt +1(UX). +We fix a primitive r-th root of unity ζr in Fp and put sr := [Fp(ζr) : Fp]. Moreover, we put +eQ,∞ := ∞, eQ,0 := 0, ev +Q,1 := ζv +r ∈ DXQ, v ∈ {0, . . . sr − 1}, +and let eQ,u ∈ DXQ, u ∈ {2, . . ., n}, such that fQ(eQ,u) = eu. Denote by +LQ,u := {eQ,u − +sr−1 +� +v=0 +buvev +Q,1 | buv ∈ Fp} ∩ {0}, u ∈ {2, . . . , n − 2}. +By applying arguments similar to the arguments given in the proof of (2) above, we have that d(X,DX) = 0 if and +only if there exists a pair (r, Q) defined above such that LQ,u ̸= ∅ for each u ∈ {2, . . . , n − 2}. Then the third and + +TOPOLOGICAL STRUCTURES OF MODULI SPACES OF CURVES AND ANABELIAN GEOMETRY IN POSITIVE CHARACTERISTIC +29 +the final paragraphs of the proof of +T2 +[T2, Lemma 3.3] implies that LQ,u, u ∈ {2, . . . , n − 2}, can be mono-anabelian +reconstructed from Q. Thus, d(X,DX) can be mono-anabelian reconstructed from πt +1(UX). This completes the proof +of the theorem. +□ +Remark 4.5. Note that Theorem +them-3 +4.4 also holds if we replace πt +1(UXi), i ∈ {1, 2}, by its maximal pro-solvable +quotient πt +1(UXi)sol. Then we obtain the following solvable version of Theorem +them-3 +4.4 which is slightly stronger than +the original theorem: +We maintain the notation introduced above. Then d(Xi,DXi), i ∈ {1, 2}, can be mono-anabelian +reconstructed from πt +1(UXi)sol. Moreover, suppose that km +1 ∼= Fp,1. Then the set of open continuous +homomorphisms +Homop +pg(πt +1(UX1)sol, πt +1(UX2)sol) +is non-empty if and only if UXm +1 +∼= UXm +2 as schemes. In particular, if this is the case, we have +km +2 ∼= Fp,2 and +Homop +pg(πt +1(UX1)sol, πt +1(UX2)sol) = Isompg(πt +1(UX1)sol, πt +1(UX2)sol). +sec-6 +4.2. The second main theorem. In this subsection, by using Theorem +them-3 +4.4, we prove a result concerning pointed +collection conjecture and the weak Hom-version conjecture (i.e. Theorem +them-4 +4.6). We maintain the notation introduced +in +moduli212 +2.1.2. +def-4 +4.2.1. +Let q ∈ M ord +0,n be an arbitrary point, k(q) an algebraic closure of k(q), and +UXq ≃ P1 +k(q) \ {a1 = 1, a2 = 0, a3 = ∞, a4, . . . , an} +as k(q)-schemes. We shall say that q is a coordinated point if either q = qgen or the following three conditions are +satisfied: +• dim(Vq) = dim(M ord +0,n ) − 1; +• there exists i ∈ {4, . . ., n} such that ai ∈ Fp; +• let ωi +n,n−1 : M ord +0,n → M ord +0,n−1 be the morphism induced by the morphism Mord +0,n → Mord +0,n−1 obtained by +forgetting the ith marked point; then ωi +n,n−1(q) is the generic point of M ord +0,n−1. +Let t be a closed point of M ord +0,n . Then there exists a set of coordinated points Pt := {qt,4, . . . , qt,n} such that +{t} = +� +qt,j∈Pt +Vqt,j . +4.2.2. +Now, we prove the second main theorem of the present paper. +Theorem 4.6. +them-4 +(1) For each closed point t ∈ M ord,cl +0,n +, the set Ct associated to t is a pointed collection (Definition +def-3 +2.4). Moreover, +for each pointed collection C ∈ Cqgen, there exists a closed point s ∈ M ord,cl +0,n +such that C = Cs. +(2) Let q ∈ M ord +0,n be an arbitrary point. Then the the natural map colleq : V cl +q +→ Cq, [t] �→ Ct, is an injection. +(3) Let q ∈ M ord +0,n be an arbitrary point. Suppose that there exists a set of coordinated points Pq such that +Vq = +� +u∈Pq +Vu. +Then the pointed collection conjecture holds for q. In particular, the pointed collection conjecture holds for +each closed point of M ord +0,n . + +30 +ZHI HU, YU YANG, AND RUNHONG ZONG +(4) Let qi ∈ M ord +0,n , i ∈ {1, 2}, be an arbitrary point. Suppose that there exists a set of coordinated points Pq1 +such that +Vq1 = +� +u∈Pq1 +Vu. +Then the weak Hom-version conjecture holds. In particular, the weak Hom-version conjecture holds when q1 +is a closed point. +Proof. Let us prove (1). We put Ft := {t′ ∈ M ord,cl +0,n +| t ∼fe t′}. Let t′′ be an arbitrary point of � +G∈πt +A(t) UG. +Then, for each G +∈ πt +A(t), Homsurj +pg (πt +1(t′′), G) is non-empty, where Homsurj +pg (−, −) denotes the subset of +Homopen +pg +(−, −) whose elements are surjections. Since πt +1(t′′) is topologically finitely generated, we obtain that the +set Homsurj +pg (πt +1(t′′), G) is finite. Then the set of open continuous homomorphisms +lim +←− +G∈πt +A(t) +Homsurj +pg (πt +1(t′′), G) = Homsurj +pg (πt +1(t′′), πt +1(t)) +is non-empty. Thus, Theorem +them-3 +4.4 implies t′′ ∈ Ft. This means +( +� +G∈πt +A(t) +UG) ∩ M ord,cl +g,n += Ft. +Since UXt can be defined over a finite field, Ft is a finite set. Then Ct is a pointed collection. +Let C ∈ Cqgen be a pointed collection and s a closed point of � +G∈C UG. By replacing t by s, and by applying +arguments similar to the arguments given in the proof above, we obtain C = Cs. +(2) follows immediately from Theorem +them-3 +4.4. Let us prove (3). If n = 4, then M ord +0,4 is a one dimensional scheme. +For each q ∈ M ord +0,4 , the pointed collection conjecture follows immediately from Theorem +them-3 +4.4. Then we may assume +n ≥ 5. To verify (iii), (ii) implies that we only need to prove that colleq is a surjection. Suppose that q is a closed point +of M ord +0,n , then (iii) follows immediately from Theorem +them-3 +4.4. +Suppose that q is a non-closed point. This means dim(Vq) ≥ 1. If q = qgen, (3) follows from (1) and (2). Let us +treat the case where q ̸= qgen. First, suppose that q is a coordinated point, and that +UXq ≃ P1 +k(q) \ {1, 0, ∞, a4, . . . , an}. +Without loss of generality, we may assume an ∈ Fp. +For each pointed collection C ⊆ Cq, by applying (1), there exists a closed point t1 ∈ M ord,cl +g,n +such that Ct1 = C. +Then we have an open continuous surjective homomorphism πt +1(q) ։ πt +1(t1). Let ω\n +n,4 : M ord +0,n → M ord +0,4 be the +morphism induced by the morphism Mord +0,n → Mord +0,4 obtained by forgetting the marked points except the first, the +second, the third, and the nth marked points. We put t′′ +1 := ω\n +n,4(t1) and q′′ := ω\n +n,4(q). Note that t′′ +1 and q′′ are closed +points of M0,4. Then Theorem +them-2 +3.18 implies that the surjection πt +1(q) ։ πt +1(t1) induces an open continuous surjective +homomorphism πt +1(q′′) ։ πt +1(t′′ +1). Thus, by Theorem +them-3 +4.4, we obtain that q′′ ∼fe t′′ +1. Then without loss of generality, +we may assume +UXt1 ≃ P1 +Fp \ {1, 0, ∞, b4, . . . , bn−1, an} +over Fp, where bi ∈ Fp for each i ∈ {4, . . . , n − 1}. +On the other hand, let ωn +n,n−1 : M ord +0,n → M ord +0,n−1 be the morphism induced by the morphism Mord +0,n → Mord +0,n−1 +obtained by forgetting the n-th marked point. We put t′ +1 := ωn +n,n−1(t1) and q′ := ωn +n,n−1(q), respectively. Since q is +a coordinated point, q′ is the generic point of M ord +0,n−1. Then we obtain t′ +1 ∈ V cl +q′ . Moreover, we see Vq = ω−1 +n,n−1(q′). +Thus, t1 = ω−1 +n,n−1(t′ +1) is a closed point of Vq. Then the pointed collection conjecture holds for q when q is a +coordinated point. + +31 +Next, we prove the general case. If Vq = � +u∈Pq Vu, then V cl +q = � +u∈Pq V cl +u and � +u∈Pq Cu = Cq. Moreover, since +we have a bijection colleu : V cl +u +≃ +−→ Cu for each u ∈ Pq, we have that +colleq : V cl +q += +� +u∈Pq +V cl +u → +� +u∈Pq +Cu = Cq +is a bijection. This completes the proof of (3). +Let us prove (4). We only need to prove the “only if" part of the weak Hom-version conjecture. Suppose that Vq2 is +not essentially contained in Vq1. This implies that there exists a closed point t2 ∈ V cl +q2 such that Ft2 ∩ Vq1 = ∅, where +Ft2 := {t′ +2 ∈ M ord,cl +0,n +| t2 ∼fe t′ +2}. By (3), we have Ct2 ̸∈ Cq1. Thus, by Lemma +lemsurj +4.1, we obtain that +Homop +pg(πt +1(q1), πt +1(t2)) = ∅. +This provides a contradiction to the assumption that Homop +pg (πt +1(q1), πt +1(q2)) is non-empty. This completes the proof +of (4). +□ +Remark 4.7. Let q ∈ Mg,n be an arbitrary point. Stevenson posed a question as follows (see +Ste +[Ste, Question 4.3] for +the case of n = 0): Does � +G∈πt +A(q) UG contain any closed points of Mg,n? By +T5 +[T5, Theorem 0.3], � +G∈πt +A(q) UG +contains a closed point of Mg,n if and only if q is a closed point of Mg,n. Furthermore, when g = 0 and q is a closed +point, the proof of Theorem +them-4 +4.6 (1) implies that +( +� +G∈πt +A(q) +UG) ∩ M cl +0,n = Fq, +where Fq := {q′ ∈ M cl +0,n | q ∼fe q′}. +REFERENCES +[FJ] +M. D. Fried, M. Jarden, Field arithmetic. Third edition. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of +Modern Surveys in Mathematics 11. Springer-Verlag, Berlin, 2008. +[G] +A. Grothendieck, Letter to G. Faltings (translation into English). Geometric Galois actions. 1. Around Grothendieck’s “Esquisse +d’un programme”. Edited by Leila Schneps and Pierre Lochak. London Mathematical Society Lecture Note Series, 242. Cambridge +University Press, Cambridge, 1997. iv+293 pp. +[K] +F. Knudsen, The projectivity of the moduli space of stable curves, II: The stacks Mg,n, Math. Scand., 52 (1983), 161¨C-199. +[M] +S. Mochizuki, The local pro-p anabelian geometry of curves. Invent. Math. 138 (1999), 319–423. +[N] +H. Nakamura, Galois rigidity of the étale fundamental groups of punctured projective lines. J. Reine Angew. Math. 411 (1990), 205- +216. +[PS] +F. Pop, M. Saïdi, On the specialization homomorphism of fundamental groups of curves in positive characteristic. Galois groups and +fundamental groups, 107–118, Math. Sci. Res. Inst. Publ., 41, Cambridge Univ. Press, Cambridge, 2003. +[R1] +M. Raynaud, Sections des fibrés vectoriels sur une courbe. Bull. Soc. math. France 110 (1982), 103–125. +[R2] +M. Raynaud, Sur le groupe fondamental d’une courbe complète en caractéristique p > 0. Arithmetic fundamental groups and non- +commutative algebra (Berkeley, CA, 1999), 335-351, Proc. Sympos. Pure Math., 70, Amer. Math. Soc., Providence, RI, 2002. +[Ste] +K. Stevenson, Galois groups of unramified covers of projective curves in characteristic p. Journal of Algebra 182 (1996), 770–804. +[T1] +A. Tamagawa, The Grothendieck conjecture for affine curves. Compositio Math. 109 (1997), 135–194. +[T2] +A. Tamagawa, On the fundamental groups of curves over algebraically closed fields of characteristic > 0. Internat. Math. Res. Notices +(1999), 853-873. +[T3] +A. Tamagawa, Fundamental groups and geometry of curves in positive characteristic. Arithmetic fundamental groups and noncom- +mutative algebra (Berkeley, CA, 1999), 297–333, Proc. Sympos. Pure Math., 70, Amer. Math. Soc., Providence, RI, 2002. +[T4] +A. Tamagawa, On the tame fundamental groups of curves over algebraically closed fields of characteristic > 0. Galois groups and +fundamental groups, 47–105, Math. Sci. Res. Inst. Publ., 41, Cambridge Univ. Press, Cambridge, 2003. +[T5] +A. Tamagawa, Finiteness of isomorphism classes of curves in positive characteristic with prescribed fundamental groups. J. Algebraic +Geom. 13 (2004), 675–724. +[Y1] +Y. Yang, On the admissible fundamental groups of curves over algebraically closed fields of characteristic p > 0, Publ. Res. Inst. +Math. Sci. 54 (2018), 649–678. + +32 +ZHI HU, YU YANG, AND RUNHONG ZONG +[Y2] +Y. Yang, On topological and combinatorial structures of pointed stable curves over algebraically closed fields of positive characteristic, +to appear in Math. Nachr. +[Y3] +Y. Yang, On the averages of generalized Hasse-Witt invariants of pointed stable curves in positive characteristic, Math. Z. 295 (2020), +1–45. +[Y4] +Y. Yang, On the existence of specialization isomorphisms of admissible fundamental groups in positive characteristic, Math. Res. Lett. +28 (2021), 1941–1959. +[Y5] +Y. Yang, Maximum generalized Hasse-Witt invariants and their applications to anabelian geometry, Selecta Math. (N.S.) 28 (2022), +Paper No. 5, 98 pp. +[Y6] +Y. +Yang, +Moduli +spaces +of +fundamental +groups +of +curves +in +positive +characteristic +I, +preprint. +See +http://www.kurims.kyoto-u.ac.jp/~yuyang/ +[Y7] +Y. Yang, Moduli spaces of fundamental groups of curves in positive characteristic II, in preparation. +[Y8] +Y. Yang, Topological and group-theoretical specializations of fundamental groups of curves in positive characteristic, preprint. See +http://www.kurims.kyoto-u.ac.jp/~yuyang/ +[Y9] +Y. Yang, On the explicit constructions of differences of tame fundamental groups of non-isomorphic curves in positive characteristic, +in preparation. +SCHOOL OF MATHEMATICS, NANJING UNIVERSITY OF SCIENCE AND TECHNOLOGY, NANJING 210094, CHINA +Email address: halfask@mail.ustc.edu.cn +RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES, KYOTO UNIVERSITY, KYOTO 606-8502, JAPAN +Email address: yuyang@kurims.kyoto-u.ac.jp +DEPARTMENT OF MATHEMATICS, NANJING UNIVERSITY, NANJING 210093, CHINA +Email address: rzong@nju.edu.cn + diff --git a/8tE4T4oBgHgl3EQfCwu_/content/tmp_files/load_file.txt b/8tE4T4oBgHgl3EQfCwu_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a8f83dd8852ef07f7db2439576a50826e5a3b1e6 --- /dev/null +++ b/8tE4T4oBgHgl3EQfCwu_/content/tmp_files/load_file.txt @@ -0,0 +1,1675 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf,len=1674 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='04864v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='AG] 12 Jan 2023 TOPOLOGICAL STRUCTURES OF MODULI SPACES OF CURVES AND ANABELIAN GEOMETRY IN POSITIVE CHARACTERISTIC ZHI HU, YU YANG, AND RUNHONG ZONG ABSTRACT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' In the present paper, we study a new kind of anabelian phenomenon concerning the smooth pointed stable curves in positive characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' It shows that the topological structures of moduli spaces of curves can be understood from the viewpoint of anabelian geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We formulate some new anabelian-geometric conjectures relating the tame fundamental groups of curves over algebraically closed fields of characteristic p > 0 to the moduli spaces of curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' These conjectures are generalized versions of the weak Isom-version of the Grothendieck conjecture for curves over algebraically closed fields of characteristic p > 0 which was formulated by Tamagawa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Moreover, we prove that the conjectures hold for certain points lying in the moduli space of curves of genus 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' CONTENTS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Introduction 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' The mystery of fundamental groups in positive characteristic 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Topology structures of moduli spaces of curves and anabelian geometry 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Main results 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Some further developments 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Structure of the present paper 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Acknowledgements 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Conjectures 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' The weak Hom-version conjecture 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' The pointed collection conjecture 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Reconstructions of marked points 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Anabelian reconstructions 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' The set of marked points 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Reconstructions of inertia subgroups 12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Reconstructions of inertia subgroups via surjections 14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Reconstructions of additive structures via surjections 21 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Main theorems 23 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' The first main theorem 23 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' The second main theorem 29 References 31 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' INTRODUCTION 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' The mystery of fundamental groups in positive characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' sec111 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let k be an algebraically closed field of characteristic p ≥ 0, and let (X, DX) be a smooth pointed stable curve of type (gX, nX) over k (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' 2gX + nX − 2 > 0, see K[K, Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1 (iv)]), where X denotes the underlying curve, DX denotes the (ordered) finite set of marked points, gX denotes the genus of X, and nX denotes the cardinality 1 2 ZHI HU, YU YANG, AND RUNHONG ZONG #(DX) of DX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We put UX := X \\ DX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' By choosing a base point of UX, we have the tame fundamental group πt 1(UX) of UX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' If p = 0, it is well-known that πt 1(UX) is isomorphic to the profinite completion of the topological fundamental group of a Riemann surface of type (gX, nX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Hence, almost no geometric information about UX can be carried out from πt 1(UX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' By contrast, if p > 0, the situation is quite different from that in characteristic 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' The tame fundamental group πt 1(UX) is very mysterious and its structure is no longer known, in particular, there exist anabelian phenomena for curves over algebraically closed fields of characteristic p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Firstly, let us explain some general background about anabelian geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' In the 1980s, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Grothendieck suggested a theory of arithmetic geometry called anabelian geometry ( G[G]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' The central question of the theory is as follows: Can we reconstruct the geometric information of a variety group-theoretically from various versions of its algebraic fundamental group?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' The original anabelian geometry suggested by Grothendieck focused on varieties over arithmetic fields, in particular, the fields finitely generated over Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' In the case of curves in characteristic 0, anabelian geometry has been deeply studied (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' N[N], T1 [T1]) and, in particular, the most important case (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=', the fields finitely generated over Q, or more general, sub-p-adic fields) has been established completely( M[M]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Note that the actions of the Galois groups of the base fields on the geometric fundamental groups play a crucial role for recovering geometric information of curves over arithmetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Next, we return to the case where k is an algebraically closed field of characteristic p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' In T2 [T2], A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Tamagawa discovered that there also exist anabelian phenomena for curves over algebraically closed fields of characteristic p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' This came rather surprisingly since it means that, in positive characteristic, the geometry of curves can be determined by their geometric fundamental groups without Galois actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Since the late 1990s, this kind of anabelian phenom- enon has been studied further by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Raynaud ( R2 [R2]), F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Pop-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Saïdi ( PS [PS]), Tamagawa ( T2 [T2], T4 [T4], T5 [T5]), and the second author of the present paper ( Y1 [Y1], Y2 [Y2], Y4 [Y4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' More precisely, they focused on the so-called weak Isom- version of Grothendieck’s anabelian conjecture for curves over algebraically closed fields of characteristic p > 0 (or the “weak Isom-version conjecture” for short) formulated by Tamagawa ( T3 [T3, Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2]), which says that curves are isomorphic if and only if their tame (or étale) fundamental groups are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' At the present, this conjecture is still wide-open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Topology structures of moduli spaces of curves and anabelian geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' In the present paper, we study a new kind of anabelian phenomenon concerning curves over algebraically closed fields of characteristic p > 0 which shows that the topological structures of moduli spaces of curves can be understood by their fundamental groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let Fp be the prime field of characteristic p > 0, and let Mord g,n,Z be the moduli stack over Z parameterizing smooth n-pointed stable curves of type (g, n) (in the sense of K[K]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We put Mord g,n,Fp := Mord g,n,Z ×Z Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Note that the set of marked points of an n-smooth pointed stable curve admits a natural action of the n-symmetric group Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Moreover, we denote by Mg,n,Fp := [Mord g,n,Fp/Sn] the quotient stack, and denote by Mg,n,Fp the coarse moduli space of Mg,n,Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let q ∈ Mg,n,Fp be an arbitrary point, k(q) the residue field of q, kq an algebraically closed field containing k(q), and Vq := {q} the topological closure of {q} in Mg,n,Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Write (Xkq, DXkq ) for the smooth pointed stable curve of type (g, n) over kq determined by the natural morphism Speckq → Mg,n,Fp and put UXkq := Xkq \\ DXkq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' In particular, we put (Xkq, DXkq ) := (Xq, DXq) and UXq := Xq \\ DXq if kq is an algebraic closure of k(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Since the isomorphism class of the tame fundamental group πt 1(UXkq ) depends only on q, we shall write πt 1(q) for the tame fundamental group πt 1(UXkq ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' sec122 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We maintain the notation introduced above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' The weak Isom-version conjecture of Tamagawa can be reformu- lated as follows: TOPOLOGICAL STRUCTURES OF MODULI SPACES OF CURVES AND ANABELIAN GEOMETRY IN POSITIVE CHARACTERISTIC 3 Weak Isom-version Conjecture .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let qi ∈ Mg,n,Fp, i ∈ {1, 2}, be an arbitrary point of Mg,n,Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' The set of continuous isomorphisms of profinite groups Isompg(πt 1(q1), πt 1(q2)) is non-empty if and only if Vq1 = Vq2 (namely, UXq1 ∼= UXq2 as schemes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' The weak Isom-version conjecture means that moduli spaces of curves can be reconstructed “as sets” from the iso- morphism classes of the tame fundamental groups of curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' This conjecture has been only confirmed by Tamagawa ( T4 [T4, Theorem 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2]) in the following case: Suppose that q1 is a closed point of M0,n,Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then the weak Isom-version conjecture holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Next, we propose a new conjecture as follows, that is the weak Hom-version of the Grothendieck conjecture for curves over algebraically closed fields of characteristic p > 0 (or is called weak Hom-version conjecture for simplic- ity), as a generalization of the weak Isom-version conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Weak Hom-version Conjecture .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let qi ∈ Mg,n,Fp, i ∈ {1, 2}, be an arbitrary point of Mg,n,Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' The set of open continuous homomorphisms of profinite groups Homop pg(πt 1(q1), πt 1(q2)) is non-empty if and only if Vq1 ⊇ Vq2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' The weak Hom-version conjecture means that the sets of deformations of a smooth pointed stable curve can be re- constructed group-theoretically from the sets of open continuous homomorphisms of their tame fundamental groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Therefore, it provides a new kind of anabelian phenomenon: The moduli spaces of curves in positive characteristic can be understood not only as sets but also “as topological spaces” from the sets of open continuous homomorphisms of tame fundamental groups of curves in positive characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Roughly speaking, this means that a smooth pointed stable curve corresponding to a geometric point over q2 can be deformed to a smooth pointed stable curve corresponding to a geometric point over q1 if and only if the set of open continuous homomorphisms of tame fundamental groups Homop pg (πt 1(q1), πt 1(q2)) is not empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' The main result of the present paper is the following (see Theorem them-4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='6 (iv) for a more general statement): maintheorem Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' The Weak Hom-version Conjecture holds when q1 is a closed point of M0,n,Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Theorem maintheorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1 follows from the following “Hom-type" anabelian result (see Theorem them-3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='4 for a more precise statement) which is a generalization of Tamagawa’s result (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' T4 [T4, Theorem 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2]): them-0-1 Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let qi ∈ Mg,n,Fp, i ∈ {1, 2}, be an arbitrary point of Mg,n,Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Suppose that q1 is a closed point of Mg,n,Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then the set of open continuous homomorphisms Homop pg(πt 1(q1), πt 1(q2)) is non-empty if and only if UXq1 ∼= UXq2 as schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Note that Theorem them-0-1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2 is essentially different from T4 [T4, Theorem 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' The reason is the following: We a priori do not know whether or not Isompg(πt 1(q1), πt 1(q2)) 4 ZHI HU, YU YANG, AND RUNHONG ZONG is non-empty even through Homop pg(πt 1(q1), πt 1(q2)) is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' In fact, for arbitrary qi ∈ Mg,n,Fp, i ∈ {1, 2}, we have Isompg(πt 1(q1), πt 1(q2)) = ∅, Homop pg(πt 1(q1), πt 1(q2)) ̸= ∅ in general ( T5 [T5, Theorem 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' On the other hand, to verify Theorem them-0-1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2, we need to establish various anabelian reconstructions from open contin- uous homomorphisms of tame fundamental groups which are much harder than the case of isomorphisms in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We explain in more detail about this point in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let us explain the main differences between the proofs of Tamagawa’s result (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' T4 [T4, Theorem 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2]) and our result (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Theorem them-0-1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2), and the new ingredient in our proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' First, we recall the key points of the proof of Tamagawa’s result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Roughly speaking, Tamagawa’s proof consists of two parts: (1) He proved that the sets of inertia subgroups of marked points and the field structures associated to inertia subgroups of marked points of smooth pointed stable curves can be reconstructed group-theoretically from tame fundamental groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' This is the most difficult part of Tamagawa’s proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' (2) By using the inertia subgroups and their associated field structures, if g = 0, he proved that the coordinates of marked points can be calculated group-theoretically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' The group-theoretical reconstructions in Tamagawa’s proofs (1) and (2) are isomorphic version reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' This means that the reconstructions should fix an isomorphism class of a tame fundamental group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' To explain this, let us show an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let UXi, i ∈ {1, 2}, be a curve of type (gX, nX) over an algebraically closed field k of characteristic p > 0 introduced above, πt 1(UXi) the tame fundamental group of UXi, φ : πt 1(UX1) → πt 1(UX2) an open continuous homomorphism, H2 ⊆ πt 1(UX2) an open subgroup, and H1 := φ−1(H2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' In Tamagawa’s proof, since φ is an isomorphism, we have H1 ≃ H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then the group-theoretical reconstruction for types implies that the type (gXH1 , nXH1 ) and the type (gXH2 , nXH2 ) of the curves corresponding to H1 and H2, respectively, are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' This is a key point in the proof of Tamagawa’s group-theoretical reconstruction of the inertia subgroups of marked points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Unfortunately, his method cannot be applied to the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' The reason is that we need to treat the case where φ is an arbitrary open continuous homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Since H1 is not isomorphic to H2 in general (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' specialization homomorphism), we do not know whether or not (gXH1 , nXH1 ) = (gXH2 , nXH2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' This is one of the main difficulties of “Hom-type” problems appeared in anabelian geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Similar difficulties for generalized Hasse-Witt invariants will appear if we try to reconstruct the field structure associated to inertia subgroups of marked points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' To overcome the difficulties mentioned above, we have the following key observation: The inequalities of Avrp(Hi) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=', the p-averages of generalized Hasse-Witt invariants (see paverage 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='3)) in- duced by φ play roles of the comparability of (outer) Galois representations in the theory of anabelian geometry of curves over algebraically closed fields of characteristic p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' In the present paper, our method for reconstructing inertia subgroups of marked points is completely different from Tamagawa’s reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We develop a new group-theoretical algorithm for reconstructing the inertia subgroups of marked points whose input datum is a profinite group which is isomorphic to πt 1(UXi), i ∈ {1, 2}, and whose output data are inertia subgroups of marked points (Theorem them-2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Moreover, we prove that the group-theoretical algorithm and the reconstructions for field structures are compatible with arbitrary surjection φ (Proposition pro-4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' By using Theorem them-2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='18 and Proposition pro-4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='19, we may prove that Tamagawa’s calculation of coordinates is compatible with our reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' This implies Theorem them-0-1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Some further developments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Moduli spaces of fundamental groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let us explain some further developments for the anabelian phenomenon concerning the weak Hom-verson conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' In Y6 [Y6],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' the second author of the present paper introduced a topological space Πg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='n (or more general,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Πg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='n) determined group-theoretically by the tame fundamental groups of smooth pointed TOPOLOGICAL STRUCTURES OF MODULI SPACES OF CURVES AND ANABELIAN GEOMETRY IN POSITIVE CHARACTERISTIC 5 stable curves (or more general,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' the geometric log étale fundamental groups of arbitrary pointed stable curves) of type (g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' n) which is called the moduli spaces of fundamental groups of curves,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' whose underlying set is the sets of isomorphism classes of fundamental groups,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' and whose topology is determined by the sets of finite quotients of fundamental groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Moreover, he posed the so-called homeomorphism conjecture, roughly speaking, which says that (by quotiening a certain equivalence relation induced by Frobenius actions) the moduli spaces of curves are homeomorphic to the moduli spaces of fundamental groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' In the present literatures, the term “anabelian” means that a geometric object can be determined by its fundamental group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Furthermore, the homeomorphism conjecture concerning moduli spaces of fundamental groups supplies a new point of view to understand anabelian phenomena as follows: The term “anabelian” means that not only a geometric object can be determined by its fundamental groups, but also a certain moduli space of geometric objects can be determined by the fundamental groups of geometric objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Under this point of view, the homeomorphism conjecture is regarded as the analogue of a famous theorem in the theory of classic Teichmüller spaces which states that the Teichmüller spaces of complex hyperbolic curves are homeomorphic to the spaces of discrete and faithful representations of topological fundamental groups of underlying surfaces into the group PSL2(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Now Theorem maintheorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1 implies that M0,4,Fp is homeomorphic to Π0,4 as topological spaces (note that Tamagawa’s result (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' T4 [T4, Theorem 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2]) only says that the natural map M0,4,Fp → Π0,4 is a bijection as sets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Moreover, based on Y1 [Y1], Y3 [Y3], Y4 [Y4], Y5 [Y5], and the main results of the present paper, the homeomorphism conjecture is confirmed for 1-dimensional moduli spaces of pointed stable curves in Y6 [Y6] and Y7 [Y7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' For the homeomorphism conjecture in the case of higher dimensional moduli spaces of curves, the weak Hom-version conjecture and the pointed collection conjecture (see Section pcc 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2 of the present paper) are also the main steps toward understanding it (see Y8 [Y8, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' The sets of finite quotients of tame fundamental groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We maintain the notation introduced in sec111 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' The techniques developed in § mpanabelian 3 of the present paper have important applications for understanding the set of finite quotients πt A(UX) of the tame fundamental groups πt 1(UX) of UX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Note that, if UX is affine, the set πét A(UX) of finite quotients of the étale fundamental groups πét 1 (UX) of UX can be completely determined by its type (gX, nX) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Abhyankar’s conjecture proved by Raynaud and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Harbater).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' However, the structure of πét 1 (UX) cannot be carried out from πét A(UX) since πét 1 (UX) is not topologically finitely generated when UX is affine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' By contrast, the isomorphism class of πt 1(UX) can be completely determined by πt A(UX) since πt 1(UX) is topolog- ically finitely generated, and one cannot expect that there exists an explicit description for the entire set πt A(UX) since there exists anabelian phenomenon mentioned above (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' πt A(UX) depends on the isomorphism class of UX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' On the other hand, for understanding more precisely the relationship between the structures of tame fundamental groups and the anabelian phenomena in positive characteristic world, it is naturally to ask the following interesting problem: How does the scheme structure of UX affect explicitly the set of finite quotients πt A(UX)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' In Y9 [Y9], by applying the techniques developed in § mpanabelian 3 of the present paper and Y5 [Y5, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2], we obtain the following result: Let q1 ∈ Mg1,n1,Fp and q2 ∈ M0,n2,Fp be arbitrary points and πt A(qi) the set of finite quotients of the tame fundamental group πt 1(qi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Suppose that q2 is a closed point of M0,n2,Fp, and that πt 1(q1) ̸∼= πt 1(q2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then we can construct explicitly a finite group Gq2 depending on q2 such that Gq2 ∈ πt A(q1) and Gq2 ̸∈ πt A(q2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' 6 ZHI HU, YU YANG, AND RUNHONG ZONG 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Structure of the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' The present paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' In Section sec-1 2, we formulate the the weak Hom-version conjecture and the pointed collection conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' In Section mpanabelian 3, we give a group-theoretical algorithm for reconstructions of inertia subgroups associated to marked points, and prove that the group-theoretical algorithm is compatible with arbitrary open surjective homomorphisms of tame fundamental groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' In Section sec-5 4, we prove our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' The second author was supported by JSPS Grant-in-Aid for Young Scientists Grant Num- bers 16J08847 and 20K14283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' CONJECTURES sec-1 In this section, we formulate two new conjectures concerning anabelian geometry of curves over algebraically closed fields of characteristic p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' The weak Hom-version conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' In this subsection, we formulate the first conjecture of the present paper which we call “the weak Hom-version conjecture”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' curves 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let k be an algebraically closed field of characteristic p > 0, and let (X, DX) be a smooth pointed stable curve of type (gX, nX) over k, where X denotes the (smooth) underlying curve of genus gX and DX denotes the (ordered) finite set of marked points with cardinality nX := #(DX) satisfying K[K, Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1 (iv)] (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' 2gX + nX − 2 > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Note that UX := X \\ DX is a hyperbolic curve over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let (Y, DY ) and (X, DX) be smooth pointed stable curves over k, and let f : (Y, DY ) → (X, DX) be a morphism of smooth pointed stable curves over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We shall say that f is étale (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' tame, Galois étale, Galois tame) if f is étale over X (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' f is étale over UX and is at most tamely ramified over DX, f is a Galois covering and is étale, f is a Galois covering and is tame).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' By choosing a base point of x ∈ UX, we have the tame fundamental group πt 1(UX, x) of UX and the étale funda- mental group π1(X, x) of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Since we only focus on the isomorphism classes of fundamental groups in the present paper, for simplicity of notation, we omit the base point and denote by πt 1(UX) and π1(X) the tame fundamental group πt 1(UX, x) of UX and the étale fundamental group π1(X, x) of X, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Note that there is a natural continuous surjective homomorphism πt 1(UX) ։ π1(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' moduli212 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let Fp be an algebraic closure of Fp, and let Mord g,n,Fp be the moduli stack over Z parameterizing smooth pointed stable curves of type (g, n) in the sense of K[K, Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' The set of marked points of a smooth pointed stable curve admits a natural action of the n-symmetric group Sn, we put Mg,n,Z := [Mord g,n,Z/Sn] the quotient stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Moreover, we denote by Mord g,n := Mg,n,Z ×Z Fp, Mg,n,Fp := Mg,n,Z ×Z Fp, and Mg,n := Mg,n,Z ×Z Fp, and denote by M ord g,n , Mg,n,Fp, and Mg,n the coarse moduli spaces of Mord g,n, Mg,n,Fp, and Mg,n, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let q ∈ M ord g,n be an arbitrary point and k(q) the residue field of q, and kq an algebraically closed field containing k(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Write (Xkq, DXkq ) for the smooth pointed stable curve of type (g, n) over kq determined by the natural mor- phism Speckq → Speck(q) → M ord g,n and UXkq for Xkq \\ DXkq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' In particular, if kq is an algebraic closure of k(q), we shall write (Xq, DXq) for (Xkq, DXkq ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Since the isomorphism class of the tame fundamental group πt 1(UXkq ) depends only on q (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=', the isomorphism class does not depend on the choices of kq), we shall write πt 1(q) and πt A(q) for πt 1(UXkq ) and the set of finite quotients of πt 1(UXkq ), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' FJ [FJ, Proposition 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='7] implies that for any points q1, q2 ∈ M ord g,n , πt 1(q1) ∼= πt 1(q2) as profinite groups if and only if πt A(q1) = πt A(q2) as sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' TOPOLOGICAL STRUCTURES OF MODULI SPACES OF CURVES AND ANABELIAN GEOMETRY IN POSITIVE CHARACTERISTIC 7 On the other hand, Let q ∈ M ord g,n and q′ ∈ Mg,n,Fp be arbitrary points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We denote by Vq ⊆ M ord g,n and Vq′ ⊆ Mg,n,Fp the topological closures of q and q′ in M ord g,n and Mg,n,Fp, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We have the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' def-2 (1) Let c1, c2 ∈ M ord,cl g,n be closed points, where (−)cl denotes the set of closed points of (−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then c1 ∼fe c2 if there exists m ∈ Z such that ν(c2) = ν(c(m) 1 ), where c(m) 1 denotes the closed point corresponding to the curve obtained by mth Frobenius twist of the curve corresponding to c1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Here “fe" means “Frobenius equivalence".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' (2) Let q1, q2 ∈ M ord g,n be arbitrary points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We denote by Vq1 ⊇fe Vq2 if, for each closed point c2 ∈ V cl q2 , there exists a closed point c1 ∈ V cl q1 such that c1 ∼fe c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Moreover, we denote by Vq1 =fe Vq2 if Vq1 ⊇fe Vq2 and Vq1 ⊆fe Vq2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Moreover, we also denote by q1 ∼fe q2 if Vq1 =fe Vq2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We have the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' pro-5 Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let ω : M ord g,n → Mg,n,Fp be the morphism induced by the natural morphism Mord g,n → Mg,n,Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let i ∈ {1, 2}, and let qi ∈ M ord g,n and q′ i := ω(qi) ∈ Mg,n,Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then we have Vq1 ⊇fe Vq2 if and only if Vq′ 1 ⊇ Vq′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' In particular, we have Vq1 =fe Vq2 if and only if Vq′ 1 = Vq′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Namely, we have Vq1 =fe Vq2 if and only if UXq1 ∼= UXq2 as schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Suppose that qi, i ∈ {1, 2}, is a closed point of M ord g,n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' If Vq1 ⊇fe Vq2, we see immeidately q1 ∼ q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Thus, we obtain UXq1 ∼= UXq2 as schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' This means q′ 1 = q′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Conversely, if Vq′ 1 ⊇ Vq′ 2, then we have q′ 1 = q′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Thus, we obtain q1 ∼ q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Suppose that qi, i ∈ {1, 2}, is an aribtrary point of M ord g,n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' If Vq1 ⊇fe Vq2, then the case of closed points implies V cl q′ 1 ⊇ V cl q′ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Since Vq′ 1 and Vq′ 2 are irreducible, we obtain Vq′ 1 ⊇ Vq′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Conversely, if Vq′ 1 ⊇ Vq′ 2, we note that Vqi is an irreducible component of (ω)−1(Vq′ i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then the case of closed points implies Vq1 ⊇fe Vq2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Denote by Homop pg(−, −) the set of open continuous homomorphisms of profinite groups, and by Isompg(−, −) the set of isomorphisms of profinite groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We have the following conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Weak Hom-version Conjecture .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let qi ∈ Mg,n (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' qi ∈ Mg,n,Fp), i ∈ {1, 2}, be an arbitrary point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then we have Homop pg(πt 1(q1), πt 1(q2)) is non-empty if and only if Vq1 ⊇fe Vq2 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Vq1 ⊇ Vq2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' The weak Hom-version conjecture means that the topological structures of the moduli spaces of smooth pointed stable curves can be understood by the tame fundamental groups of curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' In particular, the weak Hom-version conjecture implies the following conjecture which was essentially formulated by Tamagawa ( T3 [T3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Weak Isom-version Conjecture .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let qi ∈ Mg,n (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' qi ∈ Mg,n,Fp), i ∈ {1, 2}, be an arbitrary point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then we have Isompg(πt 1(q1), πt 1(q2)) is non-empty if and only if Vq1 =fe Vq2 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Vq1 = Vq2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' The weak Isom-version conjecture means that the set structures of the moduli spaces of smooth pointed stable curves can be understood by the tame fundamental groups of curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' pcc 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' The pointed collection conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' In this subsection, we formulate the second conjecture of the present paper which we call “the pointed collection conjecture”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We maintain the notation introduced in moduli212 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' 8 ZHI HU, YU YANG, AND RUNHONG ZONG 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let q be an arbitrary point of M ord g,n and G ∈ πt A(q) an arbitrary finite group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We put UG := {q′ ∈ M ord g,n | G ∈ πt A(q′)} ⊆ M ord g,n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' pro-6 Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let q be an arbitrary point of M ord g,n and G ∈ πt A(q) an arbitrary finite group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then the set UG contains an open neighborhood of q in M ord g,n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Proposition pro-6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='3 was proved by K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Stevenson when n = 0 and q is a closed point of Mg,0 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Ste [Ste, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Moreover, by similar arguments to the arguments given in the proof of Ste [Ste, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2], Proposition pro-6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='3 also holds for n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' □ def-3 Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We denote by qgen the generic point of M ord g,n , and let C ⊆ πt A(qgen) = � q∈Mord,cl g,n πt A(q) be a subset of πt A(qgen).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We shall say that C is a pointed collection if the following conditions are satisfied: (i) 0 < #((� G∈C UG) ∩ M ord,cl g,n ) < ∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' (ii) UG′ ∩ (� G∈C UG) ∩ M ord,cl g,n = ∅ for each G′ ∈ πt A(qgen) such that G′ ̸∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' On the other hand, for each closed point t ∈ M ord,cl g,n , we may define a set associated to t as follows: Ct := {G ∈ πt A(qgen) | t ∈ UG}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Note that, if t ∈ V cl q , then Ct ⊆ πt A(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Moreover, we denote by Cq := {C is a pointed collection | C ⊆ πt A(q)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' At present, no published results are known concerning the weak Hom-version conjecture (or the weak Isom- version conjecture) for non-closed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' The main difficulty of proving the weak Hom-version conjecture (or the weak Isom-version conjecture) for non-closed points of M ord g,n is the following: For each q ∈ M ord g,n , we do not know how to reconstruct the tame fundamental groups of closed points of Vq group-theoretically from πt 1(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Once the tame fundamental groups of the closed points of Vq can be reconstructed group-theoretically from πt 1(q), then the weak Hom-version conjecture for closed points of M ord g,n implies that the set of closed points of Vq can be reconstructed group-theoretically from πt 1(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Thus, the weak Hom-version conjecture for non-closed points of M ord g,n can be deduced from the weak Hom-version conjecture for closed points of M ord g,n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let q ∈ M ord g,n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Since the isomorphism class of πt 1(q) as a profinite group can be determined by the set πt A(q), the following conjecture tell us how to reconstruct group-theoretically the set of finite quotients of a closed point of Vq from πt A(q) (or πt 1(q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Pointed Collection Conjecture .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' For each t ∈ M ord,cl g,n , the set Ct associated to t is a pointed collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Moreover, let q ∈ M ord g,n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then the natural map colleq : V cl q → Cq, [t] �→ Ct, is a bijection, where [t] denotes the image of t in V cl q := V cl q / ∼fe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Write q′ ∈ Mg,n,Fp for the image ω(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then we have V cl q = V cl q′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' This means that the pointed collection conjecture holds if and only if the weak Hom-version conjecture holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' RECONSTRUCTIONS OF MARKED POINTS mpanabelian The main purposes of the present section are as follows: We will give a new mono-anabelian reconstruction of Ine(πt 1(UX)), and prove that the mono-anabelian reconstruction (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=', the group-theoretical algorithm) is compatible with any open continuous homomorphisms of tame fundamental groups of smooth pointed stable curves with a fixed type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' TOPOLOGICAL STRUCTURES OF MODULI SPACES OF CURVES AND ANABELIAN GEOMETRY IN POSITIVE CHARACTERISTIC 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Anabelian reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We maintain the notation introduced in curves 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let us recall the definitions concerning “anabelian reconstructions".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' definition 1 Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let F be a geometric object and ΠF a profinite group associated to the object F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Suppose that we are given an invariant InvF depending on the isomorphism class of F (in a certain category), and that we are given an additional structure AddF (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' a family of subgroups, a family of quotient groups) on the profinite group ΠF depending functorially on F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We shall say that InvF can be mono-anabelian reconstructed from ΠF if there exists a group-theoretical algorithm whose input datum is ΠF, and whose output datum is InvF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We shall say that AddF can be mono-anabelian recon- structed from ΠF if there exists a group-theoretical algorithm whose input datum is ΠF, and whose output datum is AddF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let Fi, i ∈ {1, 2}, be a geometric object and ΠFi a profinite group associated to Fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Suppose that we are given an additional structure AddFi on the profinite group ΠFi depending functorially on Fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We shall say that a map (or a morphism) AddF1 → AddF2 can be mono-anabelian reconstructed from an open continuous homomorphism ΠF1 → ΠF2 if there exists a group-theoretical algorithm whose input datum is ΠF1 → ΠF2, and whose output datum is AddF1 → AddF2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' unicov313 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let K be the function field of X, and let �K be the maximal Galois extension of K in a fixed separable closure of K, unramified over UX and at most tamely ramified over DX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then we may identify πt 1(UX) with Gal( �K/K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We define the universal tame covering of (X, DX) associated to πt 1(UX) to be ( � X, D � X), where � X denotes the nor- malization of X in �K, and D � X denotes the inverse image of DX in � X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then there is a natural action of πt 1(UX) on ( � X, D � X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' For each �e ∈ D � X, we denote by I�e the inertia subgroup of πt 1(UX) associated to �e (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=', the stabilizer of �e in πt 1(UX)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then we have I�e ∼= �Z(1)p′, where �Z(1)p′ denotes the prime-to-p part of �Z(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' The following result was proved by Tamagawa ( T4 [T4, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1 and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' proposition 1 (1) The type (gX, nX) can be mono-anabelian reconstructed from πt 1(UX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' (2) Let �e and �e′ be two points of D � X distinct from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then the intersection of I�e and I�e′ is trivial in πt 1(UX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Moreover, the map D � X → Sub(πt 1(UX)), �e �→ I�e, is an injection, where Sub(−) denotes the set of closed subgroups of (−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' (3) Write Ine(πt 1(UX)) for the set of inertia subgroups in πt 1(UX), namely the image of the map D � X → Sub(πt 1(UX)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then Ine(πt 1(UX)) can be mono-anabelian reconstructed from πt 1(UX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' In particular, the set of marked points DX and π1(X) can be mono-anabelian reconstructed from πt 1(UX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' sec-2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' The set of marked points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We maintain the notation introduced in curves 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Moreover, we suppose that gX ≥ 2 and nX > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We will prove that the set of marked points can be regarded as a quotient set of a set of cohomological classes of a suitable covering of curves (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Proposition pro-2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' The main idea is the following: By taking a suitable étale covering with a prime degree f : (Y, DY ) → (X, DX), for every marked point x ∈ DX, there exists a set of tame coverings with a prime degree which is totally ramified over the inverse image f −1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then x can be regarded as the set of cohomological classes corresponding to such coverings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' triple 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let h : (W, DW ) → (X, DX) be a connected Galois tame covering over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We put Ramh := {e ∈ DX | h is ramified over e}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' 10 ZHI HU, YU YANG, AND RUNHONG ZONG Let (Y, DY ) be a smooth pointed stable curve over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We shall say that (ℓ, d, f : (Y, DY ) → (X, DX)) is an mp-triple associated to (X, DX) if the following conditions hold: (i) ℓ and d are prime numbers distinct from each other such that (ℓ, p) = (d, p) = 1 and ℓ ≡ 1 (mod d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' then all dth roots of unity are contained in Fℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' (ii) f is a Galois étale covering over k whose Galois group is isomorphic to µd, where µd ⊆ F× ℓ denotes the subgroup of dth roots of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Here, “mp” means “marked points”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then we have a natural injection H1 ét(Y, Fℓ) ֒→ H1 ét(UY , Fℓ) induced by the natural surjection πt 1(UY ) ։ π1(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Note that every non-zero element of H1 ét(UY , Fℓ) induces a connected Galois tame covering of (Y, DY ) of degree ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We obtain an exact sequence 0 → H1 ét(Y, Fℓ) → H1 ét(UY , Fℓ) → Div0 DY (Y ) ⊗ Fℓ → 0 with a natural action of µd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' sec31aaa 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let (Div0 DY (Y ) ⊗ Fℓ)µd ⊆ Div0 DY (Y ) ⊗ Fℓ be the subset of elements on which µd acts via the character µd ֒→ F× ℓ and M ∗ Y ⊆ H1 ét(UY , Fℓ) the subset of elements whose images are non-zero elements of (Div0 DY (Y )⊗Fℓ)µd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' For each α ∈ M ∗ Y , write gα : (Yα, DYα) → (Y, DY ) for the tame covering induced by α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We define ǫ : M ∗ Y → Z, where ǫ(α) := #DYα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Denote by MY := {α ∈ M ∗ Y | #Ramgα = d} = {α ∈ M ∗ Y | ǫ(α) = ℓ(dnX − d) + d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Note that MY is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' For each α ∈ MY , since the image of α is contained in (Div0 DY (Y ) ⊗ Fℓ)µd, we obtain that the action of µd on Ramgα ⊆ DY is transitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Thus, there exists a unique marked point eα ∈ DX such that f(y) = eα for each y ∈ Ramgα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' For each e ∈ DX, we put MY,e := {α ∈ MY | gα is ramified over f −1(e)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then, for any marked points e, e′ ∈ DX distinct from each other, we have MY,e ∩ MY,e′ = ∅ and the disjoint union MY = � e∈DX MY,e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' 315 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Next, we define a pre-equivalence relation ∼ on MY as follows: Let α, β ∈ MY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then α ∼ β if λα + µβ ∈ MY for each λ, µ ∈ F× ℓ for which λα + µβ ∈ M ∗ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then we have the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' pro-2 Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' The pre-equivalence relation ∼ on MY is an equivalence relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Moreover, the map ϑX : MY / ∼→ DX, [α] �→ eα, is a bijection, where [α] denotes the image of α in MY / ∼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let β, γ ∈ MY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' If Ramgβ = Ramgγ, then, for each λ, µ ∈ F× ℓ for which λβ + µγ ̸= 0, we have Ramgλβ+µγ = Ramgβ = Ramgγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Thus we obtain that β ∼ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' On the other hand, if β ∼ γ, we have Ramgβ = Ramgγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Otherwise, we have #Ramgβ+γ = 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' This means that β ∼ γ if and only if Ramgβ = Ramgγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then ∼ is an equivalence relation on MY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let us prove that ϑX is a bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' It is easy to see that ϑX is an injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' On the other hand, for each e ∈ DX, the structure of the maximal pro-ℓ tame fundamental groups implies that we may construct a connected tame Galois covering of h : (Z, DZ) → (Y, DY ) such that the element of H1 ét(UY , Fℓ) induced by h is contained in MY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then ϑX is a surjection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' This completes the proof of Proposition pro-2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' □ TOPOLOGICAL STRUCTURES OF MODULI SPACES OF CURVES AND ANABELIAN GEOMETRY IN POSITIVE CHARACTERISTIC 11 rem-2-1 Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We claim that the set MY / ∼ does not depend on the choices of mp-triples associated to (X, DX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let (ℓ∗, d∗, f ∗ : (Y ∗, DY ∗) → (X, DX)) be an arbitrary mp-triple associated to (X, DX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Hence we obtain a resulting set MY ∗/ ∼ and a natural bijection ϑ∗ X : MY ∗/ ∼→ DX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We will prove that there exists a natural bijection δ : MY ∗/ ∼ ≃ −→ MY / ∼ such that ϑ∗ X = ϑX ◦ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' First, suppose that ℓ ̸= ℓ∗ and d ̸= d∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then we may construct a natural bijection δ : MY ∗/ ∼ ≃ −→ MY / ∼ as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let α ∈ MY and α∗ ∈ MY ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Write (Yα, DYα) → (Y, DY ) and (Yα∗, DYα∗) → (Y ∗, DY ∗) for the Galois tame coverings induced by α and α∗, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We consider the following fiber product in the category of smooth pointed stable curves (Yα, DYα) ×(X,DX) (Yα∗, DYα∗) which is a smooth pointed stable curve over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Thus, we obtain a connected tame covering (Yα, DYα) ×(X,DX) (Yα∗, DYα∗) → (X, DX) of degree dd∗ℓℓ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then it is easy to check that ϑX([α]) = ϑ∗ X([α∗]) if and only if the cardinality of the set of marked points of (Yα, DYα) ×(X,DX) (Yα∗, DYα∗) is equal to dd∗(ℓℓ∗(nX − 1) + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We put [α] := δ([α∗]) if ϑX([α]) = ϑ∗ X([α∗]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Moreover, by the construction above, we obtain that ϑ∗ X = ϑX ◦ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' In the general case, we may choose an mp-triple (ℓ∗∗, d∗∗, f ∗∗ : (Y ∗∗, DY ∗∗) → (X, DX)) associated to (X, DX) such that ℓ∗∗ ̸= ℓ, ℓ∗∗ ̸= ℓ∗, d∗∗ ̸= d, and d∗∗ ̸= d∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Hence we obtain a resulting set MY ∗∗/ ∼ and a natural bijection ϑ∗∗ X : MY ∗∗/ ∼→ DX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then the proof given above implies that there are natural bijections δ1 : MY ∗∗/ ∼ ≃ −→ MY / ∼ and δ2 : MY ∗∗/ ∼ ≃ −→ MY ∗/ ∼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Thus, we may put δ := δ1 ◦ δ−1 2 : MY ∗/ ∼ ≃ −→ MY / ∼ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' rem-2-2 Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let H ⊆ πt 1(UX) be an arbitrary open normal subgroup and fH : (XH, DXH) → (X, DX) the Galois tame covering over k induced by the natural inclusion H ֒→ πt 1(UX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let (ℓ, d, f : (Y, DY ) → (X, DX)) be an mp-triple associated to (X, DX) such that (#(πt 1(UX)/H), ℓ) = (#(πt 1(UX)/H), d) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then we obtain an mp-triple (ℓ, d, g : (Z, DZ) := (Y, DY ) ×(X,DX) (XH, DXH) → (XH, DXH)) associated to (XH, DXH) induced by (ℓ, d, f : (Y, DY ) → (X, DX)), where (Y, DY ) ×(X,DX) (XH, DXH) denotes the fiber product in the category of smooth pointed stable curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' The mp-triple associated to (XH, DXH) induces a set MZ/ ∼ which can be identified with the set of marked points DXH of (XH, DXH) by applying Proposition pro-2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Moreover, for each eX ∈ DX and each αY,eX ∈ MY,eX, αY,eX induces an element αZ = � eXH ∈f −1 H (eX) αZ,eXH over (Z, DZ) via the natural morphism (Z, DZ) → (Y, DY ), where αZ,eXH ∈ MZ,eXH .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' On the other hand, for each e′ XH ∈ DXH and each e′ X ∈ DX, we have that fH(e′ XH) = e′ X if and only if there exists an element αY,e′ X ∈ MY,e′ X such that the following two conditions hold: the element α′ Z, induced by αY,e′ X via the natural morphism (Z, DZ) → (Y, DY ), can be represented by a linear combination α′ Z = � eXH ∈SXH α′ Z,eXH , where SXH is a subset of DXH, and αZ,eXH ∈ MZ,eXH ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' e′ XH ∈ SXH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' 12 ZHI HU, YU YANG, AND RUNHONG ZONG lem-1 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let (ℓ, d, f : (Y, DY ) → (X, DX)) be a triple associated to (X, DX) and gY the genus of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then we have #(MY,e) = ℓ2gY +1 − ℓ2gY , e ∈ DX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Moreover, we have #(MY ) = nX(ℓ2gY +1 − ℓ2gY ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let e ∈ DX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Write De ⊆ DY for the set f −1(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then MY,e can be naturally regarded as a subset of H1 ét(Y \\ De, Fℓ) via the natural open immersion Y \\ De ֒→ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Write Le for the Fℓ-vector space generated by MY,e in H1 ét(Y \\ De, Fℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then we have MY,e = Le \\ H1 ét(Y, Fℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Write He for the quotient Le/H1 ét(Y, Fℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We have an exact sequence as follows: 0 → H1 ét(Y, Fℓ) → Le → He → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Since the action of µd on f −1(e) is transitive, we obtain dimFℓ(He) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' On the other hand, since dimFℓ(H1 ét(Y, Fℓ)) = 2gY , we obtain #(MY,e) = ℓ2gY +1 − ℓ2gY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Thus, we have #(MY ) = nX(ℓ2gY +1 − ℓ2gY ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' This completes the proof of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' □ sec-3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Reconstructions of inertia subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We maintain the notation introduced in curves 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We will prove that the inertia subgroups of marked points can be mono-anabelian reconstructed from πt 1(UX) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Proposition them-1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' The main idea is as follows: Let H ⊆ πt 1(UX) be an arbitrary normal open subgroup and (XH, DXH) → (X, DX) the tame covering corresponding to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Firstly, by using some numerical conditions induced by the Riemann-Hurwitz formula, the étale fundamental group π1(X) can be mono-anabelian reconstructed from πt 1(UX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then the results obtained in Section sec-2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2 implies that DX can be mono-anabelian reconstructed from πt 1(UX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Moreover, DXH can also be mono-anabelian reconstructed from H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Secondly, since the natural injection H ֒→ πt 1(UX) induces a map of sets of cohomological classes obtained in Section sec-2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2, we obtain that the natural map DXH → DX can be mono-anabelian reconstructed from H ֒→ πt 1(UX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Thus, by taking a cofinal system of open normal subgroups of πt 1(UX), we obtain a new mono-anabelian reconstruction of Ine(πt 1(UX)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' First, we have the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' lem-2 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' (1) The prime number p (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=', the characteristic of k) can be mono-anabelian reconstructed from πt 1(UX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' (2) The étale fundamental group π1(X) can be mono-anabelian reconstructed from πt 1(UX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' (1) Let P be the set of prime numbers, and let Q be an arbitrary open subgroup of πt 1(UX) and rQ an integer such that #{l ∈ P | rQ = dimFl(Qab ⊗ Fl)} = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then we see immediately that the characteristic of k is the unique prime number p such that there exists an open subgroup T ⊆ πt 1(UX) and rT ̸= dimFp(T ab ⊗ Fp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' (2) Let H be an arbitrary open normal subgroup of πt 1(UX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We denote by (XH, DXH) the smooth pointed stable curve of type (gXH, nXH) over k induced by H, and denote by fH : (XH, DXH) → (X, DX) the morphism of smooth pointed stable curves over k induced by the natural inclusion H ֒→ πt 1(UX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We note that fH is étale if and only if gXH − 1 = #(πt 1(UX)/H)(gX − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We put Et(πt 1(UX)) := {H ⊆ πt 1(UX) is an open normal subgroup : gXH − 1 = #(πt 1(UX)/H)(gX − 1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Moreover, Proposition proposition 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2 (1) implies that gXH and gX can be mono-anabelian reconstructed from H and πt 1(UX), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then the set Et(πt 1(UX)) can be mono-anabelian reconstructed from πt 1(UX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We obtain that π1(X) = πt 1(UX)/ � H∈Et(πt 1(UX)) H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' This completes the proof of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' □ TOPOLOGICAL STRUCTURES OF MODULI SPACES OF CURVES AND ANABELIAN GEOMETRY IN POSITIVE CHARACTERISTIC 13 gpmptriple 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Suppose gX ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let us define a group-theoretical object corresponding to an mp-triple which was introduced in triple 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We shall say that (ℓ, d, y) is an mp-triple associated to πt 1(UX) if the following two conditions hold: ℓ and d are prime numbers distinct from each other such that (ℓ, p) = (d, p) = 1 and ℓ ≡ 1 (mod d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' then all d-th roots of unity are contained in Fℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' y ∈ Hom(π1(X), µd) such that y ̸= 0, where µd ⊆ F× ℓ denotes the subgroup of d-th roots of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Moreover, by applying Lemma lem-2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='7, there is a triple (ℓ, d, y) associated to πt 1(UX) which can be mono- anabelian reconstructed from πt 1(UX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let f : (Y, DY ) → (X, DX) be a Galois étale covering induced by y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then we see immediately that (ℓ, d, f : (Y, DY ) → (X, DX)) is an mp-triple associated to (X, DX) defined in triple 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We denote by πt 1(UY ) the kernel of the composition of the surjections πt 1(UX) ։ π1(X) y։ µd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Since H1 ét(Y, Fℓ) ∼= Hom(π1(Y ), Fℓ) and H1 ét(UY , Fℓ) ∼= Hom(πt 1(UY ), Fℓ), Lemma lem-2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='7 implies immediately that the fol- lowing exact sequence 0 → H1 ét(Y, Fℓ) → H1 ét(UY , Fℓ) → Div0 DY (Y ) ⊗ Fℓ → 0 can be mono-anabelian reconstructed from πt 1(UY ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Thus, Proposition proposition 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2 (1) implies that the set MY / ∼ defined in 315 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='4 can be mono-anabelian reconstructed from πt 1(UY ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Note that, by Remark rem-2-1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='4, the set MY / ∼ does not depend on the choices of mp-triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then we put Dgp X := MY / ∼, where “gp" means “group-theoretical".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' By Proposition pro-2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='3, we may identify Dgp X with the set of marked points DX of (X, DX) via the bijection ϑX : Dgp X ≃ −→ DX defined in Proposition pro-2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' pro-3 Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let H ⊆ πt 1(UX) be an arbitrary open normal subgroup and fH : (XH, DXH) → (X, DX) the morphism of smooth pointed stable curves over k induced by the natural inclusion H ֒→ πt 1(UX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Suppose gX ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then the sets Dgp X and Dgp XH can be mono-anabelian reconstructed from πt 1(UX) and H, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Moreover, the inclusion H ֒→ πt 1(UX) induces a map γH,πt 1(UX) : Dgp XH → Dgp X such that the following commutative diagram holds: Dgp XH ϑXH −−−−→ DXH γH,πt 1(UX )\uf8e6� \uf8e6�γfH Dgp X ϑX −−−−→ DX, where γfH denotes the map of the sets of marked points induced by fH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We only need to prove the “moreover" part of Proposition pro-3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We maintain the notation introduced in Remark rem-2-2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Note that, for each eX ∈ DX and each eXH ∈ DXH, the sets MY,eX and MZ,eXH can be mono-anabelian reconstructed from πt 1(UX) and H, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then the “moreover" part follows from Remark rem-2-2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' □ rem-pro-3-1 Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We maintain the notation introduced in Proposition pro-3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let π1(XH) be the étale fundamental group of XH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then we have a natural surjection H ։ π1(XH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Note that π1(XH) admits an action of πt 1(UX)/H induced by the outer action of πt 1(UX)/H on H which is induced by the exact sequence 1 → H → πt 1(UX) → πt 1(UX)/H → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Moreover, the action of πt 1(UX)/H on π1(XH) induces an action of πt 1(UX)/H on Dgp XH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' On the other hand, it is easy to check that the action of πt 1(UX)/H on Dgp XH coincides with the natural action of πt 1(UX)/H on DXH when we identify Dgp X with DX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' 14 ZHI HU, YU YANG, AND RUNHONG ZONG 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' them-1 Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Write Ine(πt 1(UX)) for the set of inertia subgroups in πt 1(UX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then Ine(πt 1(UX)) can be mono- anabelian reconstructed from πt 1(UX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let CX := {Hi}i∈Z>0 be a set of open normal subgroups of πt 1(UX) such that lim ←−i πt 1(UX)/Hi ∼= πt 1(UX) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=', a cofinal system of open normal subgroups).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let �e ∈ D � X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' For each i ∈ Z>0, we write (XHi, DXHi) for the smooth pointed stable curve of type (gXHi , nXHi ) induced by Hi and eXHi ∈ DXHi for the image of �e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then we obtain a sequence of marked points ICX �e : · · · �→ eXH2 �→ eXH1 induced by CX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Note that the sequence ICX �e admits a natural action of πt 1(UX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We may identify the inertia subgroup I�e associated to �e with the stabilizer of ICX �e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Moreover, since Proposition proposition 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2 (1) implies that (gXHi , nXHi ) can be mono-anabelian reconstructed from Hi, by choosing a suitable set of open normal subgroups CX, we may assume that gXH1 ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' If nXH1 = 0, Proposition them-1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='10 is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then we may assume that nXH1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' On the other hand, Proposition pro-3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='8 implies that, for each Hi, i ∈ Z>0, the set Dgp XHi can be mono-anabelian reconstructed from Hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' For each eXHi ∈ DXHi , we denote by egp XHi := ϑ−1 XHi (eXHi ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then the sequence of marked points ICX �e induces a sequence ICX �egp : · · · �→ egp XH2 �→ egp XH1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' By applying the “moreover” part of Proposition pro-3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='8, we see that ICX �egp can be mono-anabelian reconstructed from CX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then Remark rem-pro-3-1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='9 implies that the stabilizer of ICX �egp is equal to the stabilizer of ICX �e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' This completes the proof of the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' □ sec-4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Reconstructions of inertia subgroups via surjections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' In this subsection, we will prove that the mono- anabelian reconstructions obtained in Proposition them-1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='10 are compatible with any open continuous homomorphisms (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Theorem them-2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' sett331 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let (Xi, DXi), i ∈ {1, 2}, be a smooth pointed stable curve of type (gX, nX) over an algebraically closed field ki of characteristic p > 0, UXi := Xi \\ DXi, πt 1(UXi) the tame fundamental group of UXi, and π1(Xi) the étale fundamental group of Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then Lemma lem-2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='7 implies that π1(Xi) can be mono-anabelian reconstructed from πt 1(UXi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Moreover, in this subsection, we suppose that nX > 0, and that φ : πt 1(UX1) ։ πt 1(UX2) is an arbitrary open continuous surjective homomorphism of profinite groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Note that, since (Xi, DXi), i ∈ {1, 2}, is a smooth pointed stable curve of type (gX, nX), φ induces a natural surjection φp′ : πt 1(UX1)p′ ։ πt 1(UX2)p′, where (−)p′ denotes the maximal prime-to-p quotient of (−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Since πt 1(UXi)p′, i ∈ {1, 2}, is topologically finitely generated, and πt 1(UX1)p′ is isomorphic to πt 1(UX2)p′ as abstract profinite groups, we obtain that φp′ : πt 1(UX1)p′ ≃ −→ πt 1(UX2)p′ is an isomorphism ( FJ [FJ, Proposition 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We explain the main idea in the proof of Theorem them-2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let H2 ⊆ πt 1(UX2) be an arbitrary open normal subgroup and H1 := φ−1(H2) ⊆ πt 1(UX1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We write (XHi, DXHi), i ∈ {1, 2}, for the smooth pointed smooth curve of type (gXHi , nXHi ) over ki induced by Hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' To prove the compatibility, we need to prove that, for any prime number ℓ ̸= p, the weight-monodromy filtration of Hab 2 ⊗ Fℓ induces the weight-monodromy filtration of Hab 1 ⊗ Fℓ via the natural surjection φ|H1 : H1 ։ H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Note that the weight 1 part of Hab i ⊗ Fℓ corresponds to π1(XHi)ab ⊗ Fℓ, and the weight 2 part of Hab i ⊗ Fℓ corresponds to the image of the subgroup of Hi generated by the inertia subgroups of the marked points of DXHi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' The key observation is as follows: TOPOLOGICAL STRUCTURES OF MODULI SPACES OF CURVES AND ANABELIAN GEOMETRY IN POSITIVE CHARACTERISTIC 15 The inequality of the limit of p-averages (see Proposition coro-p-average 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='11 (1) below) Avrp(H1) ≥ Avrp(H2) of H1 and H2 induced by the surjection φ|H1 : H1 ։ H2 plays a role of the comparability of “Galois actions” in the theory of the anabelian geometry of curves over algebraically closed fields of characteristic p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' paverage 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Firstly, we have the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' coro-p-average (1) Let (X, DX) be a pointed stable curve of type (gX, nX) over an algebraically closed field k of characteristic p > 0, UX := X \\ DX, and πt 1(UX) the tame fundamental group of UX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let r ∈ N be a natural number, and let Kpr−1 be the kernel of the natural surjection πt 1(UX) ։ πt 1(UX)ab ⊗ Z/(pr − 1)Z, where (−)ab denotes the abelianization of (−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then we have Avrp(πt 1(UX)) := lim r→∞ dimFp(Kab pr−1 ⊗ Fp) #(πt 1(UX)ab ⊗ Z/(pr − 1)Z) = � gX − 1, if nX ≤ 1, gX, if nX > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' (2) We maintain the setting introduced in sett331 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let H2 ⊆ πt 1(UX2) be an open normal subgroup such that ([πt 1(UX2) : H2], p) = 1 and H1 := φ−1(H2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Write gHi, i ∈ {1, 2}, for the genus of the smooth pointed stable curve over ki corresponding to Hi ⊆ πt 1(UXi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then we have gH1 ≥ gH2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' (1) is the Tamagawa’s result concerning the limit of p-averages of πt 1(UX) ( T4 [T4, Theorem 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let us prove (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' The surjection φ induces a surjection φp′ : πt 1(UX1)p′ ։ πt 1(UX2)p′, where (−)p′ denotes the maximal prime- to-p quotient of (−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Moreover, since πt 1(UXi)p′, i ∈ {1, 2}, is topologically finitely generated, and πt 1(UX1)p′ is isomorphic to πt 1(UX2)p′ as abstract profinite groups (since the types of (X1, DX1) and (X2, DX2) are equal to (gX, nX)), we obtain that φp′ is an isomorphism (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' FJ [FJ, Proposition 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' On the other hand, since [πt 1(UX1) : H1] = [πt 1(UX2) : H2] and ([πt 1(UX2) : H2], p) = 1, we obtain that the natural homomorphism φp′ H : Hp′ 1 ։ Hp′ 2 induced by φH := φ|H1 : H1 ։ H2 is also an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' This implies #(Hab 1 ⊗ Z/(pr − 1)Z) = #(Hab 2 ⊗ Z/(pr − 1)Z) for all r ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let KHi,pr−1, i ∈ {1, 2}, be the kernel of the natural surjection Hi ։ Hab i ⊗ Z/(pr − 1)Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then the surjection φH implies Avrp(H1) := lim r→∞ dimFp(Kab H1,pr−1 ⊗ Fp) #(Hab 1 ⊗ Z/(pr − 1)Z) ≥ Avrp(H2) := lim r→∞ dimFp(Kab H2,pr−1 ⊗ Fp) #(Hab 2 ⊗ Z/(pr − 1)Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Thus, the corollary follows from (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We have the following lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' lem-3 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let ℓ be a prime number distinct from p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then the isomorphism (φp′)−1 : πt 1(UX2)p′ ≃ −→ πt 1(UX1)p′ induces an isomorphism ψℓ X : H1 ét(X1, Fℓ) ≃ Hom(π1(X1), Fℓ) ≃ −→ Hom(π1(X2), Fℓ) ≃ H1 ét(X2, Fℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let f1 : (Y1, DY1) → (X1, DX1) be an étale covering of degree ℓ over k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Write f2 : (Y2, DY2) → (X2, DX2) for the connected Galois tame covering of degree ℓ over k2 induced by φp′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then we will prove that f2 is also an étale covering over k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Write gY1 and gY2 for the genus of Y1 and Y2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Since f1 is an étale covering of degree ℓ, the Riemann- Hurwitz formula implies gY1 = ℓ(gX1 − 1) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' On the other hand, the Riemann-Hurwitz formula implies gY2 = 16 ZHI HU, YU YANG, AND RUNHONG ZONG ℓ(gX2 − 1) + 1 + 1 2(ℓ − 1)#(Ramf2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' By applying Proposition coro-p-average 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='11 (2), the surjection φ implies gY1 ≥ gY2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' This means #(Ramf2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' So f2 is an étale covering over k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then the morphism (φp′)−1 induces an injection ψℓ X : Hom(π1(X1), Fℓ) ֒→ Hom(π1(X2), Fℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Furthermore, since dimFℓ(Hom(π1(X1), Fℓ)) = dimFℓ(Hom(π1(X2), Fℓ)) = 2gX, we obtain that ψℓ X is a bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' This completes the proof of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' □ lem-4 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Suppose gX ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then the surjection φ : πt 1(UX1) ։ πt 1(UX2) induces a bijection ρφ : Dgp X1 ≃ −→ Dgp X2, and the bijection ρφ can be mono-anabelian reconstructed from φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let (ℓ, d, y2) be an mp-triple associated to πt 1(UX2) (see gpmptriple 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then Lemma lem-3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='12 implies that φ induces an mp-triple (ℓ, d, y1) associated to πt 1(UX1), where y1 := (ψd X)−1(y2) ∈ Hom(π1(X1), µd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let fi : (Yi, DYi) → (Xi, DXi), i ∈ {1, 2}, be the étale covering of degree d over ki induced by yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then the mp-triple (ℓ, d, yi) associated to πt 1(UXi) determines an mp-triple (ℓ, d, fi : (Yi, DYi) → (Xi, DXi)) associated to (Xi, DXi) over ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Note that the types of (Y1, DY1) and (Y2, DY2) are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Write πt 1(UYi), i ∈ {1, 2}, for the kernel of πt 1(UXi) ։ π1(Xi) yi ։ µd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' By replacing (Xi, DXi) by (Yi, DYi), Lemma lem-3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='12 implies that (φ|p′ πt 1(UY1 ))−1 induces a commutative diagram as follows: 0 −−−−→ H1 ét(Y1, Fℓ) −−−−→ H1 ét(UY1, Fℓ) −−−−→ Div0 DY1 (Y1) ⊗ Fℓ −−−−→ 0 ψℓ Y \uf8e6� ψt,ℓ Y \uf8e6� \uf8e6� 0 −−−−→ H1 ét(Y2, Fℓ) −−−−→ H1 ét(UY2, Fℓ) −−−−→ Div0 DY2 (Y2) ⊗ Fℓ −−−−→ 0, where all the vertical arrows are isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We note that H1 ét(Yi, Fℓ), H1 ét(UYi, Fℓ), and Div0 DYi (Yi) ⊗ Fℓ, i, ∈ {1, 2}, are naturally isomorphic to Hom(π1(Yi), Fℓ), Hom(πt 1(UYi), Fℓ), and Hom(πt 1(UYi), Fℓ)/Hom(π1(Yi), Fℓ), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then Lemma lem-2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='7 implies that the commutative diagram above can be mono-anabelian reconstructed from φ|πt 1(UY1 ) : πt 1(UY1) ։ πt 1(UY2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Write MYi ⊆ M ∗ Yi for the subsets of H1 ét(UYi, Fℓ) defined in sec31aaa 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Since the actions of µd on the exact sequences are compatible with the isomorphisms appearing in the commutative diagram above, we have ψt,ℓ Y (M ∗ Y1) = M ∗ Y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Next, we prove ψt,ℓ Y (MY1) = MY2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let α1 ∈ MY1 and gα1 : (Yα1, DYα1) → (Y1, DY1) the Galois tame covering of degree ℓ over k1 induced by α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Write gα2 : (Yα2, DYα2) → (Y2, DY2) for the Galois tame covering of degree ℓ over k2 induced by α2 := ψt,ℓ Y (α1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Write gYα1 and gYα2 for the genus of Yα1 and Yα2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then Proposition coro-p-average 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='11 (2) and the Riemann-Hurwitz formula imply that gYα1 −gYα2 = 1 2(d−#(Ramgα2 ))(ℓ−1) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' This means d−#(Ramgα2 ) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Since α2 ∈ M ∗ Y2, we have d | #(Ramgα2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Thus, either #(Ramgα2 ) = 0 or #(Ramgα2 ) = d holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' If #(Ramgα2 ) = 0, then gα2 is an étale covering over k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then Lemma lem-3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='12 implies that gα1 is an étale covering over k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' This provides a contradiction to the fact that α1 ∈ MY1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then we have #(Ramgα2 ) = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' This means α2 ∈ MY2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Thus, we obtain ψt,ℓ Y (MY1) ⊆ MY2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' On the other hand, Lemma lem-1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='6 implies #(MY1) = #(MY2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We have ψt,ℓ Y : MY1 ≃ −→ MY2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then Proposition pro-2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='3 implies that ψt,ℓ Y induces a bijection ρφ : Dgp X1 ≃ −→ Dgp X2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Moreover, since MYi and M ∗ Yi can be mono-anabelian reconstructed from πt 1(UYi), the bijection ρφ can be mono- anabelian reconstructed from φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' This completes the proof of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' □ TOPOLOGICAL STRUCTURES OF MODULI SPACES OF CURVES AND ANABELIAN GEOMETRY IN POSITIVE CHARACTERISTIC 17 sec334 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let H2 ⊆ πt 1(UX2) be an arbitrary open normal subgroup and H1 := φ−1(H2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We write (XHi, DXHi ), i ∈ {1, 2}, for the smooth pointed stable curve of type (gXHi , nXHi ) over ki induced by Hi and fHi : (XHi, DXHi ) → (Xi, DXi) for the Galois tame coverings over ki induced by the inclusion Hi ֒→ πt 1(UXi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Moreover, Proposition pro-3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='8 implies that the inclusion Hi ֒→ πt 1(UXi) induces a map γHi,πt 1(UXi ) : Dgp XHi → Dgp Xi which fits into the following commutative diagram: Dgp XHi ϑXHi −−−−→ DXHi γHi,πt 1(UXi )\uf8e6� \uf8e6�γfHi Dgp Xi ϑXi −−−−→ DXi, where γfHi denotes the map of the sets of marked points induced by fHi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We may identify πt 1(UX1)/H1 with πt 1(UX2)/H2 via the isomorphism πt 1(UX1)/H1 ≃ −→ πt 1(UX2)/H2 induced by φ, and denote by G := πt 1(UX1)/H1 ∼= πt 1(UX2)/H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then we have the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' lem-5 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Suppose that gX ≥ 2, and that (gXH1 , nXH1 ) = (gXH2 , nXH2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then the commutative diagram of profinite groups H1 φ|H1 −−−−→ H2 \uf8e6� \uf8e6� πt 1(UX1) φ −−−−→ πt 1(UX2) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1) induces a commutative diagram Dgp XH1 ρφ|H1 −−−−→ Dgp XH2 γH1,πt 1(UX1 )\uf8e6� \uf8e6�γH2,πt 1(UX2 ) Dgp X1 ρφ −−−−→ Dgp X2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2) Moreover, the commutative diagram (2) can be mono-anabelian reconstructed from (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Proposition pro-3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='8 and Lemma lem-4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='13 imply the diagram Dgp XH1 ρφ|H1 −−−−→ Dgp XH2 γH1,πt 1(UX1 )\uf8e6� \uf8e6�γH2,πt 1(UX2 ) Dgp X1 ρφ −−−−→ Dgp X2 can be mono-anabelian reconstructed from the commutative diagram of profinite groups H1 φ|H1 −−−−→ H2 \uf8e6� \uf8e6� πt 1(UX1) φ −−−−→ πt 1(UX2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' To verify Lemma lem-5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='14, it is sufficient to check that the diagram is commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let egp XH1 ∈ Dgp XH1 , egp XH2 := ρφ|H1 (egp XH1 ) ∈ Dgp XH2 , egp X1 := γH1,πt 1(UX1 )(egp XH1 ) ∈ Dgp X1, egp X2 := (γH2,πt 1(UX2 ) ◦ ρφ|H1)(egp XH1 ) ∈ Dgp X2, and egp,∗ X1 := ρ−1 φ (egp X2) ∈ Dgp X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let us prove egp X1 = egp,∗ X1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We put Sgp XH1 := γ−1 H1,πt 1(UX1)(egp,∗ X1 ) and Sgp XH2 := γ−1 H2,πt 1(UX2 )(egp X2), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Note that egp XH2 ∈ Sgp XH2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' To verify egp X1 = egp,∗ X1 , it is sufficient to prove that egp XH1 ∈ Sgp XH1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Moreover, for each i ∈ {1, 2}, we put eXi := ϑXi(egp Xi), eXHi := ϑXHi (egp Xi), e∗ X1 := ϑX1(egp,∗ X1 ), SXi := Sgp Xi, SXHi := Sgp XHi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' 18 ZHI HU, YU YANG, AND RUNHONG ZONG Then to verify the lemma, we only need to prove that eXH1 ∈ ϑXH1 (SXH1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let (ℓ, d, y2) be an mp-triple associated to πt 1(UX2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then Lemma lem-3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='12 implies that φ induces an mp-triple (ℓ, d, y1) associated to πt 1(UX1), where y1 := (ψd X)−1(y2) ∈ Hom(π1(X1), µd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let fi : (Yi, DYi) → (Xi, DXi), i ∈ {1, 2}, be the tame covering of degree d over ki induced by yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then the mp-triple (ℓ, d, yi) associated to πt 1(UXi) induces an mp-triple (ℓ, d, fi : (Yi, DYi) → (Xi, DXi)) associated to (Xi, DXi) over ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Note that since f1 and f2 are étale, the types of (Y1, DY1) and (Y2, DY2) are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' On the other hand, we have an mp-triple (ℓ, d, g2 : (Z2, DZ2) := (Y2, DY2) ×(X2,DX2 ) (XH2, DXH2 ) → (XH2, DXH2 )) associated to (XH2, DXH2 ) induced by the natural inclusion H2 ֒→ πt 1(UX2) and the mp-triple (ℓ, d, f2 : (Y2, DY2) → (X2, DX2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' By Lemma lem-3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='12 again, we obtain an mp-triple (ℓ, d, g1 : (Z1, DZ1) := (Y1, DY1) ×(X1,DX1 ) (XH1, DXH1 ) → (XH1, DXH1 )) associated to (XH1, DXH1 ) induced by φ|H1 and the triple (ℓ, d, g2 : (Z2, DZ2) → (XH2, DXH2 )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let α2 ∈ MY2,eX2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' The final paragraph of the proof of Lemma lem-4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='13 implies that we have a bijection MY1 = � e∈DX1 MY1,e ≃ −→ MY2 = � e∈DX2 MY2,e induced by φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then α2 induces an element α1 ∈ MY1,e∗ X1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Write (Yα1, DYα1) and (Yα2, DYα2) for the smooth pointed stable curves over k1 and k2 induced by α1 and α2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Consider the connected Galois tame covering (Yα2, DYα2) ×(X2,DX2 ) (XH2, DXH2 ) → (Z2, DZ2) of degree ℓ over k2, and write β2 for an element of M ∗ Z2 corresponding to this connected Galois tame covering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then we have β2 = � c2∈SXH2 tc2βc2, where tc2 ∈ (Z/ℓZ)× and βc2 ∈ MZ2,c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' On the other hand, the proof of Lemma lem-4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='13 implies that β2 induces an element β1 := � c2∈SXH2 \\{eXH2 } tc2βρ−1 φ|H1 (c2) + teXH2 βρ−1 φ|H1 (eXH2 ) = � c2∈SXH2 \\{eXH2 } tc2βρ−1 φ|H1 (c2) + teXH2 βeXH1 ∈ M ∗ Z1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then we have that the coefficient teXH2 of βeXH1 is not equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Thus, the composition (Yα1, DYα1) ×(X1,DX1 ) (XH1, DXH1 ) → (Z1, DZ1) g1 → (XH1, DXH1) is tamely ramified over eXH1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' This means that eXH1 is contained in SXH1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' This completes the proof of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' □ rem-lem-5-1 Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Remark rem-pro-3-1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='9 implies that Dgp XHi , i ∈ {1, 2}, admits a natural action of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Moreover, the commutative diagram Dgp XH1 ρφ|H1 −−−−→ Dgp XH2 γH1,πt 1(UX1 )\uf8e6� \uf8e6�γH2,πt 1(UX2 ) Dgp X1 ρφ −−−−→ Dgp X2 is compatible with the actions of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' TOPOLOGICAL STRUCTURES OF MODULI SPACES OF CURVES AND ANABELIAN GEOMETRY IN POSITIVE CHARACTERISTIC 19 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Next, we prove that the condition (gXH1 , nXH1 ) = (gXH2 , nXH2 ) mentioned in Lemma lem-5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='14 can be omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Firstly, we treat the case of abelian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' lem-6 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We maintain the notation introduced in sec334 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Suppose that gX ≥ 2, and that G is an abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then we have (gXH1 , nXH1 ) = (gXH2 , nXH2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We write m for #G and put K2 := ker(πt 1(UX2) ։ πt 1(UX2)ab ⊗ Z/mZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then we see immediately that K2 is contained in H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let K1 := φ−1(K2) ⊆ H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Write (XKi, DXKi ) for the smooth pointed stable curves of type (gXKi , nXKi ) over ki induced by Ki and fKi : (XKi, DXKi ) → (Xi, DXi) for the tame covering over ki induced by the inclusion Ki ֒→ πt 1(UXi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We identify πt 1(UX1)/K1 with πt 1(UX2)/K2 via the isomorphism induced by φ, and denote by A := πt 1(UX1)/K1 ≃ πt 1(UX2)/K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Since each p-Galois tame covering is étale (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=', Galois tame coverings whose Galois group is a p-group), we see immediately that (gXK1 , nXK1 ) = (gXK2 , nXK2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then Lemma lem-5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='14 implies that the commutative diagram K1 φ|K1 −−−−→ K2 \uf8e6� \uf8e6� πt 1(UX1) φ −−−−→ πt 1(UX2) of profinite groups induces a commutative diagram Dgp XK1 ρφ|K1 −−−−→ Dgp XK2 γK1,πt 1(UX1 )\uf8e6� \uf8e6�γK2,πt 1(UX2 ) Dgp X1 ρφ −−−−→ Dgp X2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Moreover, Remark rem-lem-5-1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='15 implies that the commutative diagram above admits a natural action of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then, for each egp XK1 ∈ Dgp XK1 , the inertia subgroup Iegp XK1 in A associated to egp XK1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=', the stabilizer of egp XK1 under the action of A) is equal to the inertia subgroup Iegp XK2 in A associated to egp XK2 := ρφ|K1 (egp XK1 ) ∈ Dgp XK2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' On the other hand, write F for the kernel of the natural morphism A ։ G induced by the inclusion Ki ֒→ Hi, i ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Since (XHi, DXHi ) ≃ (XKi, DXKi )/F, the set of ramification indices of the Galois tame covering (XKi, DXKi ) → (XHi, DXHi ) with Galois group F are equal to {#(F ∩ Iegp XKi )}egp XKi ∈Dgp XKi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then by the Riemann-Hurwitz formula, we have (gXH1 , nXH1 ) = (gXH2 , nXH2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' This completes the proof of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' □ Next, we treat the general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' lem-7 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We maintain the notation introduced in sec334 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Suppose that gX ≥ 2 and nX ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then there exists an open normal subgroup P2 ⊆ πt 1(UX2) which is contained in H2 such that the following holds: Write (XPi, DXPi ), i ∈ {1, 2}, for the smooth pointed stable curve of type (gXPi , nXPi ) over ki induced by Pi, where P1 = φ−1(P2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We have (gXP1 , nXP1 ) = (gXP2 , nXP2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' First, suppose that G is a simple finite group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' By applying Lemma lem-6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='16, we may assume that G is non-abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Moreover, we claim that we may assume that nX is a positive even number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let us prove this claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Suppose p ̸= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let R2 ⊆ πt 1(UX2) be an open subgroup such that #(πt 1(UX2)/R2) = 2, and that R2 ⊇ ker(πt 1(UX2) ։ π1(X2)) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=', the cyclic Galois tame covering corresponding to R2 is étale).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let R1 := φ−1(R2) ⊆ πt 1(UX1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then we have that #(πt 1(UX1)/R1) = 2, and that Lemma lem-3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='12 implies R1 ⊇ ker(πt 1(UX1) ։ π1(X1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' By replacing Hi and πt 1(UXi), i ∈ {1, 2}, by Hi ∩ Ri and Ri, respectively, we may assume that nX is a even positive number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Suppose that p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let ℓ be a prime number such that (ℓ, 2) = (ℓ, #G) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' By R1 [R1, Théorème 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1], there exists an open subgroup R∗ 2 ⊆ πt 1(UX2) such that #(πt 1(UX2)/R∗ 2) = ℓ, that R∗ 2 ⊇ ker(πt 1(UX2) ։ π1(X2)), and that dimFp(R∗,ab 2 ⊗ Fp) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' 20 ZHI HU, YU YANG, AND RUNHONG ZONG Let R∗ 1 := φ−1(R∗ 2) ⊆ πt 1(UX1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then we have that #(πt 1(UX1)/R∗ 1) = ℓ, that dimFp(R∗,ab 1 ⊗ Fp) > 0, and that Lemma lem-3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='12 implies R∗ 1 ⊇ ker(πt 1(UX1) ։ π1(X1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Thus, we may take an open subgroup R′ 2 ⊆ R∗ 2 such that πt 1(UX2)/R′ 2 ≃ Z/2Z ⋊ Z/ℓZ, and that R′ 2 ⊇ ker(πt 1(UX2) ։ π1(X2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We put R′ 1 := φ−1(R′ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then the construction of R′ 1 implies πt 1(UX1)/R′ 1 ≃ Z/2Z ⋊ Z/ℓZ and R′ 1 ⊇ ker(πt 1(UX1) ։ π1(X1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' By replacing Hi and πt 1(UXi), i ∈ {1, 2}, by Hi ∩ R′ i and R′ i, respectively, we may assume that nX is a even positive number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' This completes the proof of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let #G := ptm′ such that (m′, p) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Since nX is a positive even number, we may choose a Galois tame covering f2 : (Y2, DY2) → (X2, DX2) over k2 with Galois group Z/m′Z such that f2 is totally ramified over every marked point of DX2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Write (gY2, nY2) for the type of (Y2, DY2), Q2 ⊆ πt 1(UX2) for the open normal subgroup induced by f2, Q1 := φ−1(Q2) ⊆ πt 1(UX1), f1 : (Y1, DY1) → (X1, DX1) for the Galois tame covering over k1 with Galois group Z/m′Z induced by the natural inclusion Q1 ֒→ πt 1(UX1), and (gY1, nY1) for the type of (Y1, DY1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then Lemma lem-6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='16 implies that (gY1, nY1) = (gY2, nY2) and f1 is also totally ramified over every marked point of DX1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We consider the Galois tame covering (Zi, DZi) := (XHi, DXHi ) ×(Xi,DXi) (Yi, DYi) → (Xi, DXi), i ∈ {1, 2}, over ki with Galois group G × Z/m′Z which is the composition of (Zi, DZi) → (Yi, DYi) and (Yi, DYi) → (Xi, DXi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Note that since G is a non-abelian simple finite group, (Zi, DZi) is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Moreover, by Abhyankar’s lemma, we obtain that (Zi, DZi) → (Yi, DYi) is an étale covering over ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Since (gY1, nY1) = (gY2, nY2) and (Zi, DZi) → (Yi, DYi) is unramified, the Riemann-Hurwitz formula implies (gZ1, nZ1) = (gZ2, nZ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Next, let us prove the lemma in the case where G is an arbitrary finite group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let G1 ⊆ G2 ⊆ · · · ⊆ Gn := G be a sequence of subgroups of G such that Gi/Gi−1 is a simple group for all i ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' In order to verify the lemma, we see that it is sufficient to prove the lemma when n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let N2 be the kernel of the natural homomorphism πt 1(UX2) ։ G ։ G1 and N1 := φ−1(N2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then by replacing G by G1 and by applying the lemma for the simple group G1, we obtain an open normal subgroup M2 ⊆ πt 1(UX2) which is contained in N2 such that (gXM1 , nXM1 ) = (gXM2 , nXM2 ), where M1 := φ−1(M2), and (gXMi , nXMi ), i ∈ {1, 2}, denotes the type of the smooth pointed stable curve corresponding to Mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' If Mi ⊆ Hi, i ∈ {1, 2}, then we may put Pi := Mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' If Hi, i ∈ {1, 2}, does not contain Mi, we put Oi := Mi ∩ Hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then we have Mi/Oi ≃ G/G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Note that G/G1 is a simple group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then the lemma follows from the lemma when we replace (Xi, DXi) and G by (XMi, DXMi ) and the simple group G/G1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' This completes the proof of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Now, we prove the main result of the present section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' them-2 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let ( � Xi, D � Xi), i ∈ {1, 2}, be the universal tame covering of (Xi, DXi) defined in unicov313 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let φ : πt 1(UX1) ։ πt 1(UX2) be an arbitrary open continuous surjective homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then the group-theoretical algorithm of the mono-anabelian reconstruction concerning Ine(πt 1(UXi)) obtained in Proposition them-1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='10 is compatible with the surjection φ : πt 1(UX1) ։ πt 1(UX2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Namely, the following holds: Let �e2 ∈ D � X2 and I�e2 ∈ Ine(πt 1(UX2)) the inertia subgroup associated to �e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then there exists an inertia subgroup I�e1 ∈ Ine(πt 1(UX1)) associated to a point �e1 ∈ D � X1 such that φ(I�e1) = I�e2, and that the restriction homomorphism φ|I�e1 : I�e1 ։ I�e2 is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' TOPOLOGICAL STRUCTURES OF MODULI SPACES OF CURVES AND ANABELIAN GEOMETRY IN POSITIVE CHARACTERISTIC 21 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' If nX = 0, then the theorem is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We suppose nX > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let m >> 0 be an integer such that (m, p) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We put Ki := ker(πt 1(UXi) ։ πt 1(UXi)ab ⊗ Z/mZ), i ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Write (XKi, DKi) for the smooth pointed stable curve of type (gXKi , nXKi ) over ki induced by Ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Moreover, the condition m >> 0 implies gXK1 = gXK2 ≥ 2, nXK1 = nXK2 ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' By applying Lemma lem-7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='17, we may choose a set of open subgroups CX2 := {H2,j}j∈Z>0 of πt 1(UX2) such that the following three conditions hold: H2,1 = K2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' lim ←−j πt 1(UX2)/H2,j ≃ πt 1(UX2) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' CX2 is a cofinal system);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' write {H1,j := φ−1(H2,j)}j∈Z>0 for the set of open subgroups of πt 1(UX1) induced by φ, and for each j ∈ Z>0, write (XHi,j, DXHi,j ), i ∈ {1, 2}, for the smooth pointed stable curve of type (gXHi,j , nXHi,j ) over ki induced by Hi,j, then we have (gXH1,j , nXH1,j ) = (gXH2,j , nXH2,j ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' For each j ∈ Z>0, we write eXH2,j ∈ DXH2,j for the image of �e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then we obtain a sequence of marked points I CX2 �e2 : · · · �→ eH2,2 �→ eH2,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Proposition pro-3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='8 implies that, for each H2,j, j ∈ Z>0, the set Dgp XH2,j can be mono-anabelian reconstructed from H2,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' For each eXH2,j ∈ DXH2,j , we denote by egp XH2,j := ϑ−1 XH2,j (eXH2,j ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then the sequence of marked points ICX �e2 induces a sequence ICX �egp 2 : · · · �→ egp XH2,2 �→ egp XH2,1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then Remark rem-pro-3-1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='9 implies that the inertia subgroup associated to �e2 is equal to the stabilizer of ICX �egp 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' By Lemma lem-5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='14 and Lemma lem-7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='17, I CX2 �egp 2 induces a sequence as follows: · · �→ egp XH1,2 := ρ−1 φ|H1,2 (egp XH2,2 ) ∈ Dgp XH1,2 �→ egp XH1,1 := ρ−1 φ|H1,1(egp XH2,1 ) ∈ Dgp XH1,1 with an action of I�e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then Proposition them-1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='10 implies that we have a sequence · · �→ eXH1,2 := ϑXH1,2 (egp XH1,2 ) ∈ DXH1,2 �→ eXH1,1 := ϑXH1,1 (egp XH1,1 ) ∈ DXH1,1 with an action of I�e2 Let Kker(φ) be the subfield of �K induced by the closed subgroup ker(φ) of πt 1(UX1), � X1,ker(φ) the normalization of X1 in Kker(φ), and D � X1,ker(φ) the inverse image of DX1 in � X1,ker(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then the sequence · · �→ eXH1,2 �→ eXH1,1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' determines a point �e1,ker(φ) ∈ D � X1,ker(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We choose a point of �e1 ∈ D � X1 such that the image of �e1 in D � X1,ker(φ) is �e1,ker(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then we have φ(I�e1) = I�e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Moreover, since I�e1 and I�e2 are isomorphic to �Z(1)p′, the restriction homomorphism φ|I�e1 is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' This completes the proof of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' □ sec-new6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Reconstructions of additive structures via surjections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We maintain the settings introduced in sett331 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let �e2 be an arbitrary point of D � X2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' By applying Theorem them-2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='18, there exists a point �e1 ∈ D � X1 such that φ|I�e1 : I�e1 ≃ −→ I�e2 is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Write Fp,i, i ∈ {1, 2}, for the algebraic closure of Fp in ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We put F�ei := (I�ei ⊗Z (Q/Z)p′ i ) ⊔ {∗�ei}, i ∈ {1, 2}, where {∗�ei} is an one-point set, and (Q/Z)p′ i denotes the prime-to-p part of Q/Z which can be canonically identified with � (p,m)=1 µm(ki).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Moreover, let a�ei be a generator of I�ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then we have a natural bijection I�ei ⊗Z (Q/Z)p′ i ≃ −→ Z ⊗Z (Q/Z)p′ i , a�ei ⊗ 1 �→ 1 ⊗ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' 22 ZHI HU, YU YANG, AND RUNHONG ZONG Thus, we obtain the following bijections I�ei ⊗Z (Q/Z)p′ i ≃ −→ Z ⊗Z (Q/Z)p′ i ≃ −→ � (p,m)=1 µm(ki) ≃ −→ F × p,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' This means that F�ei can be identified with Fp,i as sets, and hence admits a structure of field whose multiplicative group is I�ei ⊗Z (Q/Z)p′ i , and whose zero element is ∗�ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We will prove that φ|I�e1 : I�e1 ≃ −→ I�e2 induces an isomorphism F�e1 ≃ −→ F�e2 as fields (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Proposition pro-4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' The main idea is as follows: First, we reduce the problem to the case where nX = 3 by applying Theorem them-2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Second, the field structure of F�ei (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=', the set of isomorphisms of F�ei and Fp,i as fields) can be translated to certain problem concerning generalized Hasse-Witt invariants (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' γχi(Mχi) in the proof of Proposition pro-4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then by applying Theorem them-2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='18 again, we obtained the result by comparing γχ1(Mχ1) with γχ2(Mχ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We have the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' pro-4 Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' The field structure of F�ei, i ∈ {1, 2}, can be mono-anabelian reconstructed from πt 1(UXi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' More- over, the isomorphism φ|I�e1 : I�e1 ≃ −→ I�e2 induces an isomorphism θφ,�e1,�e2 : F�e1 ≃ −→ F�e2 as fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' First, we claim that we may assume nX = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' If gX = 0, then nX ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Suppose that gX ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Theorem them-2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='18 implies that φ : πt 1(UX1) ։ πt 1(UX2) induces an open continuous surjection φét : π1(X1) ։ π1(X2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let H′ 2 ⊆ π1(X2) be an open normal subgroup such that #(π1(X2)/H′ 2) ≥ 3 and H′ 1 := (φét)−1(H′ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Write Hi ⊆ πt 1(UXi), i ∈ {1, 2}, for the inverse image of H′ i of the natural surjection πt 1(UXi) ։ π1(Xi), and (XHi, DXHi ) for the smooth pointed stable curve of type (gXHi , nXHi ) over ki induced by Hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Note that gXH1 = gXH2 ≥ 2 and nXH1 = nXH2 ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' By replacing (Xi, DXi) by (XHi, DXHi), we may assume gX ≥ 2 and nX ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' The surjection φ induces a bijection DX1 ϑ−1 X1 −−−→ Dgp X1 ρφ −→ Dgp X2 ϑX2 −−−→ DX2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let D′ X1 := {e1,1, e1,2, e1,3} ⊆ DX1 and D′ X2 := {e2,1 := ϑX2 ◦ρφ◦ϑ−1 X1(e1,1), e2,2 := ϑX2 ◦ρφ◦ϑ−1 X1(e1,2), e2,3 := ϑX2 ◦ ρφ ◦ ϑ−1 X1(e1,3)} ⊆ DX2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then (Xi, D′ Xi), i ∈ {1, 2}, is a smooth pointed stable curve of type (gX, 3) over ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Write Ii, i ∈ {1, 2}, for the closed subgroup of πt 1(UXi) generated by the inertia subgroups associated to the elements of D � Xi whose images in DXi are contained in DXi \\ D′ Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then we have an isomorphism πt 1(Xi \\ D′ Xi) ∼= πt 1(UXi)/Ii, i ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Moreover, Theorem them-2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='18 implies that φ induces an open continuous surjective homomorphism φ′ : πt 1(X1 \\ D′ X1) ։ πt 1(X2 \\ D′ X2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Thus, by replacing (Xi, DXi), πt 1(UXi), and φ by (Xi, D′ Xi), πt 1(Xi \\ D′ Xi), and φ′, respectively, we may assume nX = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let r ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We denote by Fpr,�ei, i ∈ {1, 2}, the unique subfield of F�ei whose cardinality is equal to pr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' On the other hand, we fix any finite field Fpr of cardinality pr and an algebraic closure Fp of Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' By Proposition them-1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='10, we have that F× pr,�ei = I�ei/(pr − 1) can be mono-anabelian reconstructed from πt 1(UXi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then reconstructing the field structure of Fpr,�ei is equivalent to reconstructing Homfields(Fpr,�ei, Fpr) as a subset of Homgroup(F× pr,�ei, F× pr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Note that, in order to reconstruct the field structure of F�ei, it is sufficient to reconstruct the subset Homfields(Fpr,�ei, Fpr) for r in a cofinal subset of N with respect to division.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' TOPOLOGICAL STRUCTURES OF MODULI SPACES OF CURVES AND ANABELIAN GEOMETRY IN POSITIVE CHARACTERISTIC 23 Let χi ∈ Homgroups(πt 1(UXi)ab ⊗ Z/(pr − 1)Z, F× pr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Write Hχi for the kernel of πt 1(UXi) ։ πt 1(UXi)ab ⊗ Z/(pr − 1)Z χi → F× pr, Mχi for Hab χi ⊗ Fp, and (XHχi , DXHχi ) for the smooth pointed stable curve over ki induced by Hχi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We define Mχi[χi] := {a ∈ Mχi ⊗Fp Fp | σ(a) = χi(σ)a for all σ ∈ πt 1(UXi)ab ⊗ Z/(pr − 1)Z} and γχi(Mχi) := dimFp(Mχi[χi]) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' a generalized Hasse-Witt invariant (see Y5 [Y5, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2])).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then T4 [T4, Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='7] implies γχi(Mχi) ≤ gX + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Moreover, we define two maps Resi,r : Homgroups(πt 1(UXi)ab ⊗ Z/(pr − 1)Z, F× pr) → Homgroups(F× pr,�ei, F× pr), Γi,r : Homgroups(πt 1(UXi)ab ⊗ Z/(pr − 1)Z, F× pr) → Z≥0, χi �→ γχi(Mχi), where the map Resi,r is the restriction with respect to the natural inclusion F× pr,�ei ֒→ πt 1(UXi)ab ⊗ Z/(pr − 1)Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let m0 be the product of all prime numbers ≤ p − 2 if p ̸= 2, 3 and m0 = 1 if p = 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let r0 be the order of p in the multiplicative group (Z/m0Z)×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then T4 [T4, Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='4] implies the following result: there exists a constant C(gX) which depends only on gX such that, for each r > logp(C(gX) + 1) divisible by r0, we have Homfields(Fpr,�ei, Fpr) = Homsurj groups(F× pr,�ei, F× pr) \\ Resi,r(Γ−1 i,r ({gX + 1})), i ∈ {1, 2}, where Homsurj groups(−, −) denotes the set of surjections whose elements are contained in Homgroups(−, −).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let κ2 ∈ Homgroups(πt 1(UX2)ab ⊗ Z/(pr − 1)Z, F× pr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then φ induces a character κ1 ∈ Homgroups(πt 1(UX1)ab ⊗ Z/(pr − 1)Z, F× pr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Moreover, the surjection φ|Hκ1 induces a surjection Mκ1[κ1] ։ Mκ2[κ2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Suppose that κ2 ∈ Γ−1 2,r({gX + 1}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' The surjection Mκ1[κ1] ։ Mκ2[κ2] implies γκ1(Mκ1) = gX + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' This means κ1 ∈ Γ−1 1,r({gX + 1}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' On the other hand, the isomorphism φ|I�e1 : I�e1 ≃ −→ I�e2 induces an injection Res2,r(Γ−1 2,r({gX + 1})) ֒→ Res1,r(Γ−1 1,r({gX + 1})).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Since #(Homfields(Fpr,�e1, Fpr)) = #(Homfields(Fpr,�e2, Fpr)), we obtain that φ|I�e1 induces a bijection Homfields(Fpr,�e2, Fpr) ≃ −→ Homfields(Fpr,�e1, Fpr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Thus, φ|I�e1 induces a bijection Homfields(F�e2, Fp) ≃ −→ Homfields(F�e1, Fp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' If we choose Fp = F�e2, then the image of idF�e2 via the bijection above induces an isomorphism θφ,�e1,�e2 : F�e1 ≃ −→ F�e2 as fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' This completes the proof of the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' MAIN THEOREMS sec-5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' The first main theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' In this subsection, we apply the results obtained in the previous sections to prove that the curves of type (0, n) over Fp can be reconstructed group-theoretically from open continuous homomorphism (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Theorem them-3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' 24 ZHI HU, YU YANG, AND RUNHONG ZONG 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We fix some notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let (Xi, DXi), i ∈ {1, 2}, be a smooth pointed stable curve of type (gX, nX) over an algebraically closed field ki of characteristic p > 0, UXi := Xi \\ DXi, πt 1(UXi) the tame fundamental group of UXi, π1(Xi) the étale fundamental group of Xi, and ( � Xi, D � Xi) the universal tame covering of (Xi, DXi) associated to πt 1(UXi) ( unicov313 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let km i , i ∈ {1, 2}, be the minimal algebraically closed subfield of ki over which UXi can be defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Thus, by considering the function field of Xi, we obtain a smooth pointed stable curve (Xm i , DXm i ) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=', a minimal model of (Xi, DXi) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' T3 [T3, Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='30 and Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='31])) such that UXi ∼= UXm i ×km i ki as ki-schemes, where UXm i := Xm i \\ DXm i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Note that πt 1(UXm i ) is naturally isomorphic to πt 1(UXi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We shall denote by Fp,i the algebraic closure of Fp in ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Moreover, we put d(Xi,DXi ) := � 0, if km i ∼= Fp,i, 1, if km i ̸∼= Fp,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Firstly, we have the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' lemsurj Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let φ : πt 1(UX1) → πt 1(UX2) be an arbitrary open continuous homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then φ is a surjection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We denote by Πφ the image of φ which is an open subgroup of πt 1(UX2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let (Xφ, DXφ) be the smooth pointed stable curve of type (gXφ, nXφ) over k2 induced by Πφ and fφ : (Xφ, DXφ) → (X2, DX2) the tame covering of smooth pointed stable curves over k2 induced by the inclusion Πφ ֒→ πt 1(UX2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Since fφ is a tame covering, we have that nXφ ≥ nX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' On the other hand, if gX = 0, we have gφ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' If gX > 0, the Riemann-Hurwitz formula implies gXφ ≥ [πt 1(UX2) : Πφ](gX − 1) + 1 ≥ gX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then we have gφ ≥ gX and nXφ ≥ nX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Note that πt 1(UX1) ։ Πφ ֒→ πt 1(UX2) implies 2gX + nX − 1 ≥ 2gXφ + nXφ − 1 ≥ 2gX + nX − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then we obtain that 2gX + nX − 1 = 2gXφ + nXφ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Moreover, Proposition coro-p-average 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='11 (ii) and the natural surjection πt 1(UX1) ։ Πφ induced by φ imply that gX ≥ gXφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then we obtain that gX = gXφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Thus, we have (gX, nX) = (gXφ, nXφ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' This means that the tame covering fφ : (Xφ, DXφ) → (X2, DX2) is totally ramified over every marked point of DX2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let us prove [πt 1(UX2) : Πφ] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Suppose [πt 1(UX2) : Πφ] ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Since fφ is totally ramified, the Riemann- Hurwitz formula implies gXφ > gX if nX ̸= 0 and gX ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' This is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' If nX = 0, the Riemann-Hurwitz formula implies gX = 1 if gX ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' This contradicts the assumption that (Xi, DXi) is a pointed stable curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then we obtain gX = gXφ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Moreover, by applying the Riemann-Hurwitz formula again, since nX = nXφ, we obtain nX = nXφ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' This contradicts the assumption that (Xi, DXi) is pointed stable curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then we have [πt 1(UX2) : Πφ] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' This means that φ is a surjection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Further settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' In the remainder of this subsection, we suppose (gX, nX) = (0, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We fix two marked points e1,∞, e1,0 ∈ DX1 distinct from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Moreover, we choose any field k′ 1 ∼= k1, and choose any isomor- phism ϕ1 : X1 ≃ −→ P1 k′ 1 as schemes such that ϕ1(e1,∞) = ∞ and ϕ1(e1,0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then the set of k1-rational points X1(k1) \\ {e1,∞} is equipped with a structure of Fp-module via the bijection ϕ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Note that since any k′ 1-isomorphism of P1 k′ 1 fixing ∞ and 0 is a scalar multiplication, the Fp-module structure of X1(k1) \\ {e1,∞} does not depend on the choices of k′ 1 and ϕ1 but depends only on the choices of e1,∞ and e1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then we shall say that X1(k1) \\ {e1,∞} is equipped with a structure of Fp-module with respect to e1,∞ and e1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' By applying Theorem them-2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='18, in the next lemma, we will prove that Tamagawa’s group-theoretical criterion (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=', T2 [T2, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='3]) for linear conditions is compatible with arbitrary open continuous surjective homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' lem-8 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let φ : πt 1(UX1) ։ πt 1(UX2) be an open continuous surjective homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' By Lemma lem-4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='13, φ induces a bijection ρφ : Dgp X1 ≃ −→ Dgp X2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We may identify Dgp Xi, i ∈ {1, 2}, with DXi via the bijection ϑXi : Dgp Xi ≃ −→ TOPOLOGICAL STRUCTURES OF MODULI SPACES OF CURVES AND ANABELIAN GEOMETRY IN POSITIVE CHARACTERISTIC 25 DXi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Write e2,∞ and e2,0 for ρφ(e1,∞) and ρφ(e1,0), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let � e1∈DX1 \\{e1,∞,e1,0} be1e1 = e1,0 be a linear condition with respect to e1,∞ and e1,0 on (X1, DX1), where be1 ∈ Fp for each e1 ∈ DX1 \\ {e1,∞, e1,0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then the linear condition � e1∈DX1 \\{e1,∞,e1,0} be1ρφ(e1) = ρφ(e1,0) = e2,0 with respect to e2,∞ and e2,0 on (X2, DX2) also holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let �e2,∞ ∈ D � X2 be a point over e2,∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' The set F�e2,∞ := (I�e2,∞ ⊗Z (Q/Z)p′ 2 ) ⊔ {∗�e2,∞} admits a structure of field, and Proposition pro-4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='19 implies that the field structure can be mono-anabelian reconstructed from πt 1(UX2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Theorem them-2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='18 implies that there exists a point �e1,∞ ∈ D � X1 over e1,∞ such that φ(I�e1,∞) = �e2,∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' By Proposition pro-4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='19 again, the set F�e1,∞ := (I�e1,∞ ⊗Z (Q/Z)p′ 1 ) ⊔ {∗�e1,∞} admits a structure of field which can be mono-anabelian reconstructed from πt 1(UX1), and φ induces an isomorphism θφ,�e1,∞,�e2,∞ : F�e1,∞ ≃ −→ F�e2,∞ as fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' For each e1 ∈ DX1, we take b′ e1 ∈ Z≥0 such that b′ e1 ≡ be1 (mod p) and � e1∈DX1 \\{e1,∞,e1,0} b′ e1 ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let r ≥ 1 such that pr − 2 ≥ � e1∈DX1 \\{e1,∞,e1,0} b′ e1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' For each �e1 ∈ D � X1 over e1, write I�e1,ab for the image of the natural morphism I�e1 ֒→ πt 1(UX1) ։ πt 1(UX1)ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Moreover, since the image of I�e1,ab does not depend on the choices of �e1, we may write Ie1 for I�e1,ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' The structure of maximal prime-to-p quotient of πt 1(UX1) implies that πt 1(UX1)ab is generated by {Ie1}e1∈DX1 , and that there exists a generator ae1, e1 ∈ DX1, of Ie1 such that � e1∈DX1 ae1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We define Ie1,∞ → Z/(pr − 1)Z, ae1,∞ �→ 1, Ie1,0 → Z/(pr − 1)Z, ae1,0 �→ ( � e1∈DX1 \\{e1,∞,e1,0} b′ e1) − 1, and Ie1 → Z/(pr − 1)Z, ae1 �→ −b′ e1, e1 ∈ DX1 \\ {e1,∞, e1,0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then the homomorphisms of inertia subgroups defined above induces a surjection δ1 : πt 1(UX1) ։ πt 1(UX1)ab ։ Z/(pr − 1)Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Note that ker(δ1) does not depend on the choices of the generators {ae1}e1∈DX1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let I�e2 := φ(I�e1), �e1 ∈ D � X1, and Ie2, e2 ∈ DX2 be the image of the natural homomorphism I�e2 ֒→ πt 1(UX2) ։ πt 1(UX2)ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Since (p, pr − 1) = 1, by Theorem them-2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='18, δ1 and the isomorphism φp′ : πt 1(UX1)p′ ≃ −→ πt 1(UX2)p′ imply the following homomorphisms of inertia subgroups: Ie2,∞ → Z/(pr − 1)Z, ae2,∞ �→ 1, Ie2,0 → Z/(pr − 1)Z, ae2,0 �→ ( � e1∈DX1 \\{e1,∞,e1,0} b′ e1) − 1, and Ie2 → Z/(pr − 1)Z, ae2 �→ −b′ e1, e2 ∈ DX2 \\ {e2,∞, e2,0}, where ae2, e2 ∈ DX2, denotes the element induced by ae1, e1 ∈ DX1, via φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then the homomorphisms of inertia subgroups defined above induces a sujection δ2 : πt 1(UX2) ։ πt 1(UX2)ab ։ Z/(pr − 1)Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We put Hδi := ker(δi), Mδi := Hab δi ⊗ Fp, i ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Write (XHδi , DXHδi ) for the smooth pointed stable curve over ki induced by Hδi, where Hδ1 = φ−1(Hδ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' The Fp-vector space Mδi admits a natural action of I�ei,∞ via conjugation which coincides with the action via the following character χI�ei,∞,r : I�ei,∞ ֒→ πt 1(UXi) δi։ Z/(pr − 1)Z = I�ei,∞/(pr − 1) ֒→ F× �ei,∞, i ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' 26 ZHI HU, YU YANG, AND RUNHONG ZONG We put Mδi[χI�ei,∞ ,r] := {a ∈ Mδi ⊗Fp F�ei,∞ | σ(a) = χI�ei,∞,r(σ)a for all σ ∈ I�ei,∞} (in fact, dimF�ei,∞ (Mδi[χI�ei,∞ ,r]) is the first generalized Hasse-Witt invariant associated to the tame covering of UXi corre- sponding to Hδi ⊆ πt 1(UXi) (see Y5 [Y5, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2])).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Since the action of I�ei,∞ on Mδi is semi-simple, we obtain a surjection Mδ1[χI�e1,∞,r] ։ Mδ2[χI�e2,∞,r] induced by φ|Hδ1 and θφ,�e1,∞,�e2,∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' On the other hand, the third and the final paragraphs of the proof of T2 [T2, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='3] imply that the linear condition � e1∈DX1 \\{e1,∞,e1,0} be1e1 = e1,0 with respect to e1,∞ and e1,0 on (X1, DX1) holds if and only if Mδ1[χI�e1,∞ ,r] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Thus, we obtain Mδ2[χI�e2,∞ ,r] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then the third and the final paragraphs of the proof of T2 [T2, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='3] imply that the linear condition � e1∈DX1\\{e1,∞,e1,0} be1ρφ(e1) = e2,0 with respect to e2,∞ and e2,0 on (X2, DX2) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' This completes the proof of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' □ rem-lem-8-1 Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Note that, if X1 = P1 k, then the linear condition is as follows: � e1∈DX1 \\{∞,0} be1e1 = 0 with respect to ∞ and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Now, we prove the first main theorem of the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' them-3 Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We maintain the notation and settings introduced above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then we have the following claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' (1) d(Xi,DXi ), i ∈ {1, 2}, can be mono-anabelian reconstructed from πt 1(UXi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' (2) Suppose km 1 ∼= Fp,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then the set of open continuous homomorphisms Homop pg(πt 1(UX1), πt 1(UX2)) is non-empty if and only if UXm 1 ∼= UXm 2 as schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' In particular, if this is the case, we have km 2 ∼= Fp,2 and Homop pg(πt 1(UX1), πt 1(UX2)) = Isompg(πt 1(UX1), πt 1(UX2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Firstly, let us prove (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' The “if" part of (2) is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We treat the “only if" part of (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Suppose that Homop pg(πt 1(UX1), πt 1(UX2)) is a non-empty set, and let φ ∈ Homop pg(πt 1(UX1), πt 1(UX2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then Lemma lemsurj 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1 implies that φ is a surjection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We identify Dgp Xi, i ∈ {1, 2}, with DXi via the bijection ϑXi : Dgp Xi ≃ −→ DXi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Since φ is a surjection, Lemma lem-4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='13 implies that φ induces a bijection ρφ : DX1 ≃ −→ DX2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We put e2,0 := ρφ(e1,0) and e2,∞ := ρφ(e1,∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let �e2,0 ∈ D � X2 be a point over e2,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Theorem them-2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='18 implies that there exists a point �e1,0 ∈ D � X1 over e1,0 such that φ(I�e1,0) = I�e2,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then F�ei,0 := (I�ei,0 ⊗Z (Q/Z)p′ i )⊔{∗�ei,0}, i ∈ {1, 2}, admits a structure of field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Moreover, Proposition pro-4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='19 implies that the field structure can be mono-anabelian reconstructed from πt 1(UXi), and that φ induces a field isomorphism θφ,�e1,0,�e2,0 : F�e1,0 ≃ −→ F�e2,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Proposition proposition 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2 (1) implies that n can be mono-anabelian reconstructed from πt 1(UXi), i ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' If n = 3, (ii) is trivial, so we may assume n ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Moreover, since km 1 ∼= Fp,1, without loss of generality, we may assume k1 = Fp,1 = F�e1,0, X1 = P1 Fp,1, and DX1 = {e1,∞ = ∞, e1,0 = 0, e1,1 = 1, e1,2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' , e1,n−2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Here, e1,2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' , e1,n−2 ∈ Fp,1 \\ {e1,0, e1,1} are distinct from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Step 1: In this step, we will construct a linear condition on a certain tame covering of (X1, DX1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' TOPOLOGICAL STRUCTURES OF MODULI SPACES OF CURVES AND ANABELIAN GEOMETRY IN POSITIVE CHARACTERISTIC 27 We see that there exists a natural number r prime to p such that Fp(ζr) contains rth roots of e1,2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' , e1,n−2, where ζr denotes a fixed primitive rth root of unity in Fp,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let s := [Fp(ζr) : Fp].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' For each e1,u ∈ {e1,2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' , e1,n−2}, we fix an rth root e1/r 1,u in Fp,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then we have e1/r 1,u = s−1 � v=0 b1,uvζv r , u ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=', n − 2}, where b1,uv ∈ Fp for each u ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' , n − 2} and each v ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' , s − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let X1 \\ {e1,∞} = SpecFp,1[x1], fH1 : (XH1, DXH1) → (X1, DX1) the Galois tame covering over Fp,1 with Galois group Z/rZ determined by the equation yr 1 = x1, and H1 the open normal subgroup of πt 1(UX1) induced by the tame covering fH1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then fH1 is totally ramified over {e1,∞ = ∞, e1,0 = 0} and is étale over DX1 \\ {∞, 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Note that XH1 = P1 Fp,1, and the points of DXH1 over {e1,∞, e1,0} are {eH1,∞ := ∞, eH1,0 := 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We put eH1,u := e1/r 1,u ∈ DXH1 , u ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=', n − 2}, ev H1,1 := ζv r ∈ DXH1 , v ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' , s − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Thus, we obtain a linear condition eH1,u = s−1 � v=0 b1,uvev H1,1 with respect to eH1,∞ and eH1,0 on (XH1, DXH1 ) for each u ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' , n − 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Step 2: In this step, we will prove that the linear condition on a certain tame covering of (X1, DX1) constructed in Step 1 induces a linear condition on a certain tame covering of (X2, DX2) via the surjection φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Write H2 for φ(H1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Since (r, p) = 1, we have the following commutative diagram of profinite groups: H1 φ|H1 −−−−→ H2 \uf8e6� \uf8e6� πt 1(UX1) φ −−−−→ πt 1(UX2) \uf8e6� \uf8e6� Z/rZ Z/rZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We denote by fH2 : (XH2, DXH2) → (X2, DX2) the Galois tame covering over Fp,2 with Galois group Z/rZ induced by H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Note that Lemma lem-6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='16 implies that (XH1, DXH1) and (XH2, DXH2 ) are equal types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Moreover, Lemma lem-5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='14 implies that the following commutative diagram can be mono-anabelian reconstructed from the commutative diagram of profinite groups above: DXH1 ρφ|H1 −−−−→ DXH2 \uf8e6� \uf8e6� DX1 ρφ −−−−→ DX2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We put e2,∞ := ρφ(e1,∞), e2,u := ρφ(e1,u), u ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=', n − 2}, eH2,∞ := ρφ|H1(eH1,∞), eH2,0 := ρφ|H1(eH1,0), eH2,u := ρφ|H1 (eH1,u), u ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=', n − 2}, and ev H2,1 := ρφ|H1(ev H1,1), v ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' , s − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Remark rem-lem-5-1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='15 implies that fH2 is totally ramified over {e2,∞, e2,0} and is étale over X2 \\ {e2,∞, e2,0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then we may assume that X2 = P1 k2, and that e2,∞ = ∞, e2,0 = 0, e2,1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We regard e2,u, u ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' , n − 2}, as an element of k2 \\ {e2,0, e2,1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Moreover, we have eH2,∞ = ∞ and eH2,0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' 28 ZHI HU, YU YANG, AND RUNHONG ZONG We put ξr := θφ,�e1,0,�e2,0(ζr) which is an rth root of unity in F�e2,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Since ζr(ev H1,1) = ev+1 H1,1, we obtain ξr(ev H2,1) = ev+1 H2,1, v ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=', s − 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' By applying Lemma lem-8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2 for φ|H1 : H1 ։ H2, the following linear condition eH2,u = s−1 � v=0 b1,uvξv r(e0 H2,1) with respect to eH2,∞ and eH2,0 on (XH2, DXH2 ) holds for each u ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' , n − 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Since (eH2,u)r = e2,u, u ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' , n − 2}, we obtain e2,u = ( s−1 � v=0 b1,uvξv r (e0 H2,1))r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Moreover, if we put e0 H2,1 = 1, then we obtain that e2,u = ( s−1 � v=0 b1,uvξv r )r for each u ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' , n − 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Since θφ,�e1,0,�e2,0(ζr) = ξr, we have UX1 = UXm 1 = P1 Fp,1 \\ {e1,∞ = ∞, e1,0 = 0, e1,1 = 1, e1,2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' , e1,n−2} ≃ −→ P1 F�e2,0 \\ {e2,∞ = ∞, e2,0 = 0, e2,1 = 1, θφ,�e1,0,�e2,0(e1,2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' , θφ,�e1,0,�e2,0(e1,n−2)} ∼= P1 Fp,2 \\ {e2,∞ = ∞, e2,0 = 0, e2,1 = 1, e2,2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' , e2,n−2} and P1 Fp,2 \\ {e2,∞ = ∞, e2,0 = 0, e2,1 = 1, e2,2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' , e2,n−2} ×Fp,2 k2 ∼= UX2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' This means UXm 1 ∼= UXm 2 as schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' In particular, we have km 2 ∼= Fp,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Finally, we prove that Homop pg(πt 1(UX1), πt 1(UX2)) = Isompg(πt 1(UX1), πt 1(UX2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' The “⊇" part is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We only need to prove the “⊆" part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We may assume Homop pg(πt 1(UX1), πt 1(UX2)) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let φ′ ∈ Homop pg(πt 1(UX1), πt 1(UX2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then πt 1(UX1) is isomorphic to πt 1(UX2) as abstract profinite groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' By Lemma lemsurj 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1, φ′ is a surjection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then FJ [FJ, Proposition 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='6] implies that φ′ is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Thus, we obtain φ′ ∈ Isompro-gps(πt 1(UX1), πt 1(UX2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' This completes the proof of (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Next, let us prove (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Without loss of generality, we only treat the case where i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Moreover, let (X, DX) := (X1, DX1), DX = {e∞ = ∞, e0 = 0, e1 = 1, e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' , en−2}, k := k1, and Fp := F�e0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let (r, Q) be a pair such that the following two conditions hold: (r, p) = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Q is an open normal subgroup of πt 1(UX) such that πt 1(UX)/Q ∼= Z/rZ, and that the Galois tame covering fQ : (XQ, DXQ) → (X, DX) over k induced by Q is totally ramified over {e∞, e0} and is étale over DX \\ {e∞, e0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' By applying Theorem them-2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='18, we see immediately that the set of pairs defined above can be mono-anabelian recon- structed from πt 1(UX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We fix a primitive r-th root of unity ζr in Fp and put sr := [Fp(ζr) : Fp].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Moreover, we put eQ,∞ := ∞, eQ,0 := 0, ev Q,1 := ζv r ∈ DXQ, v ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' sr − 1}, and let eQ,u ∈ DXQ, u ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=', n}, such that fQ(eQ,u) = eu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Denote by LQ,u := {eQ,u − sr−1 � v=0 buvev Q,1 | buv ∈ Fp} ∩ {0}, u ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' , n − 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' By applying arguments similar to the arguments given in the proof of (2) above, we have that d(X,DX) = 0 if and only if there exists a pair (r, Q) defined above such that LQ,u ̸= ∅ for each u ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' , n − 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then the third and TOPOLOGICAL STRUCTURES OF MODULI SPACES OF CURVES AND ANABELIAN GEOMETRY IN POSITIVE CHARACTERISTIC 29 the final paragraphs of the proof of T2 [T2, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='3] implies that LQ,u, u ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' , n − 2}, can be mono-anabelian reconstructed from Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Thus, d(X,DX) can be mono-anabelian reconstructed from πt 1(UX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' This completes the proof of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Note that Theorem them-3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='4 also holds if we replace πt 1(UXi), i ∈ {1, 2}, by its maximal pro-solvable quotient πt 1(UXi)sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then we obtain the following solvable version of Theorem them-3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='4 which is slightly stronger than the original theorem: We maintain the notation introduced above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then d(Xi,DXi), i ∈ {1, 2}, can be mono-anabelian reconstructed from πt 1(UXi)sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Moreover, suppose that km 1 ∼= Fp,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then the set of open continuous homomorphisms Homop pg(πt 1(UX1)sol, πt 1(UX2)sol) is non-empty if and only if UXm 1 ∼= UXm 2 as schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' In particular, if this is the case, we have km 2 ∼= Fp,2 and Homop pg(πt 1(UX1)sol, πt 1(UX2)sol) = Isompg(πt 1(UX1)sol, πt 1(UX2)sol).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' sec-6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' The second main theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' In this subsection, by using Theorem them-3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='4, we prove a result concerning pointed collection conjecture and the weak Hom-version conjecture (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Theorem them-4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We maintain the notation introduced in moduli212 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' def-4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let q ∈ M ord 0,n be an arbitrary point, k(q) an algebraic closure of k(q), and UXq ≃ P1 k(q) \\ {a1 = 1, a2 = 0, a3 = ∞, a4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' , an} as k(q)-schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We shall say that q is a coordinated point if either q = qgen or the following three conditions are satisfied: dim(Vq) = dim(M ord 0,n ) − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' there exists i ∈ {4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=', n} such that ai ∈ Fp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' let ωi n,n−1 : M ord 0,n → M ord 0,n−1 be the morphism induced by the morphism Mord 0,n → Mord 0,n−1 obtained by forgetting the ith marked point;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' then ωi n,n−1(q) is the generic point of M ord 0,n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let t be a closed point of M ord 0,n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then there exists a set of coordinated points Pt := {qt,4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' , qt,n} such that {t} = � qt,j∈Pt Vqt,j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Now, we prove the second main theorem of the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' them-4 (1) For each closed point t ∈ M ord,cl 0,n , the set Ct associated to t is a pointed collection (Definition def-3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Moreover, for each pointed collection C ∈ Cqgen, there exists a closed point s ∈ M ord,cl 0,n such that C = Cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' (2) Let q ∈ M ord 0,n be an arbitrary point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then the the natural map colleq : V cl q → Cq, [t] �→ Ct, is an injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' (3) Let q ∈ M ord 0,n be an arbitrary point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Suppose that there exists a set of coordinated points Pq such that Vq = � u∈Pq Vu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then the pointed collection conjecture holds for q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' In particular, the pointed collection conjecture holds for each closed point of M ord 0,n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' 30 ZHI HU, YU YANG, AND RUNHONG ZONG (4) Let qi ∈ M ord 0,n , i ∈ {1, 2}, be an arbitrary point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Suppose that there exists a set of coordinated points Pq1 such that Vq1 = � u∈Pq1 Vu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then the weak Hom-version conjecture holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' In particular, the weak Hom-version conjecture holds when q1 is a closed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let us prove (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We put Ft := {t′ ∈ M ord,cl 0,n | t ∼fe t′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let t′′ be an arbitrary point of � G∈πt A(t) UG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then, for each G ∈ πt A(t), Homsurj pg (πt 1(t′′), G) is non-empty, where Homsurj pg (−, −) denotes the subset of Homopen pg (−, −) whose elements are surjections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Since πt 1(t′′) is topologically finitely generated, we obtain that the set Homsurj pg (πt 1(t′′), G) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then the set of open continuous homomorphisms lim ←− G∈πt A(t) Homsurj pg (πt 1(t′′), G) = Homsurj pg (πt 1(t′′), πt 1(t)) is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Thus, Theorem them-3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='4 implies t′′ ∈ Ft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' This means ( � G∈πt A(t) UG) ∩ M ord,cl g,n = Ft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Since UXt can be defined over a finite field, Ft is a finite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then Ct is a pointed collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let C ∈ Cqgen be a pointed collection and s a closed point of � G∈C UG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' By replacing t by s, and by applying arguments similar to the arguments given in the proof above, we obtain C = Cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' (2) follows immediately from Theorem them-3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let us prove (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' If n = 4, then M ord 0,4 is a one dimensional scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' For each q ∈ M ord 0,4 , the pointed collection conjecture follows immediately from Theorem them-3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then we may assume n ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' To verify (iii), (ii) implies that we only need to prove that colleq is a surjection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Suppose that q is a closed point of M ord 0,n , then (iii) follows immediately from Theorem them-3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Suppose that q is a non-closed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' This means dim(Vq) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' If q = qgen, (3) follows from (1) and (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let us treat the case where q ̸= qgen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' First, suppose that q is a coordinated point, and that UXq ≃ P1 k(q) \\ {1, 0, ∞, a4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' , an}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Without loss of generality, we may assume an ∈ Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' For each pointed collection C ⊆ Cq, by applying (1), there exists a closed point t1 ∈ M ord,cl g,n such that Ct1 = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then we have an open continuous surjective homomorphism πt 1(q) ։ πt 1(t1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let ω\\n n,4 : M ord 0,n → M ord 0,4 be the morphism induced by the morphism Mord 0,n → Mord 0,4 obtained by forgetting the marked points except the first, the second, the third, and the nth marked points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We put t′′ 1 := ω\\n n,4(t1) and q′′ := ω\\n n,4(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Note that t′′ 1 and q′′ are closed points of M0,4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then Theorem them-2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='18 implies that the surjection πt 1(q) ։ πt 1(t1) induces an open continuous surjective homomorphism πt 1(q′′) ։ πt 1(t′′ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Thus, by Theorem them-3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='4, we obtain that q′′ ∼fe t′′ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then without loss of generality, we may assume UXt1 ≃ P1 Fp \\ {1, 0, ∞, b4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' , bn−1, an} over Fp, where bi ∈ Fp for each i ∈ {4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' , n − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' On the other hand, let ωn n,n−1 : M ord 0,n → M ord 0,n−1 be the morphism induced by the morphism Mord 0,n → Mord 0,n−1 obtained by forgetting the n-th marked point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We put t′ 1 := ωn n,n−1(t1) and q′ := ωn n,n−1(q), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Since q is a coordinated point, q′ is the generic point of M ord 0,n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then we obtain t′ 1 ∈ V cl q′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Moreover, we see Vq = ω−1 n,n−1(q′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Thus, t1 = ω−1 n,n−1(t′ 1) is a closed point of Vq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Then the pointed collection conjecture holds for q when q is a coordinated point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' 31 Next, we prove the general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' If Vq = � u∈Pq Vu, then V cl q = � u∈Pq V cl u and � u∈Pq Cu = Cq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Moreover, since we have a bijection colleu : V cl u ≃ −→ Cu for each u ∈ Pq, we have that colleq : V cl q = � u∈Pq V cl u → � u∈Pq Cu = Cq is a bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' This completes the proof of (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let us prove (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' We only need to prove the “only if" part of the weak Hom-version conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Suppose that Vq2 is not essentially contained in Vq1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' This implies that there exists a closed point t2 ∈ V cl q2 such that Ft2 ∩ Vq1 = ∅, where Ft2 := {t′ 2 ∈ M ord,cl 0,n | t2 ∼fe t′ 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' By (3), we have Ct2 ̸∈ Cq1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Thus, by Lemma lemsurj 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='1, we obtain that Homop pg(πt 1(q1), πt 1(t2)) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' This provides a contradiction to the assumption that Homop pg (πt 1(q1), πt 1(q2)) is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' This completes the proof of (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Let q ∈ Mg,n be an arbitrary point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Stevenson posed a question as follows (see Ste [Ste, Question 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='3] for the case of n = 0): Does � G∈πt A(q) UG contain any closed points of Mg,n?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' By T5 [T5, Theorem 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='3], � G∈πt A(q) UG contains a closed point of Mg,n if and only if q is a closed point of Mg,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Furthermore, when g = 0 and q is a closed point, the proof of Theorem them-4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='6 (1) implies that ( � G∈πt A(q) UG) ∩ M cl 0,n = Fq, where Fq := {q′ ∈ M cl 0,n | q ∼fe q′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' REFERENCES [FJ] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Fried, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Jarden, Field arithmetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Third edition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Ergebnisse der Mathematik und ihrer Grenzgebiete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Folge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' A Series of Modern Surveys in Mathematics 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Springer-Verlag, Berlin, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' [G] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Grothendieck, Letter to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Faltings (translation into English).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Geometric Galois actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Around Grothendieck’s “Esquisse d’un programme”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Edited by Leila Schneps and Pierre Lochak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' London Mathematical Society Lecture Note Series, 242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Cambridge University Press, Cambridge, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' iv+293 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' [K] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Knudsen, The projectivity of the moduli space of stable curves, II: The stacks Mg,n, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Scand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=', 52 (1983), 161¨C-199.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' [M] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Mochizuki, The local pro-p anabelian geometry of curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Invent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' 138 (1999), 319–423.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' [N] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Nakamura, Galois rigidity of the étale fundamental groups of punctured projective lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Reine Angew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' 411 (1990), 205- 216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' [PS] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Pop, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Saïdi, On the specialization homomorphism of fundamental groups of curves in positive characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Galois groups and fundamental groups, 107–118, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=', 41, Cambridge Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Press, Cambridge, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' [R1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Raynaud, Sections des fibrés vectoriels sur une courbe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' France 110 (1982), 103–125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' [R2] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Raynaud, Sur le groupe fondamental d’une courbe complète en caractéristique p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Arithmetic fundamental groups and non- commutative algebra (Berkeley, CA, 1999), 335-351, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Sympos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Pure Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=', 70, Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=', Providence, RI, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' [Ste] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Stevenson, Galois groups of unramified covers of projective curves in characteristic p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Journal of Algebra 182 (1996), 770–804.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' [T1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Tamagawa, The Grothendieck conjecture for affine curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Compositio Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' 109 (1997), 135–194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' [T2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Tamagawa, On the fundamental groups of curves over algebraically closed fields of characteristic > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Internat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Notices (1999), 853-873.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' [T3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Tamagawa, Fundamental groups and geometry of curves in positive characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Arithmetic fundamental groups and noncom- mutative algebra (Berkeley, CA, 1999), 297–333, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Sympos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Pure Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=', 70, Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=', Providence, RI, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' [T4] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Tamagawa, On the tame fundamental groups of curves over algebraically closed fields of characteristic > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Galois groups and fundamental groups, 47–105, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=', 41, Cambridge Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Press, Cambridge, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' [T5] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Tamagawa, Finiteness of isomorphism classes of curves in positive characteristic with prescribed fundamental groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Algebraic Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' 13 (2004), 675–724.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' [Y1] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Yang, On the admissible fundamental groups of curves over algebraically closed fields of characteristic p > 0, Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' 54 (2018), 649–678.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' 32 ZHI HU, YU YANG, AND RUNHONG ZONG [Y2] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Yang, On topological and combinatorial structures of pointed stable curves over algebraically closed fields of positive characteristic, to appear in Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Nachr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' [Y3] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Yang, On the averages of generalized Hasse-Witt invariants of pointed stable curves in positive characteristic, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' 295 (2020), 1–45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' [Y4] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Yang, On the existence of specialization isomorphisms of admissible fundamental groups in positive characteristic, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' 28 (2021), 1941–1959.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' [Y5] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Yang, Maximum generalized Hasse-Witt invariants and their applications to anabelian geometry, Selecta Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=') 28 (2022), Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' 5, 98 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' [Y6] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Yang, Moduli spaces of fundamental groups of curves in positive characteristic I, preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' See http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='kurims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='kyoto-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='jp/~yuyang/ [Y7] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Yang, Moduli spaces of fundamental groups of curves in positive characteristic II, in preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' [Y8] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' Yang, Topological and group-theoretical specializations of fundamental groups of curves in positive characteristic, preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content=' See http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='kurims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='kyoto-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} +page_content='ac.' metadata={'source': 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+page_content='cn' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfCwu_/content/2301.04864v1.pdf'} diff --git a/9NAzT4oBgHgl3EQfgvyq/content/tmp_files/2301.01474v1.pdf.txt b/9NAzT4oBgHgl3EQfgvyq/content/tmp_files/2301.01474v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..924831e1937fc91a72c3c4831a7fe7907128aa5c --- /dev/null +++ b/9NAzT4oBgHgl3EQfgvyq/content/tmp_files/2301.01474v1.pdf.txt @@ -0,0 +1,1012 @@ +arXiv:2301.01474v1 [eess.SY] 4 Jan 2023 +1 +UAV-aided Metaverse over Wireless +Communications: A Reinforcement Learning +Approach +Peiyuan Si1, Wenhan Yu1, Jun Zhao1, Kwok-Yan Lam1, Qing Yang2 +1School of Computer Science & Engineering +Nanyang Technological University, Singapore +2University of North Texas, United States +{peiyuan001, wenhan002}@e.ntu.edu.sg, {junzhao, kwokyan.lam}@ntu.edu.sg, Qing.yang@unt.edu +Abstract—Metaverse is expected to create a virtual world +closely connected with reality to provide users with immersive +experience with the support of 5G high data rate communication +technique. A huge amount of data in physical world needs to be +synchronized to the virtual world to provide immersive experi- +ence for users, and there will be higher requirements on coverage +to include more users into Metaverse. However, 5G signal suffers +severe attenuation, which makes it more expensive to maintain +the same coverage. Unmanned aerial vehicle (UAV) is a promising +candidate technique for future implementation of Metaverse +as a low-cost and high-mobility platform for communication +devices. In this paper, we propose a proximal policy optimization +(PPO) based double-agent cooperative reinforcement learning +method for channel allocation and trajectory control of UAV +to collect and synchronize data from the physical world to the +virtual world, and expand the coverage of Metaverse services +economically. Simulation results show that our proposed method +is able to achieve better performance compared to the benchmark +approaches. +Index +Terms—Metaverse, +UAV, +cooperative +reinforcement +learning, PPO +I. INTRODUCTION +The proposal of Metaverse has been promoted by the im- +plementation of 5G communication technology and maturing +AR/VR devices in recent years [1]–[4]. Metaverse aims to +create a virtual world for all kinds of activities, including +education, trading and gaming, and is considered the next +generation of the Internet [?], [5], [7], [8]. With the support of +AR/VR applications, online users are provided with immersive +services that are similar to in-person activities, and the trading +of virtual items brings job opportunities. +To support the Metaverse applications, data synchronization +and wide wireless network coverage are two practical prob- +lems to be solved as the Metaverse services usually involve +wearable wireless devices. For the first problem, 5G commu- +nication technology is able to provide high-speed and low- +latency data transmission, but it is not necessary to update all +the collected data immediately, e.g., environment information +to build the background of Metaverse and offline trading +records [9]–[11]. For the second problem, 5G network suffers +higher costs for the same coverage area due to severe signal +attenuation. Thus, it is not economically efficient to deploy +base stations in suburban with low population density, and in +wild areas it is not even applicable to traditional base stations +[12]. +Unmanned aerial vehicle (UAV) is a cheaper substitution +solution to set up network coverage for Metaverse data syn- +chronization in the suburban area due to its ability to carry +communication devices. The UAV technique has been fully +studied and commercialized, and there are numerous works +on UAV-based communication scenarios for traditional appli- +cations, e.g., research on communication resource allocation, +UAV trajectory control and the internet of vehicles [13]– +[15]. The UAV-based optimization problems which take the +trajectory of UAV into consideration usually segment the flight +time of UAV into discrete time slots for the convenience of +computation. The resource allocation variables need to be +optimized in each time slot to obtain the global or local +optimal. Although these methods ensure the convergence of +the solution, the increasing number of time slots results to +the increment of algorithm complexity. Besides, the integer +characteristic of channel allocation variables results to mixed +integer programming problems, which can be hard to solve if +the variables are inseparable. +Related Work. In some cases, reinforcement learning (RL) +is more suitable for UAV-based optimization problems than +convex methods because it gives a feasible solution with +relatively good performance even if the global optimal is +extremely hard to find, and it can handle time-sequential +problems without increasing the number of variables. Cui et +al. [16] proposed multi-agent reinforcement learning resource +allocation algorithm for multi-UAV networks, and showed fast +convergence with the basic Q-learning algorithm. Luong et +al. [17] utilized the deep Q-learning algorithm to learn the +network state for the decision of the movement of UAV, and +improved the network performance by up to 70%. Rodriguez- +Ramos et al. [18] implemented a versatile Gazebo-based rein- +forcement learning framework for UAV landing on a moving +platform, which is a novel experiment of DDPG on UAV +controlling research. +For communication optimization problems with discrete +channels and continuous resource allocation, both discrete and +continuous action spaces need to be considered. To solve + +2 +discrete-continuous hybrid action space reinforcement learning +problems, multi-agent architecture is commonly adopted. Fu +et al. [19] proposed two multi-agent reinforcement learning +architectures for hybrid action spaces based on deep Q- +learning (DQN), where agents work in a parallel manner to +generate joint actions. Jiang et al. [20] designed a hybrid action +algorithm for massive access control, which optimized the +discrete action selection for back-off and distributed queuing +problems and generate continuous action for access class +barring. +The agents of most existing hybrid action space reinforce- +ment learning algorithms work in a parallel manner, which +does not build the inter-agent relationship. In this paper, +we propose a hybrid reinforcement learning architecture to +optimize the discrete channel allocation variable and the +continuous trajectory controlling variable. Two agents work +in a sequential manner motivated by the alternative optimiza- +tion algorithms, i.e., the output of an agent is the input of +another agent. Compared to the existing works, our paper +considers the inter-agent relationship for better convergence +performance. The advantage of our scenario over traditional +convex optimization is that the number of variables does not +increase when the number of time slots increases, which is +more friendly to time-sequential problems. +Contribution. The contributions of this paper are as fol- +lows: +• A PPO-based double-agent cooperative hybrid action +reinforcement learning architecture (PPO-PPO) for UAV- +enabled Metaverse data synchronization is proposed. +• Proximal policy optimization (PPO) algorithm is imple- +mented in both discrete action agents and continuous ac- +tion agents, and two agents work in a sequential manner. +• The simulation shows the comparison between the pro- +posed algorithm and two baselines (DQN and duelling +DQN), which verifies the advantage of our proposed +PPO-PPO algorithm. +The rest of this paper is organized as follows. Section +II introduces the proposed system model. The double-agent +policy generation model and its implementation are presented +in Section III and Section IV, respectively. Section V shows +the simulation results and the corresponding explanation. The +conclusion of this paper is discussed in Section VI. +II. SYSTEM MODEL +As shown in Fig. 1, we consider a UAV-based uplink data +collection system for Metaverse service. In a given L × L +area which is beyond the coverage of 5G base station, N +Metaverse data collectors (MDCs) are deployed to collect +delay-insensitive local data, such as offline digital currency +trading and weather information, which are generated by +Metaverse users or the sensors [21], [22]. The location of +MDC n is denoted by (xn, yn, 0). MDCs are assumed to have +enough energy but limited transmission power. +To synchronize the local data with the Metaverse server, +one mobile base station (MBS) carried by UAV is deployed +to collect the local data saved at MDCs through M channels. +Each MDC can occupy only one channel, but multiple MDCs +are able to share one channel. The set of MDCs in channel m is +denoted by Nm, and the number of MDCs in the set is denoted +as Nm. We assume that the UAV flies at a fixed height H, and +the location of UAV is denoted by (xuav[t], yuav[t], H). Once +the data is received by the MBS, MDCs clear the historical +data and get ready for the future data collection. In this paper, +we assume that the local data size of each receiver is U. +A. Channel Settings +According to the experimental characterization of the +vehicle-to-infrastructure radio channels in suburban environ- +ments implemented by M. Yusuf et al, the small-scale fading +of the strongest path is found to be Rician distributed [23]. +The channel gain between UAV and MDC n in channel m +and time slot t is given by [24] +hn,m[t] = +� +βn[t]gn,m[t], +(1) +where βn[t] denotes the large-scale average channel gain +at time slot t, and gn,m[t] denotes the small-scale fading +coefficient, which is modelled as Rician fading. βn[t] and +gn,m[t] are given by +βn[t] = β0d−α +n [t], +(2) +and +gn,m[t] = +� +K +K + 1g + +� +1 +K + 1 ˜g, +(3) +where β0 denotes the channel gain at the reference distance +d0 = 1m, α denotes the path loss exponent, which varies from +2 to 6 (in this paper we assume that α = 2). g denotes the +deterministic LoS channel component with |g| = 1, which +denotes the randomly scattered component. The Rician factor +is denoted by K. dn[t] denotes the distance from UAV to MDC +n in time slot t, which is given by +dn[t] = +� +(xn − xuav[t])2 + (yn − yuav[t])2 + H2. +(4) +The channel-to-noise-ratio (CNR) is given by +Γn,m[t] = hn,m[t] +Bσ2 +(5) +where σ2 denotes the power of additive white Gaussian noise +(AWGN) at the receiver. The signal to interference plus noise +ratio (SINR) of MDC n in channel m in time slot t is given +by +γn,m[t] = +pn,m[t]Γn,m[t] +1 + +|Nm|−1 +� +i=1 +pi,m[t]Γi,m[t] +, +(6) +where pn,m denotes the transmission power of MDCs. Thus, +the transmission rate of MDC n in channel m and time slot t +is given by +Rn,m[t] = Blog2(1 + γ). +(7) + +3 +Fig. 1: System model. +Channel Allocation +Trajectory Control +Environment +UAV +MDR +MDR +Discrete PPO +Continuous PPO +Combined Action +Reward +uav +uav +ˆ +{ [ ], +[ +1], +[ +1]} +I t +x +t +y +t +� +� +Critic +ch +ta +Actor +BP +Critic +ch +ta +Actor +BP +Critic +ch +ta +Actor +BP +Critic +ch +ta +Actor +BP +ch +ta +Fig. 2: Double-agent policy generation model. +III. DOUBLE-AGENT POLICY GENERATION MODEL +In this section, we introduce the double-agent policy gener- +ation model based on PPO (PPO-PPO) for channel allocation +and UAV trajectory control, which is shown in Fig. 2. +The objective is to minimize the total required time for UAV +to finish collecting the data saved at MDCs with the constraint +of maximum UAV speed by optimizing channel allocation +indicator matrix I[t], and UAV trajectory {xuav[t], yuav[t]}. +Each agent only focuses on a specific type of variable, and +the values of other variables are loaded from the results +of another agent in the previous step. In each step, the +discrete proximal policy optimization (PPO) agent generates +the channel allocation according to its policy, and forwards the +result to the continuous PPO agent for trajectory generation. +The combined action is generated by concatenating the output +of two RL agents which interact with the environment to get +reward for both RL agents. +A. Discrete Agent for Channel Allocation +In this subsection, we will introduce the action space, state +space and reward settings of the discrete agent for channel +allocation. +1) Action of the Discrete Agent: Intuitively, the channel +allocation indicator I[t] can be defined as an one-hot matrix, +i.e., In,m[t] ∈ {0, 1} denotes if channel m is selected by MDC +n. An example with the number of users N = 4 and number +of channels M = 3 is given by +I[t] = + + +I1,1[t] +I1,2[t] +I1,3[t] +I2,1[t] +I2,2[t] +I2,3[t] +I3,1[t] +I3,2[t] +I3,3[t] +I4,1[t] +I4,2[t] +I4,3[t] + + , +(8) +whose dimension is N × M. The one-hot definition of I[t] is +intuitive but increases the dimension of action space. To reduce +the dimension, we re-define the channel allocation indicator +matrix as ˆI[t], whose elements are ˆIn[t] ∈ {0, 1, .., M}. Under +this definition, ˆIn[t] = m indicates that MDC n is assigned +with channel m, and ˆIn[t] = 0 indicates that it is not assigned +with any channel. +ˆ[ ] +I t +1ˆ [ ] +I t +2ˆ [ ] +I t +3ˆ [ ] +I t +4ˆ [ ] +I t +5ˆ [ ] +I t +6ˆ [ ] +I t +Action +� +� +� +0 +M +1 +� +� +� +� +� +1 +M +1 +� +� +� +� +� +� +� +� +� +� +� +2 +M +1 +� +� +� +3 +M +1 +� +� +� +4 +M +1 +� +� +� +5 +M +1 +� +Fig. 3: Action encoding. +As shown in Fig. 3, the action of the agent is encoded +according to the channel allocation indicator matrix. The +encoded action is given by +ach +t = +N +� +n=1 +ˆIn[t](M + 1)n−1 +(9) +2) State of the Discrete Agent: The decisions of RL agents +are generated based on the current state. In this paper, the +state of the discrete agent includes the channel gain and the +remaining data at MDCs in the current step. The state of the +discrete agent is concatenated by two parts, which is given by +Sch +t = {Ures[t], h[t]}, +(10) +where Ures denotes the matrix of remaining data in MDCs, +and h[t] denotes the matrix of channel gain at tth step. +3) Reward of the Discrete Agent: The optimization objec- +tive in this paper is the required time for UAV to finish the +data collection mission, i.e., to minimize the number of steps +in each episode. Intuitively, the more steps the agent takes, the + +181(0)UAV +MDCT +BS4 +less reward it should receive. Thus, we set a time-based penalty +rtime +t +with negative value in each step to build the connection +between reward and our objective. If the agent fails to finish +the mission in given time limit Tmax, it will receive a failure +penalty rfail +t . +The time-based penalty rtime +t +is further modified according +to the data size collected by UAV in the current step to give +higher reward to the actions which result to larger transmission +rate. The reward of the discrete agent is given by +rch +t = + + + + + +rtime +t +U +N +� +n=1 +M +� +m=1 +tslotRn,m[t], if t ≤ Tmax +rfail +t , if t > Tmax +(11) +B. Continuous Agent for Trajectory Optimization +The trajectory of UAV is optimized by a continuous RL +agent, whose action, state and reward are defined as follows. +Mission Area +slot +max +2t +V +slot +max +2t +V +Action +Space (t+1) +Traj +ta +Action +Space (t) +Fig. 4: Action space of the continuous agent. +1) Action of the continuous agent: As shown in Fig. 4, the +action of the continuous agent atraj +t +determines the location of +UAV in the next step. atraj +t +is defined as +atraj +t += {ax +t , ay +t }, ax +t , ay +t ∈ [−tslotVmax, tslotVmax] , +(12) +where ax +t , ay +t denote the movement of UAV on the x-axis and +y-axis respectively. +2) State of the Continuous Agent: The state of the con- +tinuous agent is similar to the discrete agent, which includes +the current channel gain h[t] and remaining data at MDCs +Ures[t]. In addition, the current horizontal location of UAV +(xuav[t], yuav[t]) is also included in the state Straj +t , which is +given by +Straj +t += {Ures[t], h[t], (xuav[t], yuav[t])} +(13) +3) Reward of the Continuous Agent: The reward of the +continuous agent is modified based on rch +t . We give additional +penalty to the agent if the location of UAV exceeds reasonable +region to regularize the trajectory decision. The reward of the +continuous agent is given by +rtraj +t += +� +rch +t , if xuav[t] ∈ [xmin, xmax], yuav[t] ∈ [ymin, ymax] +rch +t + rpenalty +t +, otherwise +(14) +IV. IMPLEMENTATION OF PROXIMAL POLICY +OPTIMIZATION (PPO) +PPO is a state-of-art on policy reinforcement learning al- +gorithm which supports both discrete and continuous actions +spaces. In this section, we introduce the preliminary and +implementation of PPO algorithm for discrete agent (channel +allocation) and continuous agent (UAV trajectory optimiza- +tion). +A. Implementation of Continuous and Discrete PPO +1) Critic Network: The critic network is responsible to +give scores to the actor according to the current state. The +architectures of both discrete and continuous critic networks +are the same, which consists of multiple fully connected layers. +Loss function of continuous and discrete critic networks +are given by +Jtraj(φ) = +� +V traj +φ (straj +t ) − +� +rtraj +t ++ γV traj +φ′ (straj +t+1) +��2 +, +(15) +Jch(φ) = +� +V ch +φ (sch +t ) − +� +rch +t + γV ch +φ′ (sch +t+1) +��2, +(16) +where Ltraj +t (φ) and Lch +t (φ) denote the loss function for the +critic network of continuous and discrete agent respectively. +V traj +φ′ (straj +t+1) and V ch +φ′ (sch +t+1) are the state value estimations +generated by the old critic networks φ +′ +traj and φ +′ +ch respectively, +which are saved in during the interaction with environment. +V traj +φ (straj +t ) and V ch +φ (sch +t ) are the state value estimations gener- +ated by the current critic networks φtraj and φch , which are +updated in each training iteration. +2) Actor Network: As shown in Fig. 5, the architecture of +discrete and continuous actor network are different due to the +difference in action space. +The continuous actor network for trajectory control is a +network for value approximation, which outputs a µ head and +a σ head which denotes the mean and variance of Gaussian dis- +tributions respectively. Each head includes two variables, i.e., +{µx, µy} and {σx, σy}, which denotes the x-axis and y-axis +respectively. The action {ax[t], ay[t]} is generated by sampling +from the obtained distribution N(µx, σ2 +x) and N(µy, σ2 +y). +The discrete actor network for channel allocation is a +network for classification, which outputs the probabilities +Pr(a) of each action. The agent sample its action from the +obtained action probabilities with ε-greedy, i.e., the output +action is generated by sampling from Pr(a) with probability +1−ǫ, and selected randomly with probability ǫ for exploration. +The output action of the discrete actor network is encoded, +which will be decoded into one-hot indicators before being +utilized for further calculation. +Loss functions of actor networks in our implementation +adopt the trick of clipping to simplify the calculation, which +is proposed by J. Schulman et al [25]. +The PPO-PPO algorithm is summarized in Algorithm 1. + +(0)5 +Fully Conneted Layers +Softmax Layer +State +Head +� Head +� +Head +� Head +� +Trajectory +2 +( , +) +N � � +Sample From +2 +( , +) +N � � +Sample From +Fully Conneted Layers +Softmax Layer +State +Channel Allocation +Pr( ) +a +Action Probability +� +Sample with -Greedy +Fig. 5: Actor Network Architecture. +Algorithm 1 PPO-PPO +Initiate: Remaining data at MDCs, UAV location, network +parameters of discrete and continuous agent +1: for iteration t = 1, 2, .. do +2: +Discrete agent execute action according to the current +state and policy πch +θ′ +� +ach +t |sch +t +� +to obtain the channel +allocation indicator matrix ˆI[t] +3: +With given ˆI[t], the continuous agent for trajectory +control execute action according to the current state and +policy πtraj +θ′ +� +atraj +t |straj +t +� +ˆI[t] +4: +Agent interact with environment to get reward rch +t and +rtraj +t +for discrete agent and continuous agent respectively +5: +Update state straj +t +← straj +t+1, sch +t ← sch +t+1 +6: +Save +trajectory +� +sch +t , ach +t , rch +t , sch +t+1, V ch +φ′ (sch +t ) +� +and +� +straj +t , atraj +t , rtraj +t , straj +t , V traj +φ′ (straj +t ) +� +7: +for every i iterations do +8: +Shuffle data order and make batch with size bs. +9: +for j=0, 1, ..., T +bs − 1 do +10: +Calculate loss functions of critic and actor net- +works and update network parameters by gradient +ascent +11: +end for +12: +end for +13: end for +V. SIMULATION RESULTS +The performance of our proposed double-agent reinforce- +ment learning approach for Metaverse data collecting is tested +and compared with two benchmark scenarios (DQN-PPO and +duelling DQN-PPO), whose discrete agents are replaced with +DQN or duelling DQN algorithm respectively. The simulation +settings are given in Table I. +Fig. 6 presents the required time to complete data collecting +mission of our proposed algorithm and two benchmark algo- +rithms with given data size U = 50Mb. At the beginning of +the training process (0-1000 episodes), all three algorithms +TABLE I: Constant Parameter Setting +Parameter and Physical Meaning +Value +Number of channels(M) +3 +Default number of users (N) +5 +Bandwidth (B) +5MHz +Transmission power of MDCs +5W +Frequency (f) +28GHz (5G spectrum) +Power of Gaussian noise (σ2) +5 × 10−8W +Maximum speed of UAV +10m/s +Mission area size (L) +200m +are unstable because the reasonable policy has not been +established, and the agents are exploring the environment fre- +quently. From 1000 episodes to 2000 episodes, our proposed +PPO-PPO algorithm shows the tendency of convergence while +the benchmark DQN-PPO algorithm is still very unstable. +The duelling DQN-PPO algorithm also starts to finish the +mission within a shorter time period, but is less stable than +the PPO-PPO algorithm. DQN-PPO algorithm shows poor +convergence performance in this task, but both PPO-PPO and +duelling DQN-PPO algorithms are able to converge within +5000 episodes with similar performance due to their common +implementation of the advantage function. +0 +500 +1000 +1500 +2000 +2500 +3000 +3500 +4000 +4500 +5000 +Episodes +30 +35 +40 +45 +50 +55 +60 +Time (s) +DQN-PPO +Duelling DQN-PPO +PPO-PPO +Fig. 6: Comparison of required time to finish mission with +data size U = 50Mb. +Fig. 7 presents the mission completing time experiment with +a similar parameter setting as in Fig. 6, but the data size is +increased to U = 100Mb. All three algorithms need more +time to finish the data collecting mission due to larger data +size, and the PPO-PPO algorithm shows similar convergence +performance as in Fig. 6. However, the dueling DQN-PPO +algorithm becomes unstable in the training process, i.e., some +sudden increase in the required time. The superior stability +of PPO over dueling DQN can be attributed to its policy +update constraint by equipping it with a KL-divergence penalty +between the old policy (the policy for sampling data) and the +updated policy (the policy used for training and evaluating). +Fig. 8 and Fig. 9 are the corresponding rewards in the train- +ing processes of Fig. 6 and Fig. 7 respectively. We consider +the reward given to the agent as guidance but not the exact +objective function in the implementation of reinforcement + +6 +0 +500 +1000 +1500 +2000 +2500 +3000 +3500 +4000 +4500 +5000 +Episodes +60 +70 +80 +90 +100 +110 +120 +130 +140 +150 +160 +Time (s) +DQN-PPO +Duelling DQN-PPO +PPO-PPO +Fig. 7: Comparison of required time to finish mission with +data size U = 100Mb. +learning algorithm. The tendencies of the reward and the +required time are highly similar although they are generated +from different formulas, which indicates that our reward design +successfully leads the agent to learn a better policy. +0 +500 +1000 +1500 +2000 +2500 +3000 +3500 +4000 +4500 +5000 +Episodes +-3000 +-2500 +-2000 +-1500 +-1000 +-500 +0 +Reward +DQN-PPO +Duelling DQN-PPO +PPO-PPO +Fig. 8: Comparison of reward with data size U = 50Mb. +The mission completing time comparison for the case with +eight users is shown in Fig. 10. The duelling DQN-PPO +algorithm shows similar average performance as the PPO-PPO +algorithm but less stability, i.e., the required time sometimes +jumps to extremely large values. Taking the stability into +consideration, the PPO-PPO algorithm is better than duelling +DQN-PPO algorithm in general. The DQN-PPO algorithm is +obviously not able to converge in this experiment, so we do +not consider it a candidate for our double-agent reinforcement +learning algorithm. +VI. CONCLUSION +In this paper, we propose a double-agent reinforcement ar- +chitecture for data collecting and synchronization in Metavese, +and adopt PPO algorithm for both discrete and continuous +agents. Two agents with different action space and state space +work in a cascade manner for channel allocation and UAV +0 +500 +1000 +1500 +2000 +2500 +3000 +3500 +4000 +4500 +5000 +Episodes +-3000 +-2500 +-2000 +-1500 +-1000 +-500 +Reward +DQN-PPO +Duelling DQN-PPO +PPO-PPO +Fig. 9: Comparison of reward with data size U = 100Mb. +0 +500 +1000 +1500 +2000 +2500 +3000 +3500 +4000 +4500 +5000 +Episodes +50 +100 +150 +200 +Time (s) +DQN-PPO +Duelling DQN-PPO +PPO-PPO +Fig. 10: Comparison of reward with data size U = 50Mb and +8 users. +trajectory control to form a combined action in each iteration. +Our experiments indicate the advantage of the PPO-PPO in +both the required time for the mission and the stability. In +future work, we will consider transmission power allocation +and test the performance of other state-of-art reinforcement +learning algorithms in our proposed architecture. +REFERENCES +[1] Y. Wang and J. Zhao, “Mobile Edge Computing, Metaverse, 6G Wireless +Communications, Artificial Intelligence, and Blockchain: Survey and +Their Convergence,” arXiv preprint arXiv:2209.14147, 2022. +[2] L.-H. Lee, T. Braud, P. Zhou, L. Wang, D. Xu, Z. Lin, A. Kumar, +C. Bermejo, and P. Hui, “All one needs to know about metaverse: A +complete survey on technological singularity, virtual ecosystem, and +research agenda,” arXiv preprint arXiv:2110.05352, 2021. +[3] P. Si, J. Zhao, H. Han, K.-Y. Lam, and Y. Liu, “Resource Allocation and +Resolution Control in the Metaverse with Mobile Augmented Reality,” +arXiv preprint arXiv:2209.13871, 2022. +[4] T. J. Chua, W. Yu, and J. Zhao, “Resource allocation for mobile +metaverse with the Internet of Vehicles over 6G wireless commu- +nications: A deep reinforcement learning approach,” arXiv preprint +arXiv:2209.13425, 2022. +[5] H.-C. Han et al., “From visual culture in the immersive metaverse to +visual cognition in education,” in Cognitive and Affective Perspectives +on Immersive Technology in Education. IGI Global, 2020, pp. 67–84. +[6] J.-E. Yu, “Exploration of educational possibilities by four metaverse +types in physical education,” Technologies, vol. 10, no. 5, p. 104, 2022. + +7 +[7] Murat Yilmaz, Tuna Hacalo˘glu and Paul Clarke , “Examining the use of +non-fungible tokens (NFTs) as a trading mechanism for the metaverse,” +in European Conference on Software Process Improvement. Springer, +2022, pp. 18–28. +[8] A. Jungherr and D. B. Schlarb, “The Extended Reach of Game Engine +Companies: How Companies Like Epic Games and Unity Technologies +Provide Platforms for Extended Reality Applications and the Metaverse,” +Social Media+ Society, vol. 8, no. 2, 2022. +[9] M. Agiwal, A. Roy, and N. Saxena, “Next generation 5G wireless +networks: A comprehensive survey,” IEEE Communications Surveys & +Tutorials, vol. 18, no. 3, pp. 1617–1655, 2016. +[10] J. Refonaa, G. G. Sebastian, D. Ramanan, and M. Lakshmi, “Effective +identification of black money and fake currency using NFC, IoT and an- +droid,” in 2018 International Conference on Communication, Computing +and Internet of Things (IC3IoT). IEEE, 2018, pp. 275–278. +[11] J. Kohli and A. Kohli, “A Study of Preference for Online Trading +Account versus Offline Trading Account by Different Age Group,” From +the Desk of Editor, p. 42, 2020. +[12] P. V. Mane-Deshmukh, “Designing of Wireless Sensor Network to +Protect Agricultural Farm from Wild Animals,” i-Manager’s Journal on +Information Technology, vol. 7, no. 4, p. 30, 2018. +[13] Q. Wu, Y. Zeng, and R. Zhang, “Joint trajectory and communication +design for multi-UAV enabled wireless networks,” IEEE Transactions +on Wireless Communications, vol. 17, no. 3, pp. 2109–2121, 2018. +[14] W. Lu, P. Si, Y. Gao, H. Han, Z. Liu, Y. Wu, and Y. Gong, “Trajectory +and Resource Optimization in OFDM-Based UAV-Powered IoT Net- +work,” IEEE Transactions on Green Communications and Networking, +vol. 5, no. 3, pp. 1259–1270, 2021. +[15] J. S. Ng, W. Y. B. Lim, H.-N. Dai, Z. Xiong, J. Huang, D. Niyato, X.-S. +Hua, C. Leung, and C. Miao, “Joint auction-coalition formation frame- +work for communication-efficient federated learning in UAV-enabled +internet of vehicles,” IEEE Transactions on Intelligent Transportation +Systems, vol. 22, no. 4, pp. 2326–2344, 2020. +[16] J. Cui, Y. Liu, and A. Nallanathan, “Multi-agent reinforcement learning- +based resource allocation for UAV networks,” IEEE Transactions on +Wireless Communications, vol. 19, no. 2, pp. 729–743, 2019. +[17] P. Luong, F. Gagnon, L.-N. Tran, and F. Labeau, “Deep reinforcement +learning-based resource allocation in cooperative UAV-assisted wireless +networks,” IEEE Transactions on Wireless Communications, vol. 20, no. +11, pp. 7610–7625, 2021. +[18] A. Rodriguez-Ramos, C. Sampedro, H. Bavle, P. De La Puente, and P. +Campoy, “A deep reinforcement learning strategy for UAV autonomous +landing on a moving platform,” Journal of Intelligent & Robotic Sys- +tems, vol. 93, no. 1, pp. 351–366, 2019. +[19] H. Fu, H. Tang, J. Hao, Z. Lei, Y. Chen, and C. Fan, “Deep multi-agent +reinforcement learning with discrete-continuous hybrid action spaces,” +arXiv preprint arXiv:1903.04959, 2019. +[20] N. Jiang, Y. Deng, and A. Nallanathan, “Deep reinforcement learning +for discrete and continuous massive access control optimization,” in ICC +2020-2020 IEEE International Conference on Communications (ICC). +IEEE, 2020, pp. 1–7. +[21] Y. Chu, J. Lee, S. Kim, H. Kim, Y. Yoon, and H. Chung, “Review of Of- +fline Payment Function of CBDC Considering Security Requirements,” +Applied Sciences, vol. 12, no. 9, p. 4488, 2022. +[22] M. Christodorescu, W. C. Gu, R. Kumaresan, M. Minaei, M. Ozdayi, +B. Price, S. Raghuraman, M. Saad, C. Sheffield, M. Xu et al., “Towards +a two-tier hierarchical infrastructure: An offline payment system for +Central Bank digital currencies,” arXiv preprint arXiv:2012.08003, 2020. +[23] M. Yusuf, E. Tanghe, F. Challita et al., “Experimental characterization of +V2I radio channel in a suburban environment,” in 2019 13th European +Conference on Antennas and Propagation (EuCAP). IEEE, 2019, pp. +1–5. +[24] C. You and R. Zhang, “3D trajectory optimization in Rician fading for +UAV-enabled data harvesting,” IEEE Transactions on Wireless Commu- +nications, vol. 18, no. 6, pp. 3192–3207, 2019. +[25] J. Schulman, F. Wolski, P. Dhariwal, A. Radford, and O. Klimov, “Prox- +imal policy optimization algorithms,” arXiv preprint arXiv:1707.06347, +2017. + diff --git a/9NAzT4oBgHgl3EQfgvyq/content/tmp_files/load_file.txt b/9NAzT4oBgHgl3EQfgvyq/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..775515de7c55fff3ff1595de052892ead584d839 --- /dev/null +++ b/9NAzT4oBgHgl3EQfgvyq/content/tmp_files/load_file.txt @@ -0,0 +1,419 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf,len=418 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content='01474v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content='SY] 4 Jan 2023 1 UAV-aided Metaverse over Wireless Communications: A Reinforcement Learning Approach Peiyuan Si1, Wenhan Yu1, Jun Zhao1, Kwok-Yan Lam1, Qing Yang2 1School of Computer Science & Engineering Nanyang Technological University, Singapore 2University of North Texas, United States {peiyuan001, wenhan002}@e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content='ntu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content='sg, {junzhao, kwokyan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content='lam}@ntu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content='sg, Qing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content='yang@unt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content='edu Abstract—Metaverse is expected to create a virtual world closely connected with reality to provide users with immersive experience with the support of 5G high data rate communication technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' A huge amount of data in physical world needs to be synchronized to the virtual world to provide immersive experi- ence for users, and there will be higher requirements on coverage to include more users into Metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' However, 5G signal suffers severe attenuation, which makes it more expensive to maintain the same coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Unmanned aerial vehicle (UAV) is a promising candidate technique for future implementation of Metaverse as a low-cost and high-mobility platform for communication devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' In this paper, we propose a proximal policy optimization (PPO) based double-agent cooperative reinforcement learning method for channel allocation and trajectory control of UAV to collect and synchronize data from the physical world to the virtual world, and expand the coverage of Metaverse services economically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Simulation results show that our proposed method is able to achieve better performance compared to the benchmark approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Index Terms—Metaverse, UAV, cooperative reinforcement learning, PPO I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' INTRODUCTION The proposal of Metaverse has been promoted by the im- plementation of 5G communication technology and maturing AR/VR devices in recent years [1]–[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Metaverse aims to create a virtual world for all kinds of activities, including education, trading and gaming, and is considered the next generation of the Internet [?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' ], [5], [7], [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' With the support of AR/VR applications, online users are provided with immersive services that are similar to in-person activities, and the trading of virtual items brings job opportunities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' To support the Metaverse applications, data synchronization and wide wireless network coverage are two practical prob- lems to be solved as the Metaverse services usually involve wearable wireless devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' For the first problem, 5G commu- nication technology is able to provide high-speed and low- latency data transmission, but it is not necessary to update all the collected data immediately, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=', environment information to build the background of Metaverse and offline trading records [9]–[11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' For the second problem, 5G network suffers higher costs for the same coverage area due to severe signal attenuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Thus, it is not economically efficient to deploy base stations in suburban with low population density, and in wild areas it is not even applicable to traditional base stations [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Unmanned aerial vehicle (UAV) is a cheaper substitution solution to set up network coverage for Metaverse data syn- chronization in the suburban area due to its ability to carry communication devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' The UAV technique has been fully studied and commercialized, and there are numerous works on UAV-based communication scenarios for traditional appli- cations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=', research on communication resource allocation, UAV trajectory control and the internet of vehicles [13]– [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' The UAV-based optimization problems which take the trajectory of UAV into consideration usually segment the flight time of UAV into discrete time slots for the convenience of computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' The resource allocation variables need to be optimized in each time slot to obtain the global or local optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Although these methods ensure the convergence of the solution, the increasing number of time slots results to the increment of algorithm complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Besides, the integer characteristic of channel allocation variables results to mixed integer programming problems, which can be hard to solve if the variables are inseparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Related Work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' In some cases, reinforcement learning (RL) is more suitable for UAV-based optimization problems than convex methods because it gives a feasible solution with relatively good performance even if the global optimal is extremely hard to find, and it can handle time-sequential problems without increasing the number of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Cui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' [16] proposed multi-agent reinforcement learning resource allocation algorithm for multi-UAV networks, and showed fast convergence with the basic Q-learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Luong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' [17] utilized the deep Q-learning algorithm to learn the network state for the decision of the movement of UAV, and improved the network performance by up to 70%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Rodriguez- Ramos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' [18] implemented a versatile Gazebo-based rein- forcement learning framework for UAV landing on a moving platform, which is a novel experiment of DDPG on UAV controlling research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' For communication optimization problems with discrete channels and continuous resource allocation, both discrete and continuous action spaces need to be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' To solve 2 discrete-continuous hybrid action space reinforcement learning problems, multi-agent architecture is commonly adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' [19] proposed two multi-agent reinforcement learning architectures for hybrid action spaces based on deep Q- learning (DQN), where agents work in a parallel manner to generate joint actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' [20] designed a hybrid action algorithm for massive access control, which optimized the discrete action selection for back-off and distributed queuing problems and generate continuous action for access class barring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' The agents of most existing hybrid action space reinforce- ment learning algorithms work in a parallel manner, which does not build the inter-agent relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' In this paper, we propose a hybrid reinforcement learning architecture to optimize the discrete channel allocation variable and the continuous trajectory controlling variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Two agents work in a sequential manner motivated by the alternative optimiza- tion algorithms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=', the output of an agent is the input of another agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Compared to the existing works, our paper considers the inter-agent relationship for better convergence performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' The advantage of our scenario over traditional convex optimization is that the number of variables does not increase when the number of time slots increases, which is more friendly to time-sequential problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' The contributions of this paper are as fol- lows: A PPO-based double-agent cooperative hybrid action reinforcement learning architecture (PPO-PPO) for UAV- enabled Metaverse data synchronization is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Proximal policy optimization (PPO) algorithm is imple- mented in both discrete action agents and continuous ac- tion agents, and two agents work in a sequential manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' The simulation shows the comparison between the pro- posed algorithm and two baselines (DQN and duelling DQN), which verifies the advantage of our proposed PPO-PPO algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Section II introduces the proposed system model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' The double-agent policy generation model and its implementation are presented in Section III and Section IV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Section V shows the simulation results and the corresponding explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' The conclusion of this paper is discussed in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' SYSTEM MODEL As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 1, we consider a UAV-based uplink data collection system for Metaverse service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' In a given L × L area which is beyond the coverage of 5G base station, N Metaverse data collectors (MDCs) are deployed to collect delay-insensitive local data, such as offline digital currency trading and weather information, which are generated by Metaverse users or the sensors [21], [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' The location of MDC n is denoted by (xn, yn, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' MDCs are assumed to have enough energy but limited transmission power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' To synchronize the local data with the Metaverse server, one mobile base station (MBS) carried by UAV is deployed to collect the local data saved at MDCs through M channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Each MDC can occupy only one channel, but multiple MDCs are able to share one channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' The set of MDCs in channel m is denoted by Nm, and the number of MDCs in the set is denoted as Nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' We assume that the UAV flies at a fixed height H, and the location of UAV is denoted by (xuav[t], yuav[t], H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Once the data is received by the MBS, MDCs clear the historical data and get ready for the future data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' In this paper, we assume that the local data size of each receiver is U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Channel Settings According to the experimental characterization of the vehicle-to-infrastructure radio channels in suburban environ- ments implemented by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Yusuf et al, the small-scale fading of the strongest path is found to be Rician distributed [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' The channel gain between UAV and MDC n in channel m and time slot t is given by [24] hn,m[t] = � βn[t]gn,m[t], (1) where βn[t] denotes the large-scale average channel gain at time slot t, and gn,m[t] denotes the small-scale fading coefficient, which is modelled as Rician fading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' βn[t] and gn,m[t] are given by βn[t] = β0d−α n [t], (2) and gn,m[t] = � K K + 1g + � 1 K + 1 ˜g, (3) where β0 denotes the channel gain at the reference distance d0 = 1m, α denotes the path loss exponent, which varies from 2 to 6 (in this paper we assume that α = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' g denotes the deterministic LoS channel component with |g| = 1, which denotes the randomly scattered component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' The Rician factor is denoted by K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' dn[t] denotes the distance from UAV to MDC n in time slot t, which is given by dn[t] = � (xn − xuav[t])2 + (yn − yuav[t])2 + H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' (4) The channel-to-noise-ratio (CNR) is given by Γn,m[t] = hn,m[t] Bσ2 (5) where σ2 denotes the power of additive white Gaussian noise (AWGN) at the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' The signal to interference plus noise ratio (SINR) of MDC n in channel m in time slot t is given by γn,m[t] = pn,m[t]Γn,m[t] 1 + |Nm|−1 � i=1 pi,m[t]Γi,m[t] , (6) where pn,m denotes the transmission power of MDCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Thus, the transmission rate of MDC n in channel m and time slot t is given by Rn,m[t] = Blog2(1 + γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' (7) 3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 1: System model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Channel Allocation Trajectory Control Environment UAV MDR MDR Discrete PPO Continuous PPO Combined Action Reward uav uav ˆ { [ ], [ 1], [ 1]} I t x t y t � � Critic ch ta Actor BP Critic ch ta Actor BP Critic ch ta Actor BP Critic ch ta Actor BP ch ta Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 2: Double-agent policy generation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' DOUBLE-AGENT POLICY GENERATION MODEL In this section, we introduce the double-agent policy gener- ation model based on PPO (PPO-PPO) for channel allocation and UAV trajectory control, which is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' The objective is to minimize the total required time for UAV to finish collecting the data saved at MDCs with the constraint of maximum UAV speed by optimizing channel allocation indicator matrix I[t], and UAV trajectory {xuav[t], yuav[t]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Each agent only focuses on a specific type of variable, and the values of other variables are loaded from the results of another agent in the previous step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' In each step, the discrete proximal policy optimization (PPO) agent generates the channel allocation according to its policy, and forwards the result to the continuous PPO agent for trajectory generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' The combined action is generated by concatenating the output of two RL agents which interact with the environment to get reward for both RL agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Discrete Agent for Channel Allocation In this subsection, we will introduce the action space, state space and reward settings of the discrete agent for channel allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 1) Action of the Discrete Agent: Intuitively, the channel allocation indicator I[t] can be defined as an one-hot matrix, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=', In,m[t] ∈ {0, 1} denotes if channel m is selected by MDC n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' An example with the number of users N = 4 and number of channels M = 3 is given by I[t] = \uf8ee \uf8ef\uf8ef\uf8f0 I1,1[t] I1,2[t] I1,3[t] I2,1[t] I2,2[t] I2,3[t] I3,1[t] I3,2[t] I3,3[t] I4,1[t] I4,2[t] I4,3[t] \uf8f9 \uf8fa\uf8fa\uf8fb , (8) whose dimension is N × M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' The one-hot definition of I[t] is intuitive but increases the dimension of action space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' To reduce the dimension, we re-define the channel allocation indicator matrix as ˆI[t], whose elements are ˆIn[t] ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content='., M}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Under this definition, ˆIn[t] = m indicates that MDC n is assigned with channel m, and ˆIn[t] = 0 indicates that it is not assigned with any channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' ˆ[ ] I t 1ˆ [ ] I t 2ˆ [ ] I t 3ˆ [ ] I t 4ˆ [ ] I t 5ˆ [ ] I t 6ˆ [ ] I t Action � � � 0 M 1 � � � � � 1 M 1 � � � � � � � � � � � 2 M 1 � � � 3 M 1 � � � 4 M 1 � � � 5 M 1 � Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 3: Action encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 3, the action of the agent is encoded according to the channel allocation indicator matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' The encoded action is given by ach t = N � n=1 ˆIn[t](M + 1)n−1 (9) 2) State of the Discrete Agent: The decisions of RL agents are generated based on the current state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' In this paper, the state of the discrete agent includes the channel gain and the remaining data at MDCs in the current step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' The state of the discrete agent is concatenated by two parts, which is given by Sch t = {Ures[t], h[t]}, (10) where Ures denotes the matrix of remaining data in MDCs, and h[t] denotes the matrix of channel gain at tth step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 3) Reward of the Discrete Agent: The optimization objec- tive in this paper is the required time for UAV to finish the data collection mission, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=', to minimize the number of steps in each episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Intuitively, the more steps the agent takes, the 181(0)UAV MDCT BS4 less reward it should receive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Thus, we set a time-based penalty rtime t with negative value in each step to build the connection between reward and our objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' If the agent fails to finish the mission in given time limit Tmax, it will receive a failure penalty rfail t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' The time-based penalty rtime t is further modified according to the data size collected by UAV in the current step to give higher reward to the actions which result to larger transmission rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' The reward of the discrete agent is given by rch t = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 rtime t U N � n=1 M � m=1 tslotRn,m[t], if t ≤ Tmax rfail t , if t > Tmax (11) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Continuous Agent for Trajectory Optimization The trajectory of UAV is optimized by a continuous RL agent, whose action, state and reward are defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Mission Area slot max 2t V slot max 2t V Action Space (t+1) Traj ta Action Space (t) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 4: Action space of the continuous agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 1) Action of the continuous agent: As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 4, the action of the continuous agent atraj t determines the location of UAV in the next step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' atraj t is defined as atraj t = {ax t , ay t }, ax t , ay t ∈ [−tslotVmax, tslotVmax] , (12) where ax t , ay t denote the movement of UAV on the x-axis and y-axis respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 2) State of the Continuous Agent: The state of the con- tinuous agent is similar to the discrete agent, which includes the current channel gain h[t] and remaining data at MDCs Ures[t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' In addition, the current horizontal location of UAV (xuav[t], yuav[t]) is also included in the state Straj t , which is given by Straj t = {Ures[t], h[t], (xuav[t], yuav[t])} (13) 3) Reward of the Continuous Agent: The reward of the continuous agent is modified based on rch t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' We give additional penalty to the agent if the location of UAV exceeds reasonable region to regularize the trajectory decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' The reward of the continuous agent is given by rtraj t = � rch t , if xuav[t] ∈ [xmin, xmax], yuav[t] ∈ [ymin, ymax] rch t + rpenalty t , otherwise (14) IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' IMPLEMENTATION OF PROXIMAL POLICY OPTIMIZATION (PPO) PPO is a state-of-art on policy reinforcement learning al- gorithm which supports both discrete and continuous actions spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' In this section, we introduce the preliminary and implementation of PPO algorithm for discrete agent (channel allocation) and continuous agent (UAV trajectory optimiza- tion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Implementation of Continuous and Discrete PPO 1) Critic Network: The critic network is responsible to give scores to the actor according to the current state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' The architectures of both discrete and continuous critic networks are the same, which consists of multiple fully connected layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Loss function of continuous and discrete critic networks are given by Jtraj(φ) = � V traj φ (straj t ) − � rtraj t + γV traj φ′ (straj t+1) ��2 , (15) Jch(φ) = � V ch φ (sch t ) − � rch t + γV ch φ′ (sch t+1) ��2, (16) where Ltraj t (φ) and Lch t (φ) denote the loss function for the critic network of continuous and discrete agent respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' V traj φ′ (straj t+1) and V ch φ′ (sch t+1) are the state value estimations generated by the old critic networks φ ′ traj and φ ′ ch respectively, which are saved in during the interaction with environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' V traj φ (straj t ) and V ch φ (sch t ) are the state value estimations gener- ated by the current critic networks φtraj and φch , which are updated in each training iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 2) Actor Network: As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 5, the architecture of discrete and continuous actor network are different due to the difference in action space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' The continuous actor network for trajectory control is a network for value approximation, which outputs a µ head and a σ head which denotes the mean and variance of Gaussian dis- tributions respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Each head includes two variables, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=', {µx, µy} and {σx, σy}, which denotes the x-axis and y-axis respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' The action {ax[t], ay[t]} is generated by sampling from the obtained distribution N(µx, σ2 x) and N(µy, σ2 y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' The discrete actor network for channel allocation is a network for classification, which outputs the probabilities Pr(a) of each action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' The agent sample its action from the obtained action probabilities with ε-greedy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=', the output action is generated by sampling from Pr(a) with probability 1−ǫ, and selected randomly with probability ǫ for exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' The output action of the discrete actor network is encoded, which will be decoded into one-hot indicators before being utilized for further calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Loss functions of actor networks in our implementation adopt the trick of clipping to simplify the calculation, which is proposed by J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Schulman et al [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' The PPO-PPO algorithm is summarized in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' (0)5 Fully Conneted Layers Softmax Layer State Head � Head � Head � Head � Trajectory 2 ( , ) N � � Sample From 2 ( , ) N � � Sample From Fully Conneted Layers Softmax Layer State Channel Allocation Pr( ) a Action Probability � Sample with -Greedy Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 5: Actor Network Architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Algorithm 1 PPO-PPO Initiate: Remaining data at MDCs, UAV location, network parameters of discrete and continuous agent 1: for iteration t = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' do 2: Discrete agent execute action according to the current state and policy πch θ′ � ach t |sch t � to obtain the channel allocation indicator matrix ˆI[t] 3: With given ˆI[t],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' the continuous agent for trajectory control execute action according to the current state and policy πtraj θ′ � atraj t |straj t � ˆI[t] 4: Agent interact with environment to get reward rch t and rtraj t for discrete agent and continuous agent respectively 5: Update state straj t ← straj t+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' sch t ← sch t+1 6: Save trajectory � sch t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' ach t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' rch t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' sch t+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' V ch φ′ (sch t ) � and � straj t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' atraj t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' rtraj t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' straj t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' V traj φ′ (straj t ) � 7: for every i iterations do 8: Shuffle data order and make batch with size bs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 9: for j=0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=', T bs − 1 do 10: Calculate loss functions of critic and actor net- works and update network parameters by gradient ascent 11: end for 12: end for 13: end for V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' SIMULATION RESULTS The performance of our proposed double-agent reinforce- ment learning approach for Metaverse data collecting is tested and compared with two benchmark scenarios (DQN-PPO and duelling DQN-PPO), whose discrete agents are replaced with DQN or duelling DQN algorithm respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' The simulation settings are given in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 6 presents the required time to complete data collecting mission of our proposed algorithm and two benchmark algo- rithms with given data size U = 50Mb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' At the beginning of the training process (0-1000 episodes),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' all three algorithms TABLE I: Constant Parameter Setting Parameter and Physical Meaning Value Number of channels(M) 3 Default number of users (N) 5 Bandwidth (B) 5MHz Transmission power of MDCs 5W Frequency (f) 28GHz (5G spectrum) Power of Gaussian noise (σ2) 5 × 10−8W Maximum speed of UAV 10m/s Mission area size (L) 200m are unstable because the reasonable policy has not been established,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' and the agents are exploring the environment fre- quently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' From 1000 episodes to 2000 episodes, our proposed PPO-PPO algorithm shows the tendency of convergence while the benchmark DQN-PPO algorithm is still very unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' The duelling DQN-PPO algorithm also starts to finish the mission within a shorter time period, but is less stable than the PPO-PPO algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' DQN-PPO algorithm shows poor convergence performance in this task, but both PPO-PPO and duelling DQN-PPO algorithms are able to converge within 5000 episodes with similar performance due to their common implementation of the advantage function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Episodes 30 35 40 45 50 55 60 Time (s) DQN-PPO Duelling DQN-PPO PPO-PPO Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 6: Comparison of required time to finish mission with data size U = 50Mb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 7 presents the mission completing time experiment with a similar parameter setting as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 6, but the data size is increased to U = 100Mb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' All three algorithms need more time to finish the data collecting mission due to larger data size, and the PPO-PPO algorithm shows similar convergence performance as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' However, the dueling DQN-PPO algorithm becomes unstable in the training process, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=', some sudden increase in the required time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' The superior stability of PPO over dueling DQN can be attributed to its policy update constraint by equipping it with a KL-divergence penalty between the old policy (the policy for sampling data) and the updated policy (the policy used for training and evaluating).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 8 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 9 are the corresponding rewards in the train- ing processes of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 6 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 7 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' We consider the reward given to the agent as guidance but not the exact objective function in the implementation of reinforcement 6 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Episodes 60 70 80 90 100 110 120 130 140 150 160 Time (s) DQN-PPO Duelling DQN-PPO PPO-PPO Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 7: Comparison of required time to finish mission with data size U = 100Mb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' The tendencies of the reward and the required time are highly similar although they are generated from different formulas, which indicates that our reward design successfully leads the agent to learn a better policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Episodes 3000 2500 2000 1500 1000 500 0 Reward DQN-PPO Duelling DQN-PPO PPO-PPO Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 8: Comparison of reward with data size U = 50Mb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' The mission completing time comparison for the case with eight users is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' The duelling DQN-PPO algorithm shows similar average performance as the PPO-PPO algorithm but less stability, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=', the required time sometimes jumps to extremely large values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Taking the stability into consideration, the PPO-PPO algorithm is better than duelling DQN-PPO algorithm in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' The DQN-PPO algorithm is obviously not able to converge in this experiment, so we do not consider it a candidate for our double-agent reinforcement learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' CONCLUSION In this paper, we propose a double-agent reinforcement ar- chitecture for data collecting and synchronization in Metavese, and adopt PPO algorithm for both discrete and continuous agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Two agents with different action space and state space work in a cascade manner for channel allocation and UAV 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Episodes 3000 2500 2000 1500 1000 500 Reward DQN-PPO Duelling DQN-PPO PPO-PPO Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 9: Comparison of reward with data size U = 100Mb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Episodes 50 100 150 200 Time (s) DQN-PPO Duelling DQN-PPO PPO-PPO Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 10: Comparison of reward with data size U = 50Mb and 8 users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' trajectory control to form a combined action in each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Our experiments indicate the advantage of the PPO-PPO in both the required time for the mission and the stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' In future work, we will consider transmission power allocation and test the performance of other state-of-art reinforcement learning algorithms in our proposed architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' REFERENCES [1] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Wang and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Zhao, “Mobile Edge Computing, Metaverse, 6G Wireless Communications, Artificial Intelligence, and Blockchain: Survey and Their Convergence,” arXiv preprint arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content='14147, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' [2] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Lee, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Braud, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Zhou, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Wang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Xu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Lin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Kumar, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Bermejo, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Hui, “All one needs to know about metaverse: A complete survey on technological singularity, virtual ecosystem, and research agenda,” arXiv preprint arXiv:2110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content='05352, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' [3] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Si, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Zhao, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Han, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Lam, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Liu, “Resource Allocation and Resolution Control in the Metaverse with Mobile Augmented Reality,” arXiv preprint arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content='13871, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' [4] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Chua, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Yu, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Zhao, “Resource allocation for mobile metaverse with the Internet of Vehicles over 6G wireless commu- nications: A deep reinforcement learning approach,” arXiv preprint arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content='13425, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' [5] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Han et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=', “From visual culture in the immersive metaverse to visual cognition in education,” in Cognitive and Affective Perspectives on Immersive Technology in Education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' IGI Global, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 67–84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' [6] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content='-E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Yu, “Exploration of educational possibilities by four metaverse types in physical education,” Technologies, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 10, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 5, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 104, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 7 [7] Murat Yilmaz, Tuna Hacalo˘glu and Paul Clarke , “Examining the use of non-fungible tokens (NFTs) as a trading mechanism for the metaverse,” in European Conference on Software Process Improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Springer, 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 18–28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' [8] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Jungherr and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Schlarb, “The Extended Reach of Game Engine Companies: How Companies Like Epic Games and Unity Technologies Provide Platforms for Extended Reality Applications and the Metaverse,” Social Media+ Society, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 8, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 2, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' [9] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Agiwal, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Roy, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Saxena, “Next generation 5G wireless networks: A comprehensive survey,” IEEE Communications Surveys & Tutorials, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 18, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 1617–1655, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' [10] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Refonaa, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Sebastian, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Ramanan, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Lakshmi, “Effective identification of black money and fake currency using NFC, IoT and an- droid,” in 2018 International Conference on Communication, Computing and Internet of Things (IC3IoT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' IEEE, 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 275–278.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' [11] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Kohli and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Kohli, “A Study of Preference for Online Trading Account versus Offline Trading Account by Different Age Group,” From the Desk of Editor, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 42, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' [12] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Mane-Deshmukh, “Designing of Wireless Sensor Network to Protect Agricultural Farm from Wild Animals,” i-Manager’s Journal on Information Technology, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 7, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 4, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 30, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' [13] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Wu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Zeng, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Zhang, “Joint trajectory and communication design for multi-UAV enabled wireless networks,” IEEE Transactions on Wireless Communications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 17, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 2109–2121, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' [14] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Lu, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Si, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Gao, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Han, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Wu, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Gong, “Trajectory and Resource Optimization in OFDM-Based UAV-Powered IoT Net- work,” IEEE Transactions on Green Communications and Networking, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 5, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 1259–1270, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' [15] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Ng, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Lim, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content='-N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Dai, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Xiong, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Huang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Niyato, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Hua, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Leung, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Miao, “Joint auction-coalition formation frame- work for communication-efficient federated learning in UAV-enabled internet of vehicles,” IEEE Transactions on Intelligent Transportation Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 22, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 2326–2344, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' [16] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Cui, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Liu, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Nallanathan, “Multi-agent reinforcement learning- based resource allocation for UAV networks,” IEEE Transactions on Wireless Communications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 19, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 729–743, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' [17] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Luong, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Gagnon, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content='-N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Tran, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Labeau, “Deep reinforcement learning-based resource allocation in cooperative UAV-assisted wireless networks,” IEEE Transactions on Wireless Communications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 20, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 11, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 7610–7625, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' [18] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Rodriguez-Ramos, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Sampedro, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Bavle, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' De La Puente, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Campoy, “A deep reinforcement learning strategy for UAV autonomous landing on a moving platform,” Journal of Intelligent & Robotic Sys- tems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 93, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 351–366, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' [19] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Fu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Tang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Hao, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Lei, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Chen, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Fan, “Deep multi-agent reinforcement learning with discrete-continuous hybrid action spaces,” arXiv preprint arXiv:1903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content='04959, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' [20] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Jiang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Deng, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Nallanathan, “Deep reinforcement learning for discrete and continuous massive access control optimization,” in ICC 2020-2020 IEEE International Conference on Communications (ICC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' IEEE, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 1–7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' [21] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Chu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Lee, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Kim, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Kim, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Yoon, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Chung, “Review of Of- fline Payment Function of CBDC Considering Security Requirements,” Applied Sciences, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 12, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 9, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 4488, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' [22] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Christodorescu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Gu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Kumaresan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Minaei, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Ozdayi, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Price, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Raghuraman, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Saad, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Sheffield, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=', “Towards a two-tier hierarchical infrastructure: An offline payment system for Central Bank digital currencies,” arXiv preprint arXiv:2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content='08003, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' [23] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Yusuf, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Tanghe, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Challita et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=', “Experimental characterization of V2I radio channel in a suburban environment,” in 2019 13th European Conference on Antennas and Propagation (EuCAP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' IEEE, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 1–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' [24] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' You and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Zhang, “3D trajectory optimization in Rician fading for UAV-enabled data harvesting,” IEEE Transactions on Wireless Commu- nications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 18, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' 3192–3207, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' [25] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Schulman, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Wolski, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Dhariwal, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Radford, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content=' Klimov, “Prox- imal policy optimization algorithms,” arXiv preprint arXiv:1707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} +page_content='06347, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'} diff --git a/9NE4T4oBgHgl3EQf3Q1N/vector_store/index.faiss b/9NE4T4oBgHgl3EQf3Q1N/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..65c58f30db45716bd49649bdc926bf884bfb3aa7 --- 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b/BtAzT4oBgHgl3EQfh_1H/content/tmp_files/2301.01492v1.pdf.txt @@ -0,0 +1,1191 @@ +arXiv:2301.01492v1 [cs.IT] 4 Jan 2023 +A Pulse-Shape Binary Multiplex Modulation +Pavel Loskot, Senior Member, IEEE +Abstract +The root raised-cosine pulse commonly used in linear digital modulations yields exactly two +intersymbol interference components from the preceding and the subsequent data symbols, provided +that the roll-off factor is 100% and the modulation packing factor is set to 50%. This can be exploited +to symmetrically multiplex two data streams of transmitted symbols. Hence, the proposed scheme is +referred to as pulse-shape binary multiplex modulation. The demodulation of the two multiplexed +data streams at the receiver can be aided by making the streams mutually orthogonal. It can be +achieved by superposition modulation with symbol-by-symbol interference cancellation, proper design +of transmission sequences interleaving pilot and data symbols in order to also enable channel estimation, +and using orthogonal spreading sequences. The presented numerical results indicate that the proposed +modulation scheme can outperform Nyquist signaling in terms of transmission reliability or the time +required for transmitting the whole sequence of data symbols. For instance, differentially encoded +modulation symbols can be transmitted twice as fast by the proposed modulation scheme with a 3 +dB penalty in signal-to-noise ratio over additive white Gaussian noise channels. +Index Terms +Intersymbol-interference; linear modulation; Nyquist signaling; partial response signaling; root raise +cosine pulse; sequence multiplexing. +I. INTRODUCTION +The spectrum scarcity necessities the use of spectrally efficient modulations. The Nyquist +signaling is a well established and robust technique for constructing linear digital modulations +which are employed in a vast majority of today’s communication systems. These modulation +schemes are often combined with channel encoding to improve the transmission reliability and +even approach the channel capacity. An alternative strategy is to assume modulations having +a controlled level of intersymbol interference (ISI), which can increase the rate of information +transmission as well as act as a form of information encoding for improving the transmission +The author is with ZJU-UIUC Institute, Haining, China (e-mail: pavelloskot@intl.zju.edu.cn). +This work was supported by a research grant from Zhejiang University. + +PULSE-SHAPE BINARY MULTIPLEX MODULATION +1 +reliability [1], albeit at the cost of increased detection complexity at the receiver. Such so-called +faster-than-Nyquist (FTN) schemes are linear modulations that can be used over band-limited +channels [2]. +The renewed interest in FTN signaling schemes goes back to the early 2000’s [3]. However, +a closely related idea of partial response linear modulations with controlled ISI appeared much +earlier [4]. The achievable spectral efficiency of coded and uncoded FTN schemes is evaluated +in [2], [5], and [6]. The observation that up to 25% increase in the transmission rate is possible +without deteriorating the error performance is known as the Mazo limit [3], [2], [7], [6]. The +energy and complexity costs of FTN signaling are reviewed in [2]. +The FTN schemes can be implemented both in time and in frequency domains [2], [8]. An +orthogonal FTN scheme based on OFDM was designed in [9]. Alternatively, Nyquist signaling +with dual root raised-cosine (RRC) pulses akin to duobinary modulation has been investigated +in [10]. This scheme was further refined for the RRC pulses with zero roll-off in [11]. The link +between duobinary modulation and FTN signaling has been pointed out in [1]. +An important issue is how to efficiently perform the detection of transmitted symbols at the +receiver. Unlike the ISI due to multipath propagation, the ISI created by FTN signaling also +correlates samples of additive noise. The optimum detection necessitates the use of whitening +matched filter (WMF) prior to symbol decisions. The ISI at the detector input can be equivalently +represented as an auxiliary channel [5], [6]. The output signal of such channel has a trellis-like +structure, which can be optimally equalized by the Viterbi, BCJR and other such algorithms +with varying complexity [2], [7], [6]. These decoding methods can approach the performance +of zero-ISI (Nyquist) modulations over additive white Gaussian noise (AWGN) channels [2]. +The symbol-by-symbol detector for FTN signals was devised in [6] and [12]. The detection of +FTN signals with oversampling and one-bit quantization was developed in [5]. A low complexity +linear equalization for FTN signaling was designed in [13]. The joint channel estimation and +decoding of FTN signals was studied in [14] and in [15]. +Nearly all investigations of FTN signaling schemes in the literature assume the RRC modulation +pulse. The RRC pulse is parameterized by a time period, Tp, and a roll-off factor, α. Linear +modulations combine the RRC pulses weighted by data symbols, which are then transmitted +once every symbol period, Ts. The packing factor defines the relationship between Tp and Ts, +i.e., τ = 1 − Ts/Tp. The design and analysis of FTN signaling in the literature usually assumes +arbitrary values of 0 ≤ α ≤ 1 and 0 ≤ τ < 1. The search for good values of α and τ over an + +PULSE-SHAPE BINARY MULTIPLEX MODULATION +2 +entire α − τ plane to allow symbol-by-symbol decisions was carried out in [12]. However and +importantly, the case of 100% bandwidth roll-off is rarely explicitly considered in the literature +[6]. The authors in [5] noticed that, for α = 1 and an arbitrary value of τ, the ISI is approximately +limited to the two previous and the two subsequent symbol samples. +In this paper, we show that the RRC pulse with 100% roll-off and 50% packing has a well- +defined ISI, which is exactly and symmetrically constrained to one previous and one subsequent +symbol. Such a unique property of the RRC pulse appears to remain unnoticed in the literature. +Interestingly, reference [1] states that ISI with only two components can be obtained with 100% +roll-off and 50% packing assuming prolate spheroidal wave pulses, but not RRC pulses. Although +such a modulation scheme can be assumed to be a special case of FTN signaling, it is argued +that RRC pulses having 100% roll-off and 50% packing offers symmetric multiplexing of the +two transmitted data streams. For this reason, such a partial response signaling is referred to +in this paper as a pulse-shape binary multiplexing (PSBM) modulation. The main task then +is how to separate the two multiplexed data streams at the receiver with acceptable reliability +and complexity. As with other partial response signalings, the modulation constellation and +the dependency between transmitted symbols must be carefully selected in order to trade- +off the performance and the decoding complexity. We design several transmission sequences +interleaving pilot and data symbols, discuss superposition modulation with symbol-by-symbol +sequential interference cancellation (SIC), and also consider orthogonal spreading sequences to +aid separation of the data streams at the receiver. In addition, the performance of multiplexed +differentially encoded phase-shift keying (PSK) modulation symbols is evaluated numerically. +The numerical results identify several cases when the proposed PSBM modulation outperforms +the Nyquist signaling in terms of either transmission reliability or the time required to transmit +a given number of data symbols. +The rest of this paper is organized as follows. Linear modulation schemes that are related to +the proposed pulse-shape multiplexing signaling are outlined in Section II. System model and +the received signal structure are described in Section III. The proposed pulse-shape multiplexing +modulation is defined in Section IV including the design of transmitted symbol sequences. +Numerical results are presented in Section V. Section VI concludes the paper. +We adopt the following notations: E[·] is expectation, ⊛ is convolution, | · | is absolute value, +(·)∗ is complex conjugate, Re{·} and Im{·}, respectively, denote the real and imaginary part +of a complex number, Card{·} is cardinality of a set, (·)T is matrix transpose, (·)−1 is matrix + +PULSE-SHAPE BINARY MULTIPLEX MODULATION +3 +inverse, and ∥·∥ is the Euclidean norm of a matrix or vector. +II. RELATED LINEAR MODULATION SCHEMES +A linearly modulation signal is constructed as, +x(t) = +� +k +sk p(t − kTs) +(1) +where sk are M-ary modulation symbols transmitted every symbol period, Ts, and p(t) denotes +a deterministic pulse-shape, which is also known at the receiver. The stationary sequence of +transmitted symbols, sk, has zero-mean, and the variance, E[|sk|2] = Es. The symbols are usually +obtained as output of a finite-state modulator, i.e., +sk = s(qk, ck) +(2) +where the states, qk, represent modulation memory, and the data symbols, ck, each carry, log2 M, +bits of input information. In this paper, p(t) is assumed to be the unit-energy RRC pulse, [4] +p(t) = rrcα(t/Ts) +√Ts +(3) +where +rrcα(t) = +1 +1 − 16α2t2 +�sin ((1 − α)πt) +πt ++ 4α cos ((1 + α)πt) +π +� +. +(4) +The roll-off factor, 0 ≤ α ≤ 1, however, it is possible to also consider pulse shapes having a +roll-off greater than 100%. +Since the sequence of symbols, sk, is stationary, the auto-correlation, Rs(i − j) = E +� +sis∗ +j +� +. +The corresponding power-spectrum density (PSD) of signal (1) is computed as, [4] +Sx(f) = 1 +Ts +|P(f)|2 � +k +Rs(k) ej2πfkTs +(5) +where P(f) denotes the Fourier transform of p(t). +Correlative coding assumes the discrete modulator (2) to be a finite impulse response (FIR) +filter, i.e., +sk = +K−1 +� +i=0 +vick−i. +(6) + +PULSE-SHAPE BINARY MULTIPLEX MODULATION +4 +The filter weights, vi, are normalized, so that, � +i |vi|2 = 1. More importantly, with a change in +indices, modulated signal (1) with symbols (6) can be rewritten as, +x(t) = +� +k +K−1 +� +i=0 +vi ck−ip(t − kTs) += +� +k +ck +K−1 +� +i=0 +vi p(t − (k + i)Ts) = +� +k +ck ˜p(t − kTs) +(7) +where the compound pulse, ˜p(t) = �K−1 +i=0 vi p(t − iTs). +Duobinary modulation is a special case of correlative coding, such that the FIR filter has +only two non-zero weights, vo = v1 = 1/ +√ +2, the modulation symbols are binary, i.e., ck ∈ +{−√Es, +√Es}, and the RRC pulse has the smallest possible roll-off, α = 0. Modified duobinary +modulation assumes instead the weights, v0 = 1/ +√ +2, v1 = 0, and v2 = −1/ +√ +2. +The following modulations assume the RRC pulse-shape with an arbitrary roll-off value. +Differential PSK constructs the transmitted symbols as, +sk = ck sk−1 +(8) +where the data symbols, ck ∈ {√Es ej2π(i−1)/M}, i = 1, 2, . . . , M. Generalized shift-keying +extends the modulation alphabet of amplitude or phase shift-keying modulations with a zero +symbol [16]. Offset-quadrature (M = 4) PSK delays the imaginary part of the modulated signal +by half a symbol period, i.e., +x(t) = +� +k +Re{ck} p(t − kTs) + j Im{ck} p(t − kTs − Ts/2). +(9) +Finally, FTN signaling is a linear modulation described by eq. (1). More importantly, the RRC +pulse-shape in (3) can now be scaled by, Tp = Ts/(1 − τ), instead of Ts, where 0 ≤ τ < 1 is +so-called the packing factor, i.e., +x(t) = +� +k +skp(t − k(1 − τ)Tp) = +� +k +skp(t − kTs) +(10) +so that Tp is a design parameter of the pulse, p(t), whereas, Ts = (1 −τ)Tp, denotes the symbol +period. Thus, τ = 0 packing corresponds to a conventional Nyquist signaling, whereas τ = 1 +packing would completely overlap the transmitted symbols. More importantly, the PSD of (10) +is still given by eq. (5), and it is otherwise completely independent of the packing factor, τ. + +PULSE-SHAPE BINARY MULTIPLEX MODULATION +5 +III. RECEIVED SIGNAL +The standard wireless channel model with L propagation paths is an FIR filter with the impulse +response, +˜h(t) = +L +� +l=1 +hl(t)δ(t − τl). +(11) +The signal delays, τl, are assumed to be constant. The path attenuations, hl(t), are zero-mean +circularly symmetric Gaussian processes. These processes are stationary, and generally mutually +correlated. They have a defined auto-correlation, Rh(∆t), which determines the coherence bandwidth. +For narrow-band signals, the number of paths, L, is small. For L = 1, the channel model (11) +becomes frequency non-selective. In low-mobility scenarios, the channel attenuations, hl(t), are +often assumed to be constant over blocks of transmitted symbols, and independent between the +successive blocks, which is often referred to as a block fading model. +The received signal corresponding to multi-path propagation model (11) is written as, +y(t) = ˜h(t) ⊛ x(t) + w(t) += +L +� +l=1 +hl(t)x(t − τl) + w(t) +(12) +where w(t) is a zero-mean stationary circularly symmetric AWGN with the variance, σ2 +w = +E[|w(t)|2]. +The received signal is filtered through a filter matched to the transmitted pulse, p(t), and +synchronously sampled at a rate, 1/Ts. In particular, assuming RRC pulses, the matched filter, +p∗(−t) = p(t), and provided that the channel attenuations are constant over blocks of transmitted + +PULSE-SHAPE BINARY MULTIPLEX MODULATION +6 +symbols, the received samples are modeled as, +rn =y(t) ⊛ p∗(−t) +��� +t=nTs+τ0 += +L +� +l=1 +hl x(t − τl) ⊛ p(t) +��� +t=nTs+τ0 + w(t) ⊛ p(t) +��� +t=nTs+τ0 += +L +� +l=1 +hl +� +k +sk +� ∞ +−∞ +p(ζ + (n − k)Ts − τl)p(ζ − τ0) dζ ++ +� ∞ +−∞ +w(ζ + nTs)p(ζ − τ0) dζ += +� +k +sk +L +� +l=1 +hl pn−k,l + wn = +� +k +sk˜pn−k + wn +=sk˜p0 + +� +k +n̸=k +sk˜pn−k +� +�� +� +ISI ++wn. +(13) +The timing offset, τ0, at the receiver can be optimized to minimize the ISI term (in some sense) +in (13) defined as, +˜pn−k = +L +� +l=1 +hl +� ∞ +−∞ +p(ζ + (n − k)Ts − τl)p(ζ − τ0) dζ, n ̸= k. +(14) +Thus, the ISI arises when the orthogonality between the transmitter and the receiver pulses +is violated, for example, due to multi-path propagation, time-synchronization errors between +transmitter and receiver, and also due to symbol-period compression in FTN signaling schemes +[4]. +An interesting question is how much ISI is produced for different combinations of parameters +α and τ in FTN signaling schemes using RRC pulses. Hence, define the function, +ISI(µ) = Card{|˜pk| > µ, k ̸= 0} +(15) +to be the number of ISI components that are greater than a given threshold, µ. Note that, +ISI(µ) ∈ {0, 2, 4, . . .}, due to even symmetry of the RRC pulses. Assuming different thresholds, +µ, the roll-off, 0 ≤ α ≤ 2, and the RRC pulses truncated to (−4Ts, +4Ts), the values ISI(µ) = 0 +(red points) and ISI(µ) = 2 (blue points) in the α − τ plane are shown in Fig. 1. The empty +(white) spaces in Fig. 1 indicate the values, ISI(µ) > 2. It can be observed that by decreasing +the threshold, µ, several cases of interest for designing FTN signaling schemes start to emerge. +In particular, exactly two ISI components can be obtained for these parameters: α = 1.0 and +τ = 0.5, α = 1.07 and τ ∈ (0.70, 0.71), and α ∈ (1.65, 1.85) and τ ∈ (0.47, 0.50). + +PULSE-SHAPE BINARY MULTIPLEX MODULATION +7 +0 +� �� +1 +� +� +2 +=0.01 +� +�� � +� + + + +=0.005 + +  + + + + +=0.003 +0 +0.5 +1 +0 +1 +2 +=0.002 +1−τ +1−τ +α +α +Fig. 1. The ISI(µ) = 0 components (red points) and ISI(µ) = 2 components (blue points) for four different thresholds, µ. +IV. PULSE-SHAPE BINARY MULTIPLEX MODULATION +As indicated in Fig. 1, the RRC pulse with 100% roll-off and 50% packing has well-defined +and finite ISI components. In particular, the RRC pulse (4) for α = 1 becomes, +rrc1(t) = 4 cos(2πt) +π(1 − 16t2). +(16) +This pulse has the following ISI components in an AWGN channel without multi-path. Such a +fundamental property appears to remain unnoticed in the literature. +Lemma 1: Let n be a non-negative integer. The ISI integral involving the RRC pulse, rrc1(t), +with 100% roll-off has the exact solution, +� ∞ +−∞ +rrc1(t) × rrc1(t − n/4) dt += + + + + + + + + + + + + + + + + + + + +8/(3π) +n = 1 +8 +π(n−2)n(n+2) +n − odd, n > 1 +1 +n = 0 +1/2 +n = 2 +0 +n − even, n > 2. +(17) + +PULSE-SHAPE BINARY MULTIPLEX MODULATION +8 +n=0 +n=2 +n=4 +1 +2 +3 +4 +5 +0.998 +0.999 +1.000 +1.001 +1.002 +d +truncated integral values +Fig. 2. Numerically computed integral (17) truncated to interval, (−d, +d), as a function of d (solid lines). The exact values +for an infinite interval are shifted to be all equal to unity in order to enable comparison. The dashed lines are mirrored solid +lines about the unit value. +Lemma 1 can be proved by solving the integral for the first few values of n (for example, using +Mathematica software), and then using induction. +However, the result (17) is exact only when the integration is performed over an infinite +interval. In practice, the pulse shapes must be truncated to a finite interval. The numerically +computed values of integral (17) when the interval of integration is truncated to (−d, +d) are +shown in Fig. 2. It can be observed that the RRC pulse shape, rrc1(t), should not be truncated +to the intervals shorter than, (−4, +4), in order to achieve the RRC property given in Lemma 1 +with at least 99.9% accuracy. +Definition 2: The modulated signal of pulse-shape binary multiplex modulation is written as, +x(t) = +� +k +sk +rrc1 +� +t−kTs +2Ts +� +√2Ts +. +(18) +The synchronously sampled matched filter output of modulated signal (18) received in AWGN, +w(t), is, +rn =(x(t) + w(t)) ⊛ +rrc1 +� +t +2Ts +� +√2Ts +��� +t=nTs += +� +k +sk +� ∞ +−∞ +rrc1(ζ + (n − k)/2) rrc1(ζ) dζ + wn += +�1 +2sn−1 + sn + 1 +2sn+1 +� ++ wn. +(19) + +PULSE-SHAPE BINARY MULTIPLEX MODULATION +9 +a1 +a1+a2 +2 +a3 +a2+a3 +2 +a4 +a3+a4 +2 +a2 +Stream #1: +Stream #2: +b1 +b1+b2 +2 +b3 +b2+b3 +2 +b3+b4 +2 +b2 +T1 +T2 +T3 +T4 +T5 +T6 +T7 +Fig. 3. A visualization of pulse-shape multiplex modulated signal. +The noise samples, wn, in (19) are zero-mean, have the variance, E[|wn|2] = E[|w(t)|2] = σ2 +w, +and their stationary auto-correlation is, +Rw(n − m) = E[wnw∗ +m] = + + + + + + + + + +σ2 +w +n = m +σ2 +w/2 +|n − m| = 1 +0 +|n − m| > 1. +(20) +Such noise samples can be equivalently modeled by a simple FIR filter, +wn = un + un−1 +√ +2 +(21) +where un are the samples of a zero-mean, circularly symmetric Gaussian process having the +variance, E[|un|2] = σ2 +w. In addition, it is straightforward to show that the variance of the sum +of N noise samples having the correlations (20) is, +var +� N +� +n=1 +wn +� += (2N − 1)σ2 +w +(22) +which is greater than the variance, Nσ2 +w, of the sum of N uncorrelated samples. +The modulated signal (18) in Definition 2 can be visualized as shown in Fig. 3. In particular, +the transmitted data symbols can be viewed as consisting of two multiplexed streams of data +symbols, ak, and, bk, which are each transmitted with a period 2Ts, but mutually shifted by Ts. +The corresponding received symbol samples after the matched filtering are, +rn = + + + +an + bn−1+bn +2 ++ wn +n − odd +bn + an+an+1 +2 ++ wn +n − even. +(23) +Using (5), the PSD of modulated signal (18) is computed as, +Sx(f) = 2|RRC1(2Tsf)|2 � +k +E[s0s∗ +k] ej2πfkTs +(24) +where the Fourier transform of the pulse, rrc1(t), is, +RRC1(f) = + + + +cos(πf/2) +|f| ≤ 1 +0 +otherwise. +(25) + +PULSE-SHAPE BINARY MULTIPLEX MODULATION +10 +A. Transmitted Sequence Design +The optimum detection of transmitted symbols in the presence of ISI must consider complete +sequences of received samples. However, in the absence of multi-path, the received samples +have structure (19), and the transmitted sequences can be designed, so that the complexity of +detection at the receiver can be reduced. +The key strategy for reducing the detection complexity is to exploit orthogonality among +sub-sequences of transmitted symbols. Offset-quadrature PSK modulation (9) alternates one- +dimensional modulation symbols along the in-phase and quadrature components, which allows +the optimum symbol-by-symbol decisions. +Multiplexing two data streams as described by (23) can exploit the design principles of +superposition modulation and multiuser detection. In such a case, symbol-by-symbol decisions +can be performed by SIC. Specifically, provided that symbols, bn, can be reliably detected, even +if the symbols, (an + an+1)/2, are not yet known, then the symbol, an, can be reliably detected +after canceling the ISI term, (bn−1 + bn)/2. +In the sequel, three other sequence design strategies are discussed in more detail. The first +strategy combines pilot and data symbols to aid the data detection and channel estimation. The +second strategy employs orthogonal spreading codes in order to separate the two multiplexed +data sequences. The third strategy adopts the differential encoding of transmitted symbols. +B. Sequences with Interleaved Pilot Symbols +In general, pilot symbols for channel estimation can be interleaved with data symbols or +superimposed onto data symbols [17]. Here, the more common former approach is adopted. +Thus, consider a transmitted sequence consisting of alternating groups of Ld data symbols and +Lp ≪ Ld pilot symbols, which are separated by a single zero-symbol as shown in Fig. 4. +For instance, the following sub-sequences with reduced or no ISI can be considered with pilot +symbol, p, and arbitrary data symbols, d1, and, d2: (0, p, 0), (d1, p, −d1), (d1, p, −d1, −p, d1), +and (d1, p, −d1, −p, d2, p, −d2). These sub-sequences enable ISI-free data detection and channel +estimation, as can be deduced from eq. (23) and Fig. 3. Recall also that the noise samples, wn, +and, wn±2, are uncorrelated, i.e., independent. + +PULSE-SHAPE BINARY MULTIPLEX MODULATION +11 +0 pilots +0 +data +0 pilots +0 +data +0 +Lp +separator +Ld +Fig. 4. The transmitted sequence with interleaved sub-sequences of data and pilot symbols and a single zero-symbol separator. +In order to illustrate the ISI-free channel estimation, consider the sequence, (d1, p, p, −d1). +The received samples corresponding to the two pilot symbols in the middle are, +rn =h3 +2p + h1 +2d1 + wn +rn+1 =h3 +2p − h1 +2d1 + wn+1 +(26) +where h denotes the complex-valued channel attenuation (i.e., frequency non-selective slow +fading). The samples, rn, and, rn+1, can be simply combined as, +rn + rn+1 = 3hp + wn + wn+1 +(27) +where the total variance of the additive noise samples is equal to 3σ2 +w due to correlations (20). +More generally, the transmitted sequence, +(−p, d1, p, d2, −p, d3, p, d4, −p, d5, . . . , dN, ±p) +(28) +where the last pilot symbol is p, if N is odd, and −p, if N is even, allows the ISI-free symbol- +by-symbol decisions of all data symbols. Moreover, assuming again a slow fading channel, the +received samples corresponding to the pilot symbols can be summed up to obtain, +N +� +n=1 +(−1)n r2n−1 = N h p + +√ +Nw +where the noise sample, w, has the variance, σ2 +w, so the signal-to-noise ratio (SNR) for estimating +the channel coefficient, h, has been improved N-times. Note also that once the channel has been +estimated, the pilot symbols can be subtracted from the received samples in order to aid decisions +of the remaining data symbols. +Finally, consider the case of a symbol repetition diversity. The transmitted sequence, (d, 0, d, 0, . . ., 0, d), +of a data symbol, d, repeated (N ≥ 2)-times corresponds to the canonical Nyquist signaling. +The pulse-shape multiplex modulation instead transmits the sequence, (d, d, . . . , d), of N1-times +repeated data symbol, d. For the same sequence length, N1 = 2N − 1. Assuming slowly + +PULSE-SHAPE BINARY MULTIPLEX MODULATION +12 +fading channel, the detector combines the received samples for the two modulation schemes, +respectively, as, +r =N h d + +√ +Nw +r =(2 · 3/2 + 2(N1 − 2)) h d/ +√ +2 + +� +2N1 − 1w +(29) +where the scaling by +√ +2 was introduced for the second modulation in order to account for the +larger number of symbols in its transmitted sequence. The resulting SNR of these two schemes +is proportional to, γ ∝ N, and, γ ∝ 2N −3/2, respectively. Consequently, for symbol repetition +diversity, the SNR gain of the pulse-shape binary multiplexing is asymptotically 3 dB larger than +for the Nyquist signaling. +C. Sequences with Orthogonal Spreading +Another strategy for transmitting interleaved, but orthogonal symbols in modulated signal (18) +is to use orthogonal spreading codes. In particular, assume transmitted symbols, +an = d1c(1) +n , +bn = d2c(2) +n +(30) +where d1 and d2 are two data symbols, and, c(1) +n and c(2) +n , n = 1, 2, . . . , N, are generally complex- +valued, orthogonal spreading sequences, so that, +N +� +n=1 +c(i) +n c∗(j) +n += + + + +N, +i = j +0, +i ̸= j. +(31) +Then, the sequences of received samples (23) are linearly combined as, +N +� +n=1 +r2n−1c∗(1) +n +=d1 + d2 +N +� +n=1 +c(2) +n + c(2) +n+1 +2 +c∗(1) +n ++ +N +� +n=1 +w2n−1c∗(1) +n +=d1 + ˜w1 +N +� +n=1 +r2nc∗(2) +n +=d2 + d1 +N +� +n=1 +c(1) +n + c(1) +n+1 +2 +c∗(2) +n ++ +N +� +n=1 +w2nc∗(2) +n +=d2 + ˜w2 +(32) +provided that the spreading sequences, c(1) +n , and, c(2) +n , are exactly orthogonal. In such a case, the +SNR improvement for transmitting two data symbols with orthogonal spreading sequences using +the pulse-shape binary multiplex modulation (18) is proportional to, +γ ∝ +N2 +2N − 1. +(33) + +PULSE-SHAPE BINARY MULTIPLEX MODULATION +13 +0 +50 +100 +150 +200 +250 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +r=10% +r=5% +N +Probability +Fig. 5. The probability (35) vs. the spreading sequence length, N, assuming κ = 5% and κ = 10%, respectively. +For instance, if the spreading symbols, cn, are generated independently at random and with an +equal probability from the set, {−1, +1}, the probability that two such sequences are orthogonal +is, +Pr +� N +� +n=1 +c(1) +n c∗(2) +n += 0 +� += +� N +N/2 +� �1 +2 +�N/2 �1 +2 +�N−N/2 += +� N +N/2 +� +2−N. +(34) +Since the probability (34) of exact orthogonality asymptotically goes to zero with large N, +consider instead the probability, +Pr +� +−⌈κ N/2⌋ ≤ +N +� +n=1 +c(1) +n c∗(2) +n +≤ ⌈κ N/2⌋ +� += +⌈κ N/2⌋ +� +n=−⌈κ N/2⌋ +�N +n +� �1 +2 +�N +(35) +for some small κ ≥ 0. The probabilities (35) as a function of N for two different values of +factor, κ, are shown in Fig. 5. These probabilities are indicative of how many random spreading +sequences need to be generated in order to select the required number of such sequences having +an acceptable level of mutual orthogonality. + +PULSE-SHAPE BINARY MULTIPLEX MODULATION +14 +c0 +c0 +2 (1 + c1) +c0 +2 (1 + c1c2 + 2c1) +c0 +2 (1 + c0c1 + 2c0) +c0 +2 (1 + c1c2) +c0c1 +c0c1 +2 (1 + c2c3 + 2c2) +c0c1 +2 (1 + c2c3) +c0c1c2 +c0c1c2 +2 +(1 + c3c4 + 2c3) +c0c1c2c3 +c0c1c2 +2 +(1 + c3c4) +c0 +c0c1c2c3c4 +T1 +T0 +T2 +T3 +T5 +T4 +Stream #1: +Stream #2: +Tx symbols: +Fig. 6. Differentially encoded M-ary PSK symbols transmitted via pulse-shape binary multiplex modulation. +D. Sequences with Differential Encoding +Differential PSK is a popular modulation scheme for fast fading channels, since it alleviates +the need for recovering the absolute phase reference. Fig. 6 shows differentially encoded M-ary +PSK symbols (8) transmitted via pulse-shape binary multiplex modulation. In particular, the n-th +transmitted symbol is, +sn = +�n−2 +� +k=0 +ck +� +1 + cn−1cn + 2cn−1 +2 +=1 +2 +�n−2 +� +k=0 +ck +� ++ 1 +2 +�n−1 +� +k=0 +ck +� +cn + +�n−1 +� +k=0 +ck +� +. +(36) +Consequently, the differential decoding can be performed as, +cn = +� +2sn − +�n−2 +� +k=0 +ck +� +− 2 +�n−1 +� +k=0 +ck +�� �n−1 +� +k=0 +ck +�∗ +=2sn +�n−1 +� +k=0 +c∗ +k +� +− c∗ +n−1 − 2. +(37) +The performance of this modulation scheme is evaluated in the next section. +V. NUMERICAL EXAMPLES +It is convenient to use a vector notation to generate samples of pulse-shape binary multiplex +modulation (18) received over a frequency non-selective fading channel. The vector, r, of N +received samples corresponding to the vector, s, of N transmitted symbols can be obtained as, +r = s A diag(h) + w AT +0 + +PULSE-SHAPE BINARY MULTIPLEX MODULATION +15 +where h is a vector of fading channel coefficients, w are samples of AWGN, and the (N × N) +ISI matrix, +A = + + +1 +1/2 +1/2 +1 +1/2 +... +... +1/2 +1 + + += A0 AT +0 . +The optimum detection requires that the additive noise is first whitened as, [4] +r A−T +0 += s A diag(h) A−T +0 ++ w. +Then the maximum likelihood (ML) detection of sequence s is, +ˆs = arg min +s +���r A−T +0 +− s A diag +� +ˆh +� +A−T +0 +��� +2 +(38) +where ˆh is the estimate of h representing channel state information (CSI). +An uncoded binary phase shift keying (BPSK) modulation and Rayleigh-distributed fading +amplitudes, h, are assumed for simplicity. The transmitted sequence interleaves pilot symbols and +data symbols as shown in Fig. 4. The pilot symbols are used to estimate the channel coefficients, +h, by linear minimum mean-square error (LMMSE) algorithm. The spectral efficiency of pulse- +shape binary multiplexing is, 2, which is always larger than the spectral efficiency of the Nyquist +signaling being equal to, 2/(1 + α). +The BER curves, Pe, for short data sequences of Ld = 4 and Ld = 8 binary symbols, +respectively, separated by a single zero-symbol are shown in Fig. 7 and Fig. 8. The SNR is +defined as, γb = 1/(2σ2 +w). Both cases of perfect and estimated CSI are considered. The Nyquist +signaling (no ISI) with symbol-by-symbol decisions is assumed as a reference. The ML data +detector (38) is used for pulse-shape multiplexing signaling. +It can be observed that the performance penalty due to channel estimation is much larger for +pulse-shape multiplexing than for the Nyquist signaling, which is to be expected. The WMF +improves the performance by several dB’s for both signaling schemes. More importantly, the +performance of pulse-shape multiplexing improves with the data block length by exploiting the +time diversity over a fading channel, so it can significantly outperform the Nyquist signaling +at medium to large SNR values. It is likely that by employing more sophisticated channel +estimation and equalization techniques, the performance of pulse-shape multiplexing can be +further improved. In order to demonstrate the effect of time diversity, Fig. 9 shows that, over + +PULSE-SHAPE BINARY MULTIPLEX MODULATION +16 +0 +5 +10 +15 +20 +10-5 +10-4 +10-3 +10-2 +10-1 +100 +estim. +CSI +perfect +CSI +No-ISI +ISI +estim. CSI +perfect CSI +ML +WMF-ML +γb [dB] +Pe +LLLddd === 444 +Fig. 7. The BER of BPSK vs. SNR over Rayleigh fading channel for sequences of 4 binary symbols. +an AWGN channel, the performance of pulse-shape multiplexing is worse than that of Nyquist +signaling, even though some performance loss can be recovered by WMF. +Lastly, the BER performance of Nyquist modulation and pulse-shape multiplex modulation +transmitting differentially encoded quadrature PSK (QPSK) symbols over an AWGN channel is +compared in Fig. 10. It can be observed that even though the pulse-shape multiplexing suffers +asymptotically a 3 dB penalty in SNR, it reduces the time required for transmitting the whole +symbol sequence to one half. +VI. CONCLUSION +The paper introduced a pulse-shape binary multiplex modulation. Such a modulation scheme is +akin to partial-response signaling, correlative coding, offset-QPSK modulation and FTN signaling. +It combines two data streams under controlled ISI created by the RRC pulses having 100% +roll-off, and transmitted at twice the Nyquist rate. The ISI analysis showed that this is unique +property among all the roll-off factors being at most 100% and the packing factors greater +than 5%. However, the successive samples of additive noises at the output of matched filter at + +PULSE-SHAPE BINARY MULTIPLEX MODULATION +17 +0 +5 +10 +15 +20 +10-5 +10-4 +10-3 +10-2 +10-1 +100 +estim. +CSI +perfect +CSI +No-ISI +ISI +estim. CSI +perfect CSI +ML +WMF-ML +γb [dB] +Pe +LLLddd === 888 +Fig. 8. The BER of BPSK vs. SNR over Rayleigh fading channel for sequences of 8 binary symbols. +the receiver become correlated, which incurs a SNR performance penalty. This penalty could +be reduced or even removed by using more complex sequence-based detection schemes as +shown elsewhere in the literature. The BER performance as well as decoding complexity of +the proposed pulse-shape binary multiplexing modulation scheme is critically affected by the +choice of transmitted sequences. One can consider superposition modulation with SIC decoding, +interleave data symbols with pilot and zero-symbols to aid channel estimation and data decoding, +and also employ orthogonal spreading sequences to separate the multiplexed data streams. The +numerical results indicate that pulse-shape binary multiplexing can exploit time-diversity in +fading channels to outperform the Nyquist signaling. In addition, it has been shown numerically +that a sequence of differentially encoded PSK symbols can be transmitted twice as fast by the +proposed modulation scheme compared to canonical Nyquist signaling, although with a 3 dB +SNR penalty over AWGN channels. + +PULSE-SHAPE BINARY MULTIPLEX MODULATION +18 +4 +5 +6 +7 +8 +9 +10 +11 +10 -5 +N=2 +ML +WHF-ML +BPSK +4 +5 +6 +7 +8 +9 +10 +11 +10-5 +N=4 +γb [dB] +Pe +Pe +Fig. 9. The BER of BPSK vs. SNR over AWGN channel for sequences of 2 and 4 binary symbols, respectively. +6 +8 +10 +12 +14 +16 +18 +10-6 +10-5 +10-4 +10-3 +10-2 +DQPSK +DQPSK-PSBM +γb [dB] +Pe +Fig. 10. +The BER comparison of differentially encoded QPSK with Nyquist and pulse-shape binary multiplexing (PSBM) +modulation transmitted over an AWGN channel. + +PULSE-SHAPE BINARY MULTIPLEX MODULATION +19 +REFERENCES +[1] J. Zhou, D. Li, and X. Wang, “Generalized Faster-Than-Nyquist signaling,” in ISIT, 2012, pp. 1478–1482. +[2] J. B. Anderson, F. Rusek, and V. Öwall, “Faster-Than-Nyquist signaling,” Proc. of the IEEE, vol. 101, no. 8, pp. 1817–1830, +August 2013. +[3] A. Liveris and C. Georghiades, “Exploiting Faster-Than-Nyquist signaling,” IEEE Transactions Communications, vol. 51, +no. 9, pp. 1502–1511, September 2003. +[4] J. G. Proakis and M. Salehi, Digital Communications, 5th ed. +McGraw-Hill Education, NY, USA, 2008. +[5] L. Landau, M. Dörpinghaus, and G. P. Fettweis, “1-bit quantization and oversampling at receiver: Communication over +bandlimited channels with noise,” IEEE Comm. Letters, vol. 21, no. 5, pp. 1007–1010, May 2017. +[6] A. Modenini, G. Colavolpe, and N. Alagha, “How to significantly improve the spectral efficiency of linear modulations +through time-frequency packing and advanced processing,” in Proc. ICC, 2012, pp. 3260–3264. +[7] J. Fan, S. Guo, X. Zhou, Y. Ren, G. Y. Li, and X. Chen, “Faster-Than-Nyquist signaling: An overview,” IEEE Access, +vol. 5, pp. 1925–1940, February 2017. +[8] Y. Yamada, M. Sawahashi, and K. Saito, “Performance of time and frequency compression of Faster-than-Nyquist signaling +in frequency-selective fading channels,” in APCC, 2015, pp. 550–554. +[9] T. E. Bogale, L. B. Le, X. Wang, and L. Vandendorpe, “Multipath multiplexing for capacity enhancement in SIMO wireless +systems,” IEEE Transactions Wireless Communications, vol. 16, no. 10, pp. 6895–6911, October 2017. +[10] H. Zhang, X. Huang, J. A. Zhang, and Y. J. Guo, “Dual pulse shaping transmission and equalization for high-speed +wideband wireless communication systems,” IEEE Transactions on Circuits and Systems I, vol. 67, no. 7, pp. 1549–8328, +July 2020. +[11] H. Li, X. Huang, J. A. Zhang, H. Zhang, and Z. Cheng, “Dual pulse shaping transmission with sinc-function based +complementary Nyquist pulses,” IET Communications, vol. 16, no. 17, pp. 2091–2104, October 2022. +[12] E. Bedeer, M. H. Ahmed, and H. Yanikomeroglu, “A very low complexity successive symbol-by-symbol sequence estimator +for Faster-Than-Nyquist signaling,” IEEE Access, vol. 5, pp. 7414–7422, June 2017. +[13] J. Bas and A. A. Dowhuszko, “Linear time-packing detectors for optical feeder link in high throughput satellite systems,” +in GC-ElecEng, 2020, pp. 21–26. +[14] Q. Shi, N. Wu, X. Ma, and H. Wang, “Frequency-domain joint channel estimation and decoding for Faster-Than-Nyquist +signaling,” IEEE Transactions Communications, vol. 66, no. 2, pp. 781–795, February 2018. +[15] N. Wu, W. Yuan, Q. Guo, and J. Kuang, “A hybrid BP-EP-VMP approach to joint channel estimation and decoding for +FTN signaling over frequency selective fading channels,” IEEE Access, vol. 5, pp. 6849–6858, May 2017. +[16] P. Loskot, “A generalized FSK-based PHY layer design for wireless sensor networks,” in Chinacom, 2012, pp. 362–367. +[17] A. K. Jagannatham and B. D. Rao, “Superimposed pilots vs. conventional pilots for channel estimation,” in ACSSC, 2006, +pp. 767–771. + diff --git a/BtAzT4oBgHgl3EQfh_1H/content/tmp_files/load_file.txt b/BtAzT4oBgHgl3EQfh_1H/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4a76048b97635bae07a5ea0d84e33ddfeba110bd --- /dev/null +++ b/BtAzT4oBgHgl3EQfh_1H/content/tmp_files/load_file.txt @@ -0,0 +1,467 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf,len=466 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content='01492v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content='IT] 4 Jan 2023 A Pulse-Shape Binary Multiplex Modulation Pavel Loskot, Senior Member, IEEE Abstract The root raised-cosine pulse commonly used in linear digital modulations yields exactly two intersymbol interference components from the preceding and the subsequent data symbols, provided that the roll-off factor is 100% and the modulation packing factor is set to 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' This can be exploited to symmetrically multiplex two data streams of transmitted symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Hence, the proposed scheme is referred to as pulse-shape binary multiplex modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The demodulation of the two multiplexed data streams at the receiver can be aided by making the streams mutually orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' It can be achieved by superposition modulation with symbol-by-symbol interference cancellation, proper design of transmission sequences interleaving pilot and data symbols in order to also enable channel estimation, and using orthogonal spreading sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The presented numerical results indicate that the proposed modulation scheme can outperform Nyquist signaling in terms of transmission reliability or the time required for transmitting the whole sequence of data symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' For instance, differentially encoded modulation symbols can be transmitted twice as fast by the proposed modulation scheme with a 3 dB penalty in signal-to-noise ratio over additive white Gaussian noise channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Index Terms Intersymbol-interference;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' linear modulation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Nyquist signaling;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' partial response signaling;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' root raise cosine pulse;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' sequence multiplexing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' INTRODUCTION The spectrum scarcity necessities the use of spectrally efficient modulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The Nyquist signaling is a well established and robust technique for constructing linear digital modulations which are employed in a vast majority of today’s communication systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' These modulation schemes are often combined with channel encoding to improve the transmission reliability and even approach the channel capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' An alternative strategy is to assume modulations having a controlled level of intersymbol interference (ISI), which can increase the rate of information transmission as well as act as a form of information encoding for improving the transmission The author is with ZJU-UIUC Institute, Haining, China (e-mail: pavelloskot@intl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content='zju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' This work was supported by a research grant from Zhejiang University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' PULSE-SHAPE BINARY MULTIPLEX MODULATION 1 reliability [1], albeit at the cost of increased detection complexity at the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Such so-called faster-than-Nyquist (FTN) schemes are linear modulations that can be used over band-limited channels [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The renewed interest in FTN signaling schemes goes back to the early 2000’s [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' However, a closely related idea of partial response linear modulations with controlled ISI appeared much earlier [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The achievable spectral efficiency of coded and uncoded FTN schemes is evaluated in [2], [5], and [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The observation that up to 25% increase in the transmission rate is possible without deteriorating the error performance is known as the Mazo limit [3], [2], [7], [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The energy and complexity costs of FTN signaling are reviewed in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The FTN schemes can be implemented both in time and in frequency domains [2], [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' An orthogonal FTN scheme based on OFDM was designed in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Alternatively, Nyquist signaling with dual root raised-cosine (RRC) pulses akin to duobinary modulation has been investigated in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' This scheme was further refined for the RRC pulses with zero roll-off in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The link between duobinary modulation and FTN signaling has been pointed out in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' An important issue is how to efficiently perform the detection of transmitted symbols at the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Unlike the ISI due to multipath propagation, the ISI created by FTN signaling also correlates samples of additive noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The optimum detection necessitates the use of whitening matched filter (WMF) prior to symbol decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The ISI at the detector input can be equivalently represented as an auxiliary channel [5], [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The output signal of such channel has a trellis-like structure, which can be optimally equalized by the Viterbi, BCJR and other such algorithms with varying complexity [2], [7], [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' These decoding methods can approach the performance of zero-ISI (Nyquist) modulations over additive white Gaussian noise (AWGN) channels [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The symbol-by-symbol detector for FTN signals was devised in [6] and [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The detection of FTN signals with oversampling and one-bit quantization was developed in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' A low complexity linear equalization for FTN signaling was designed in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The joint channel estimation and decoding of FTN signals was studied in [14] and in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Nearly all investigations of FTN signaling schemes in the literature assume the RRC modulation pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The RRC pulse is parameterized by a time period, Tp, and a roll-off factor, α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Linear modulations combine the RRC pulses weighted by data symbols, which are then transmitted once every symbol period, Ts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The packing factor defines the relationship between Tp and Ts, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=', τ = 1 − Ts/Tp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The design and analysis of FTN signaling in the literature usually assumes arbitrary values of 0 ≤ α ≤ 1 and 0 ≤ τ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The search for good values of α and τ over an PULSE-SHAPE BINARY MULTIPLEX MODULATION 2 entire α − τ plane to allow symbol-by-symbol decisions was carried out in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' However and importantly, the case of 100% bandwidth roll-off is rarely explicitly considered in the literature [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The authors in [5] noticed that, for α = 1 and an arbitrary value of τ, the ISI is approximately limited to the two previous and the two subsequent symbol samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' In this paper, we show that the RRC pulse with 100% roll-off and 50% packing has a well- defined ISI, which is exactly and symmetrically constrained to one previous and one subsequent symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Such a unique property of the RRC pulse appears to remain unnoticed in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Interestingly, reference [1] states that ISI with only two components can be obtained with 100% roll-off and 50% packing assuming prolate spheroidal wave pulses, but not RRC pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Although such a modulation scheme can be assumed to be a special case of FTN signaling, it is argued that RRC pulses having 100% roll-off and 50% packing offers symmetric multiplexing of the two transmitted data streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' For this reason, such a partial response signaling is referred to in this paper as a pulse-shape binary multiplexing (PSBM) modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The main task then is how to separate the two multiplexed data streams at the receiver with acceptable reliability and complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' As with other partial response signalings, the modulation constellation and the dependency between transmitted symbols must be carefully selected in order to trade- off the performance and the decoding complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' We design several transmission sequences interleaving pilot and data symbols, discuss superposition modulation with symbol-by-symbol sequential interference cancellation (SIC), and also consider orthogonal spreading sequences to aid separation of the data streams at the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' In addition, the performance of multiplexed differentially encoded phase-shift keying (PSK) modulation symbols is evaluated numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The numerical results identify several cases when the proposed PSBM modulation outperforms the Nyquist signaling in terms of either transmission reliability or the time required to transmit a given number of data symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Linear modulation schemes that are related to the proposed pulse-shape multiplexing signaling are outlined in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' System model and the received signal structure are described in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The proposed pulse-shape multiplexing modulation is defined in Section IV including the design of transmitted symbol sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Numerical results are presented in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Section VI concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' We adopt the following notations: E[·] is expectation, ⊛ is convolution, | · | is absolute value, (·)∗ is complex conjugate, Re{·} and Im{·}, respectively, denote the real and imaginary part of a complex number, Card{·} is cardinality of a set, (·)T is matrix transpose, (·)−1 is matrix PULSE-SHAPE BINARY MULTIPLEX MODULATION 3 inverse, and ∥·∥ is the Euclidean norm of a matrix or vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' RELATED LINEAR MODULATION SCHEMES A linearly modulation signal is constructed as, x(t) = � k sk p(t − kTs) (1) where sk are M-ary modulation symbols transmitted every symbol period, Ts, and p(t) denotes a deterministic pulse-shape, which is also known at the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The stationary sequence of transmitted symbols, sk, has zero-mean, and the variance, E[|sk|2] = Es.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The symbols are usually obtained as output of a finite-state modulator, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=', sk = s(qk, ck) (2) where the states, qk, represent modulation memory, and the data symbols, ck, each carry, log2 M, bits of input information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' In this paper, p(t) is assumed to be the unit-energy RRC pulse, [4] p(t) = rrcα(t/Ts) √Ts (3) where rrcα(t) = 1 1 − 16α2t2 �sin ((1 − α)πt) πt + 4α cos ((1 + α)πt) π � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' (4) The roll-off factor, 0 ≤ α ≤ 1, however, it is possible to also consider pulse shapes having a roll-off greater than 100%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Since the sequence of symbols, sk, is stationary, the auto-correlation, Rs(i − j) = E � sis∗ j � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The corresponding power-spectrum density (PSD) of signal (1) is computed as, [4] Sx(f) = 1 Ts |P(f)|2 � k Rs(k) ej2πfkTs (5) where P(f) denotes the Fourier transform of p(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Correlative coding assumes the discrete modulator (2) to be a finite impulse response (FIR) filter, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=', sk = K−1 � i=0 vick−i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' (6) PULSE-SHAPE BINARY MULTIPLEX MODULATION 4 The filter weights, vi, are normalized, so that, � i |vi|2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' More importantly, with a change in indices, modulated signal (1) with symbols (6) can be rewritten as, x(t) = � k K−1 � i=0 vi ck−ip(t − kTs) = � k ck K−1 � i=0 vi p(t − (k + i)Ts) = � k ck ˜p(t − kTs) (7) where the compound pulse, ˜p(t) = �K−1 i=0 vi p(t − iTs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Duobinary modulation is a special case of correlative coding, such that the FIR filter has only two non-zero weights, vo = v1 = 1/ √ 2, the modulation symbols are binary, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=', ck ∈ {−√Es, +√Es}, and the RRC pulse has the smallest possible roll-off, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Modified duobinary modulation assumes instead the weights, v0 = 1/ √ 2, v1 = 0, and v2 = −1/ √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The following modulations assume the RRC pulse-shape with an arbitrary roll-off value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Differential PSK constructs the transmitted symbols as, sk = ck sk−1 (8) where the data symbols, ck ∈ {√Es ej2π(i−1)/M}, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' , M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Generalized shift-keying extends the modulation alphabet of amplitude or phase shift-keying modulations with a zero symbol [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Offset-quadrature (M = 4) PSK delays the imaginary part of the modulated signal by half a symbol period, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=', x(t) = � k Re{ck} p(t − kTs) + j Im{ck} p(t − kTs − Ts/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' (9) Finally, FTN signaling is a linear modulation described by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' More importantly, the RRC pulse-shape in (3) can now be scaled by, Tp = Ts/(1 − τ), instead of Ts, where 0 ≤ τ < 1 is so-called the packing factor, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=', x(t) = � k skp(t − k(1 − τ)Tp) = � k skp(t − kTs) (10) so that Tp is a design parameter of the pulse, p(t), whereas, Ts = (1 −τ)Tp, denotes the symbol period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Thus, τ = 0 packing corresponds to a conventional Nyquist signaling, whereas τ = 1 packing would completely overlap the transmitted symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' More importantly, the PSD of (10) is still given by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' (5), and it is otherwise completely independent of the packing factor, τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' PULSE-SHAPE BINARY MULTIPLEX MODULATION 5 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' RECEIVED SIGNAL The standard wireless channel model with L propagation paths is an FIR filter with the impulse response, ˜h(t) = L � l=1 hl(t)δ(t − τl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' (11) The signal delays, τl, are assumed to be constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The path attenuations, hl(t), are zero-mean circularly symmetric Gaussian processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' These processes are stationary, and generally mutually correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' They have a defined auto-correlation, Rh(∆t), which determines the coherence bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' For narrow-band signals, the number of paths, L, is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' For L = 1, the channel model (11) becomes frequency non-selective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' In low-mobility scenarios, the channel attenuations, hl(t), are often assumed to be constant over blocks of transmitted symbols, and independent between the successive blocks, which is often referred to as a block fading model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The received signal corresponding to multi-path propagation model (11) is written as, y(t) = ˜h(t) ⊛ x(t) + w(t) = L � l=1 hl(t)x(t − τl) + w(t) (12) where w(t) is a zero-mean stationary circularly symmetric AWGN with the variance, σ2 w = E[|w(t)|2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The received signal is filtered through a filter matched to the transmitted pulse, p(t), and synchronously sampled at a rate, 1/Ts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' In particular,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' assuming RRC pulses,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' the matched filter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' p∗(−t) = p(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' and provided that the channel attenuations are constant over blocks of transmitted PULSE-SHAPE BINARY MULTIPLEX MODULATION 6 symbols,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' the received samples are modeled as,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' rn =y(t) ⊛ p∗(−t) ��� t=nTs+τ0 = L � l=1 hl x(t − τl) ⊛ p(t) ��� t=nTs+τ0 + w(t) ⊛ p(t) ��� t=nTs+τ0 = L � l=1 hl � k sk � ∞ −∞ p(ζ + (n − k)Ts − τl)p(ζ − τ0) dζ + � ∞ −∞ w(ζ + nTs)p(ζ − τ0) dζ = � k sk L � l=1 hl pn−k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content='l + wn = � k sk˜pn−k + wn =sk˜p0 + � k n̸=k sk˜pn−k � �� � ISI +wn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' (13) The timing offset, τ0, at the receiver can be optimized to minimize the ISI term (in some sense) in (13) defined as, ˜pn−k = L � l=1 hl � ∞ −∞ p(ζ + (n − k)Ts − τl)p(ζ − τ0) dζ, n ̸= k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' (14) Thus, the ISI arises when the orthogonality between the transmitter and the receiver pulses is violated, for example, due to multi-path propagation, time-synchronization errors between transmitter and receiver, and also due to symbol-period compression in FTN signaling schemes [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' An interesting question is how much ISI is produced for different combinations of parameters α and τ in FTN signaling schemes using RRC pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Hence, define the function, ISI(µ) = Card{|˜pk| > µ, k ̸= 0} (15) to be the number of ISI components that are greater than a given threshold, µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Note that, ISI(µ) ∈ {0, 2, 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' }, due to even symmetry of the RRC pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Assuming different thresholds, µ, the roll-off, 0 ≤ α ≤ 2, and the RRC pulses truncated to (−4Ts, +4Ts), the values ISI(µ) = 0 (red points) and ISI(µ) = 2 (blue points) in the α − τ plane are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The empty (white) spaces in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 1 indicate the values, ISI(µ) > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' It can be observed that by decreasing the threshold, µ, several cases of interest for designing FTN signaling schemes start to emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' In particular, exactly two ISI components can be obtained for these parameters: α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content='0 and τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content='5, α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content='07 and τ ∈ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content='70, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content='71), and α ∈ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content='65, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content='85) and τ ∈ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content='47, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content='50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' PULSE-SHAPE BINARY MULTIPLEX MODULATION 7 0 � �� 1 � � 2 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content='01 � �� � � =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content='005 \x0e \x0f\x10 \x11 \x12 \x13 \x14 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content='003 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content='5 1 0 1 2 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content='002 1−τ 1−τ α α Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The ISI(µ) = 0 components (red points) and ISI(µ) = 2 components (blue points) for four different thresholds, µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' PULSE-SHAPE BINARY MULTIPLEX MODULATION As indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 1, the RRC pulse with 100% roll-off and 50% packing has well-defined and finite ISI components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' In particular, the RRC pulse (4) for α = 1 becomes, rrc1(t) = 4 cos(2πt) π(1 − 16t2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' (16) This pulse has the following ISI components in an AWGN channel without multi-path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Such a fundamental property appears to remain unnoticed in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Lemma 1: Let n be a non-negative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The ISI integral involving the RRC pulse, rrc1(t), with 100% roll-off has the exact solution, � ∞ −∞ rrc1(t) × rrc1(t − n/4) dt = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 8/(3π) n = 1 8 π(n−2)n(n+2) n − odd, n > 1 1 n = 0 1/2 n = 2 0 n − even, n > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' (17) PULSE-SHAPE BINARY MULTIPLEX MODULATION 8 n=0 n=2 n=4 1 2 3 4 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content='998 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content='999 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content='001 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content='002 d truncated integral values Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Numerically computed integral (17) truncated to interval, (−d, +d), as a function of d (solid lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The exact values for an infinite interval are shifted to be all equal to unity in order to enable comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The dashed lines are mirrored solid lines about the unit value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Lemma 1 can be proved by solving the integral for the first few values of n (for example, using Mathematica software), and then using induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' However, the result (17) is exact only when the integration is performed over an infinite interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' In practice, the pulse shapes must be truncated to a finite interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The numerically computed values of integral (17) when the interval of integration is truncated to (−d, +d) are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' It can be observed that the RRC pulse shape, rrc1(t), should not be truncated to the intervals shorter than, (−4, +4), in order to achieve the RRC property given in Lemma 1 with at least 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content='9% accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Definition 2: The modulated signal of pulse-shape binary multiplex modulation is written as, x(t) = � k sk rrc1 � t−kTs 2Ts � √2Ts .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' (18) The synchronously sampled matched filter output of modulated signal (18) received in AWGN, w(t), is, rn =(x(t) + w(t)) ⊛ rrc1 � t 2Ts � √2Ts ��� t=nTs = � k sk � ∞ −∞ rrc1(ζ + (n − k)/2) rrc1(ζ) dζ + wn = �1 2sn−1 + sn + 1 2sn+1 � + wn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' (19) PULSE-SHAPE BINARY MULTIPLEX MODULATION 9 a1 a1+a2 2 a3 a2+a3 2 a4 a3+a4 2 a2 Stream #1: Stream #2: b1 b1+b2 2 b3 b2+b3 2 b3+b4 2 b2 T1 T2 T3 T4 T5 T6 T7 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' A visualization of pulse-shape multiplex modulated signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The noise samples, wn, in (19) are zero-mean, have the variance, E[|wn|2] = E[|w(t)|2] = σ2 w, and their stationary auto-correlation is, Rw(n − m) = E[wnw∗ m] = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 σ2 w n = m σ2 w/2 |n − m| = 1 0 |n − m| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' (20) Such noise samples can be equivalently modeled by a simple FIR filter, wn = un + un−1 √ 2 (21) where un are the samples of a zero-mean, circularly symmetric Gaussian process having the variance, E[|un|2] = σ2 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' In addition, it is straightforward to show that the variance of the sum of N noise samples having the correlations (20) is, var � N � n=1 wn � = (2N − 1)σ2 w (22) which is greater than the variance, Nσ2 w, of the sum of N uncorrelated samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The modulated signal (18) in Definition 2 can be visualized as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' In particular, the transmitted data symbols can be viewed as consisting of two multiplexed streams of data symbols, ak, and, bk, which are each transmitted with a period 2Ts, but mutually shifted by Ts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The corresponding received symbol samples after the matched filtering are, rn = \uf8f1 \uf8f2 \uf8f3 an + bn−1+bn 2 + wn n − odd bn + an+an+1 2 + wn n − even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' (23) Using (5), the PSD of modulated signal (18) is computed as, Sx(f) = 2|RRC1(2Tsf)|2 � k E[s0s∗ k] ej2πfkTs (24) where the Fourier transform of the pulse, rrc1(t), is, RRC1(f) = \uf8f1 \uf8f2 \uf8f3 cos(πf/2) |f| ≤ 1 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' (25) PULSE-SHAPE BINARY MULTIPLEX MODULATION 10 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Transmitted Sequence Design The optimum detection of transmitted symbols in the presence of ISI must consider complete sequences of received samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' However, in the absence of multi-path, the received samples have structure (19), and the transmitted sequences can be designed, so that the complexity of detection at the receiver can be reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The key strategy for reducing the detection complexity is to exploit orthogonality among sub-sequences of transmitted symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Offset-quadrature PSK modulation (9) alternates one- dimensional modulation symbols along the in-phase and quadrature components, which allows the optimum symbol-by-symbol decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Multiplexing two data streams as described by (23) can exploit the design principles of superposition modulation and multiuser detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' In such a case, symbol-by-symbol decisions can be performed by SIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Specifically, provided that symbols, bn, can be reliably detected, even if the symbols, (an + an+1)/2, are not yet known, then the symbol, an, can be reliably detected after canceling the ISI term, (bn−1 + bn)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' In the sequel, three other sequence design strategies are discussed in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The first strategy combines pilot and data symbols to aid the data detection and channel estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The second strategy employs orthogonal spreading codes in order to separate the two multiplexed data sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The third strategy adopts the differential encoding of transmitted symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Sequences with Interleaved Pilot Symbols In general, pilot symbols for channel estimation can be interleaved with data symbols or superimposed onto data symbols [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Here, the more common former approach is adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Thus, consider a transmitted sequence consisting of alternating groups of Ld data symbols and Lp ≪ Ld pilot symbols, which are separated by a single zero-symbol as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' For instance, the following sub-sequences with reduced or no ISI can be considered with pilot symbol, p, and arbitrary data symbols, d1, and, d2: (0, p, 0), (d1, p, −d1), (d1, p, −d1, −p, d1), and (d1, p, −d1, −p, d2, p, −d2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' These sub-sequences enable ISI-free data detection and channel estimation, as can be deduced from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' (23) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Recall also that the noise samples, wn, and, wn±2, are uncorrelated, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=', independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' PULSE-SHAPE BINARY MULTIPLEX MODULATION 11 0 pilots 0 data 0 pilots 0 data 0 Lp separator Ld Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The transmitted sequence with interleaved sub-sequences of data and pilot symbols and a single zero-symbol separator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' In order to illustrate the ISI-free channel estimation, consider the sequence, (d1, p, p, −d1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The received samples corresponding to the two pilot symbols in the middle are, rn =h3 2p + h1 2d1 + wn rn+1 =h3 2p − h1 2d1 + wn+1 (26) where h denotes the complex-valued channel attenuation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=', frequency non-selective slow fading).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The samples, rn, and, rn+1, can be simply combined as, rn + rn+1 = 3hp + wn + wn+1 (27) where the total variance of the additive noise samples is equal to 3σ2 w due to correlations (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' More generally, the transmitted sequence, (−p, d1, p, d2, −p, d3, p, d4, −p, d5, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' , dN, ±p) (28) where the last pilot symbol is p, if N is odd, and −p, if N is even, allows the ISI-free symbol- by-symbol decisions of all data symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Moreover, assuming again a slow fading channel, the received samples corresponding to the pilot symbols can be summed up to obtain, N � n=1 (−1)n r2n−1 = N h p + √ Nw where the noise sample, w, has the variance, σ2 w, so the signal-to-noise ratio (SNR) for estimating the channel coefficient, h, has been improved N-times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Note also that once the channel has been estimated, the pilot symbols can be subtracted from the received samples in order to aid decisions of the remaining data symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Finally, consider the case of a symbol repetition diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The transmitted sequence, (d, 0, d, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=', 0, d), of a data symbol, d, repeated (N ≥ 2)-times corresponds to the canonical Nyquist signaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The pulse-shape multiplex modulation instead transmits the sequence, (d, d, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' , d), of N1-times repeated data symbol, d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' For the same sequence length, N1 = 2N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Assuming slowly PULSE-SHAPE BINARY MULTIPLEX MODULATION 12 fading channel, the detector combines the received samples for the two modulation schemes, respectively, as, r =N h d + √ Nw r =(2 · 3/2 + 2(N1 − 2)) h d/ √ 2 + � 2N1 − 1w (29) where the scaling by √ 2 was introduced for the second modulation in order to account for the larger number of symbols in its transmitted sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The resulting SNR of these two schemes is proportional to, γ ∝ N, and, γ ∝ 2N −3/2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Consequently, for symbol repetition diversity, the SNR gain of the pulse-shape binary multiplexing is asymptotically 3 dB larger than for the Nyquist signaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Sequences with Orthogonal Spreading Another strategy for transmitting interleaved, but orthogonal symbols in modulated signal (18) is to use orthogonal spreading codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' In particular, assume transmitted symbols, an = d1c(1) n , bn = d2c(2) n (30) where d1 and d2 are two data symbols, and, c(1) n and c(2) n , n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' , N, are generally complex- valued, orthogonal spreading sequences, so that, N � n=1 c(i) n c∗(j) n = \uf8f1 \uf8f2 \uf8f3 N, i = j 0, i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' (31) Then, the sequences of received samples (23) are linearly combined as, N � n=1 r2n−1c∗(1) n =d1 + d2 N � n=1 c(2) n + c(2) n+1 2 c∗(1) n + N � n=1 w2n−1c∗(1) n =d1 + ˜w1 N � n=1 r2nc∗(2) n =d2 + d1 N � n=1 c(1) n + c(1) n+1 2 c∗(2) n + N � n=1 w2nc∗(2) n =d2 + ˜w2 (32) provided that the spreading sequences, c(1) n , and, c(2) n , are exactly orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' In such a case, the SNR improvement for transmitting two data symbols with orthogonal spreading sequences using the pulse-shape binary multiplex modulation (18) is proportional to, γ ∝ N2 2N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' (33) PULSE-SHAPE BINARY MULTIPLEX MODULATION 13 0 50 100 150 200 250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content='9 1 r=10% r=5% N Probability Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The probability (35) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' the spreading sequence length, N, assuming κ = 5% and κ = 10%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' For instance, if the spreading symbols, cn, are generated independently at random and with an equal probability from the set, {−1, +1}, the probability that two such sequences are orthogonal is, Pr � N � n=1 c(1) n c∗(2) n = 0 � = � N N/2 � �1 2 �N/2 �1 2 �N−N/2 = � N N/2 � 2−N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' (34) Since the probability (34) of exact orthogonality asymptotically goes to zero with large N, consider instead the probability, Pr � −⌈κ N/2⌋ ≤ N � n=1 c(1) n c∗(2) n ≤ ⌈κ N/2⌋ � = ⌈κ N/2⌋ � n=−⌈κ N/2⌋ �N n � �1 2 �N (35) for some small κ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The probabilities (35) as a function of N for two different values of factor, κ, are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' These probabilities are indicative of how many random spreading sequences need to be generated in order to select the required number of such sequences having an acceptable level of mutual orthogonality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' PULSE-SHAPE BINARY MULTIPLEX MODULATION 14 c0 c0 2 (1 + c1) c0 2 (1 + c1c2 + 2c1) c0 2 (1 + c0c1 + 2c0) c0 2 (1 + c1c2) c0c1 c0c1 2 (1 + c2c3 + 2c2) c0c1 2 (1 + c2c3) c0c1c2 c0c1c2 2 (1 + c3c4 + 2c3) c0c1c2c3 c0c1c2 2 (1 + c3c4) c0 c0c1c2c3c4 T1 T0 T2 T3 T5 T4 Stream #1: Stream #2: Tx symbols: Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Differentially encoded M-ary PSK symbols transmitted via pulse-shape binary multiplex modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Sequences with Differential Encoding Differential PSK is a popular modulation scheme for fast fading channels, since it alleviates the need for recovering the absolute phase reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 6 shows differentially encoded M-ary PSK symbols (8) transmitted via pulse-shape binary multiplex modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' In particular, the n-th transmitted symbol is, sn = �n−2 � k=0 ck � 1 + cn−1cn + 2cn−1 2 =1 2 �n−2 � k=0 ck � + 1 2 �n−1 � k=0 ck � cn + �n−1 � k=0 ck � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' (36) Consequently, the differential decoding can be performed as, cn = � 2sn − �n−2 � k=0 ck � − 2 �n−1 � k=0 ck �� �n−1 � k=0 ck �∗ =2sn �n−1 � k=0 c∗ k � − c∗ n−1 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' (37) The performance of this modulation scheme is evaluated in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' NUMERICAL EXAMPLES It is convenient to use a vector notation to generate samples of pulse-shape binary multiplex modulation (18) received over a frequency non-selective fading channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The vector, r, of N received samples corresponding to the vector, s, of N transmitted symbols can be obtained as, r = s A diag(h) + w AT 0 PULSE-SHAPE BINARY MULTIPLEX MODULATION 15 where h is a vector of fading channel coefficients, w are samples of AWGN, and the (N × N) ISI matrix, A = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 1 1/2 1/2 1 1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 1/2 1 \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb = A0 AT 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The optimum detection requires that the additive noise is first whitened as, [4] r A−T 0 = s A diag(h) A−T 0 + w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Then the maximum likelihood (ML) detection of sequence s is, ˆs = arg min s ���r A−T 0 − s A diag � ˆh � A−T 0 ��� 2 (38) where ˆh is the estimate of h representing channel state information (CSI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' An uncoded binary phase shift keying (BPSK) modulation and Rayleigh-distributed fading amplitudes, h, are assumed for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The transmitted sequence interleaves pilot symbols and data symbols as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The pilot symbols are used to estimate the channel coefficients, h, by linear minimum mean-square error (LMMSE) algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The spectral efficiency of pulse- shape binary multiplexing is, 2, which is always larger than the spectral efficiency of the Nyquist signaling being equal to, 2/(1 + α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The BER curves, Pe, for short data sequences of Ld = 4 and Ld = 8 binary symbols, respectively, separated by a single zero-symbol are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 7 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The SNR is defined as, γb = 1/(2σ2 w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Both cases of perfect and estimated CSI are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The Nyquist signaling (no ISI) with symbol-by-symbol decisions is assumed as a reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The ML data detector (38) is used for pulse-shape multiplexing signaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' It can be observed that the performance penalty due to channel estimation is much larger for pulse-shape multiplexing than for the Nyquist signaling, which is to be expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The WMF improves the performance by several dB’s for both signaling schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' More importantly, the performance of pulse-shape multiplexing improves with the data block length by exploiting the time diversity over a fading channel, so it can significantly outperform the Nyquist signaling at medium to large SNR values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' It is likely that by employing more sophisticated channel estimation and equalization techniques, the performance of pulse-shape multiplexing can be further improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' In order to demonstrate the effect of time diversity, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 9 shows that, over PULSE-SHAPE BINARY MULTIPLEX MODULATION 16 0 5 10 15 20 10-5 10-4 10-3 10-2 10-1 100 estim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' CSI perfect CSI No-ISI ISI estim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' CSI perfect CSI ML WMF-ML γb [dB] Pe LLLddd === 444 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The BER of BPSK vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' SNR over Rayleigh fading channel for sequences of 4 binary symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' an AWGN channel, the performance of pulse-shape multiplexing is worse than that of Nyquist signaling, even though some performance loss can be recovered by WMF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Lastly, the BER performance of Nyquist modulation and pulse-shape multiplex modulation transmitting differentially encoded quadrature PSK (QPSK) symbols over an AWGN channel is compared in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' It can be observed that even though the pulse-shape multiplexing suffers asymptotically a 3 dB penalty in SNR, it reduces the time required for transmitting the whole symbol sequence to one half.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' CONCLUSION The paper introduced a pulse-shape binary multiplex modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Such a modulation scheme is akin to partial-response signaling, correlative coding, offset-QPSK modulation and FTN signaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' It combines two data streams under controlled ISI created by the RRC pulses having 100% roll-off, and transmitted at twice the Nyquist rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The ISI analysis showed that this is unique property among all the roll-off factors being at most 100% and the packing factors greater than 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' However, the successive samples of additive noises at the output of matched filter at PULSE-SHAPE BINARY MULTIPLEX MODULATION 17 0 5 10 15 20 10-5 10-4 10-3 10-2 10-1 100 estim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' CSI perfect CSI No-ISI ISI estim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' CSI perfect CSI ML WMF-ML γb [dB] Pe LLLddd === 888 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The BER of BPSK vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' SNR over Rayleigh fading channel for sequences of 8 binary symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' the receiver become correlated, which incurs a SNR performance penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' This penalty could be reduced or even removed by using more complex sequence-based detection schemes as shown elsewhere in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The BER performance as well as decoding complexity of the proposed pulse-shape binary multiplexing modulation scheme is critically affected by the choice of transmitted sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' One can consider superposition modulation with SIC decoding, interleave data symbols with pilot and zero-symbols to aid channel estimation and data decoding, and also employ orthogonal spreading sequences to separate the multiplexed data streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The numerical results indicate that pulse-shape binary multiplexing can exploit time-diversity in fading channels to outperform the Nyquist signaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' In addition, it has been shown numerically that a sequence of differentially encoded PSK symbols can be transmitted twice as fast by the proposed modulation scheme compared to canonical Nyquist signaling, although with a 3 dB SNR penalty over AWGN channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' PULSE-SHAPE BINARY MULTIPLEX MODULATION 18 4 5 6 7 8 9 10 11 10 -5 N=2 ML WHF-ML BPSK 4 5 6 7 8 9 10 11 10-5 N=4 γb [dB] Pe Pe Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The BER of BPSK vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' SNR over AWGN channel for sequences of 2 and 4 binary symbols, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 6 8 10 12 14 16 18 10-6 10-5 10-4 10-3 10-2 DQPSK DQPSK-PSBM γb [dB] Pe Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' The BER comparison of differentially encoded QPSK with Nyquist and pulse-shape binary multiplexing (PSBM) modulation transmitted over an AWGN channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' PULSE-SHAPE BINARY MULTIPLEX MODULATION 19 REFERENCES [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Zhou, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Li, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Wang, “Generalized Faster-Than-Nyquist signaling,” in ISIT, 2012, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 1478–1482.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' [2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Anderson, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Rusek, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Öwall, “Faster-Than-Nyquist signaling,” Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' of the IEEE, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 101, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 8, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 1817–1830, August 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Liveris and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Georghiades, “Exploiting Faster-Than-Nyquist signaling,” IEEE Transactions Communications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 51, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 9, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 1502–1511, September 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' [4] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Proakis and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Salehi, Digital Communications, 5th ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' McGraw-Hill Education, NY, USA, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' [5] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Landau, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Dörpinghaus, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Fettweis, “1-bit quantization and oversampling at receiver: Communication over bandlimited channels with noise,” IEEE Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Letters, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 21, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 1007–1010, May 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' [6] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Modenini, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Colavolpe, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Alagha, “How to significantly improve the spectral efficiency of linear modulations through time-frequency packing and advanced processing,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' ICC, 2012, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 3260–3264.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' [7] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Fan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Guo, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Zhou, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Ren, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Li, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Chen, “Faster-Than-Nyquist signaling: An overview,” IEEE Access, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 1925–1940, February 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' [8] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Yamada, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Sawahashi, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Saito, “Performance of time and frequency compression of Faster-than-Nyquist signaling in frequency-selective fading channels,” in APCC, 2015, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 550–554.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' [9] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Bogale, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Le, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Wang, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Vandendorpe, “Multipath multiplexing for capacity enhancement in SIMO wireless systems,” IEEE Transactions Wireless Communications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 16, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 10, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 6895–6911, October 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' [10] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Huang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Zhang, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Guo, “Dual pulse shaping transmission and equalization for high-speed wideband wireless communication systems,” IEEE Transactions on Circuits and Systems I, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 67, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 1549–8328, July 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' [11] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Huang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Zhang, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Cheng, “Dual pulse shaping transmission with sinc-function based complementary Nyquist pulses,” IET Communications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 16, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 17, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 2091–2104, October 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' [12] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Bedeer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Ahmed, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Yanikomeroglu, “A very low complexity successive symbol-by-symbol sequence estimator for Faster-Than-Nyquist signaling,” IEEE Access, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 7414–7422, June 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' [13] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Bas and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Dowhuszko, “Linear time-packing detectors for optical feeder link in high throughput satellite systems,” in GC-ElecEng, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 21–26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' [14] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Shi, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Wu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Ma, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Wang, “Frequency-domain joint channel estimation and decoding for Faster-Than-Nyquist signaling,” IEEE Transactions Communications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 66, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 781–795, February 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' [15] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Wu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Yuan, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Guo, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Kuang, “A hybrid BP-EP-VMP approach to joint channel estimation and decoding for FTN signaling over frequency selective fading channels,” IEEE Access, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 6849–6858, May 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' [16] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Loskot, “A generalized FSK-based PHY layer design for wireless sensor networks,” in Chinacom, 2012, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 362–367.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' [17] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Jagannatham and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' Rao, “Superimposed pilots vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' conventional pilots for channel estimation,” in ACSSC, 2006, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} +page_content=' 767–771.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAzT4oBgHgl3EQfh_1H/content/2301.01492v1.pdf'} diff --git a/D9E4T4oBgHgl3EQf6Q7Z/content/tmp_files/2301.05331v1.pdf.txt b/D9E4T4oBgHgl3EQf6Q7Z/content/tmp_files/2301.05331v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..827b7e5ba14706fe840bcc261383784610ce1cf0 --- /dev/null +++ b/D9E4T4oBgHgl3EQf6Q7Z/content/tmp_files/2301.05331v1.pdf.txt @@ -0,0 +1,7877 @@ +Detection problems in the spiked matrix models +Ji Hyung Jung∗, Hye Won Chung†, and Ji Oon Lee‡ +January 16, 2023 +Abstract +We study the statistical decision process of detecting the low-rank signal from various signal-plus- +noise type data matrices, known as the spiked random matrix models. We first show that the principal +component analysis can be improved by entrywise pre-transforming the data matrix if the noise is non- +Gaussian, generalizing the known results for the spiked random matrix models with rank-1 signals. As +an intermediate step, we find out sharp phase transition thresholds for the extreme eigenvalues of spiked +random matrices, which generalize the Baik-Ben Arous-P´ech´e (BBP) transition. +We also prove the +central limit theorem for the linear spectral statistics for the spiked random matrices and propose a +hypothesis test based on it, which does not depend on the distribution of the signal or the noise. When +the noise is non-Gaussian noise, the test can be improved with an entrywise transformation to the data +matrix with additive noise. We also introduce an algorithm that estimates the rank of the signal when +it is not known a priori. +1 +Introduction +One of the most natural approach to ‘signal-plus-noise’ type data is to consider spiked random matrices, +which are the low-rank deformation of large random matrices. Most notable examples of spiked random +matrices include spiked Wigner matrix and spiked Wishart matrix, where the signals are given as a low-rank +mean matrix (spiked Wigner matrix) and a low-rank perturbation of the identity in its covariance matrix +(spiked Wishart matrix). In this paper, we focus on the following three types of noisy data matrices, known +as spiked random matrices, which generalize spiked Wigner/Wishart matrices: +• Spiked Wigner matrix: the data matrix is of the form +UΛ1/2U T + W, +(1.1) +where U = [u(1), u(2), . . . , u(k)] ∈ RN×k with U T U = Ik, and W is an N × N Wigner matrix. The +signal-to-noise ratio (SNR) Λ = diag(λ1, λ2, . . . , λk) with λ1 ≥ λ2 ≥ . . . λk > 0 for some positive integer +k, independent of N. +∗Department of Mathematical Sciences, KAIST, Daejeon, 34141, Korea +email: jhjung66@kaist.ac.kr +†School of Electrical Engineering, KAIST, Daejeon, 34141, Korea +email: hwchung@kaist.ac.kr +‡Department of Mathematical Sciences, KAIST, Daejeon, 34141, and School of Mathematics, KIAS, Seoul, 02455, Korea +email: jioon.lee@kaist.edu +1 +arXiv:2301.05331v1 [math.ST] 12 Jan 2023 + +• Rectangular matrix with spiked mean (additive model): the data matrix is of the form +UΛ1/2V T + X, +(1.2) +where U = [u(1), u(2), . . . , u(k)] ∈ RM×k, V = [v(1), v(2), . . . , v(k)] ∈ RN×k with U T U = V T V = +Ik, and X is an M × N random i.i.d. matrix whose entries are centered with variance N −1. The SNR +Λ is given as in the spiked Wigner matrix. +• Rectangular matrix with spiked covariance (multiplicative model): the data matrix is of the form +(I + UΛU T )1/2X, +(1.3) +where U = [u(1), u(2), . . . , u(k)] with U T U = Ik and X is an M × N random i.i.d. matrix whose +entries are centered with variance N −1. The SNR Λ is given as in the spiked Wigner matrix. +Here, Ik is the identity matrix with rank k and we allow the case k = 0 where no signal is present. Throughout +the paper, for the ease of notation, we denote by W an N × N Wigner matrix, and X an M × N random +i.i.d. matrix. +To describe the detection problems we consider in this paper, we first review the known results for the +simplest case of the spiked random matrix models with rank-1 spike, i.e, k = 1 in (1.1), (1.2), and (1.3). +Signal detection problem in rank-1 spiked random matrices: Many problems concerning the +signal detection can be answered in the case with Gaussian noise and rank-1 spike. In this case, the spikes +U = u and V = v are vectors, and SNR λ1 = λ, hence the spiked random matrices are of the following +forms: +√ +λuuT + W +(1.4) +√ +λuvT + X +(1.5) +(I + λuuT )1/2X. +(1.6) +For this case, reliable detection of the signal, i.e., detection with probability 1 − o(1) as M, N → ∞, is +impossible if the SNR λ is below a certain threshold [44, 47]. The threshold is 1 as N → ∞ for spiked +Wigner matrices; for spiked rectangular matrices, with additional assumption M/N → d0 as N → ∞, the +threshold is √d0 for a general class of priors [50]. On the other hand, the signal can be reliably detected +by the principal component analysis (PCA) if the SNR is above the threshold in which case the signal can +actually be estimated [24, 41, 43]. +In the subcritical case where the signal is not reliably detectable, it is natural to consider a hypothesis +test on the presence of the signal between H0 : λ = 0 and H1 : λ = ω, commonly referred to as the +weak detection, which is also known as the sphericity test in the case the spike is drawn from the uniform +distribution on the unit sphere, known as the spherical prior. As asserted by Neyman–Pearson lemma, the +likelihood ratio (LR) test is optimal in the sense that it minimizes the sum of the Type-I error and the +Type-II error. It was proved for several distributions of the spikes, called priors, that this sum for a spiked +Wigner matrix converges to +erfc +�1 +4 +� +− log (1 − λ) +� +(1.7) +when H is Gaussian Orthogonal Ensemble (GOE), and for a spiked Wishart matrix +erfc +� +1 +4 +� +− log +� +1 − λ2 +d0 +�� +(1.8) +2 + +when XXT is a Wishart Ensemble; see, e.g., [47, 29, 28]. Here, erfc(·) is the complementary error function +defined as +erfc(x) = +� ∞ +x +e−t2dt. +(1.9) +Though optimal, the LR test is not efficient, and it is desirable to construct a test that does not depend +on information about the prior, which is typically not known in many practical applications. In [22], an +optimal and universal test for spiked Wigner matrices was proposed, which is based on the linear spectral +statistics (LSS) of the data matrix, a linear functional defined as +LN(f) = +N +� +i=1 +f(µi) +(1.10) +for a given function f, where µ1, · · · µN are the eigenvalues of the data matrix. The test is extended to spiked +rectangular matrices in [34], where the singular values of the data matrix is used instead of the eigenvalues. +If the noise is non-Gaussian, it is possible to improve the PCA by transforming the data matrix entrywise +for spiked Wigner matrices [42, 50] and for spiked rectangular matrices [34]. In this improved PCA, the +threshold is lowered by a certain factor that depends on the Fisher information of the noise distribution. +Below this threshold, the LSS-based test proposed in [22] for spiked Wigner matrices can also be improved by +applying the entrywise transformation for the improved PCA. It is not known whether the reliable detection +is impossible below the threshold except for the case of the spiked Wigner matrix with Rademacher prior +[21]. +Spiked random matrices with general rank: The more relevant structure for application is that +the latent signal contains multiple spikes, or a spike with a higher rank. For such models of spiked random +matrices, similar to the cases with rank-1 spikes, it is natural to ask the following questions: +• What is the spectral threshold for a reliable detection lower than the existing one for Gaussian noise +if the noise is non-Gaussian? +• Can we design an efficient algorithm to weakly detect the presence of signal (i.e., better than a random +guess) when a reliable detection is not feasible? +Contrary to the rank-1 spike case, the questions addressed above have never been answered, even for the +simplest case of Gaussian noise. Furthermore, for the spikes with general rank, we need to consider another +important problem of finding the rank of the spike in case it is not known a priori. While viable solutions to +resolve the issue in the context of the community detection were suggested in [40, 16] for any spiked Wigner +matrices and [49, 25] for spiked rectangular matrices, these methods are not applicable in the sub-critical +case. To the best of our knowledge, there are no spectral algorithms for estimating the rank of signal in the +sub-critical regime. We thus aim to the following question as well: +• Can we design an efficient algorithm to estimate the rank of signal when a reliable detection is not +feasible? +Main contributions +Our main contributions are mainly divided into three parts as follows: +• (Strong detection) We prove that the PCA can be improved by an entrywise transformation if the +noise is non-Gaussian, under a mild assumption on the distribution (prior) of the spike. +3 + +• (Weak detection I) We propose a universal test to detect the presence of signal with low computational +complexity, based on the linear spectral statistics (LSS). The test does not require any prior information +on the signal, and if the noise is Gaussian the error of the proposed test is optimal. For the spiked +Wigner matrix or the additive model of the spiked rectangular matrix with the non-Gaussian noise, +we suggest an improved test via an entrywise transformation. +• (Weak detection II) We present an LSS-based test for estimating the rank of a signal when Λ = λI. +Heuristically, it is possible to increase the SNR via an entrywise transformation. Here, we illustrate the +main idea of the entrywise transformation for the spiked Wigner matrix of the form M = UΛ1/2U T + W. +If |uiuT +j |, |uivT +j | ≪ Wij, then by applying a function q entrywise to +√ +NY , we obtain a transformed matrix +whose entries are +q( +√ +NMij) = q( +√ +NWij + +√ +NuiΛ1/2uT +j ) ≈ q( +√ +NWij) + +√ +Nq′( +√ +NWij)uiΛ1/2uT +j , +where ui and vi denote i−th row vector of the signal matrix U and V , respectively. With negligible error, it +is possible to approximate the coefficient q′( +√ +NWij) in the second term in the right side by its expectation. +(See Appendix B.3 for the proof) Then, +q( +√ +NMij) = q( +√ +NWij + +√ +NuiΛ1/2uT +j ) ≈ +√ +N +� +q( +√ +NWij) +√ +N ++ E[q′( +√ +NWij)]uiΛ1/2uT +j +� +and the transformed matrix is approximately of the form U(Λ′)1/2U T + Q after a proper normalization, +which becomes another spiked Wigner matrix with different SNR. By optimizing the transformation q, we +find that the SNR is effectively increased (or equivalently, the threshold √d0 is lowered) in the PCA for the +transformed matrix. The change of the threshold and a BBP-type transition for the largest eigenvalues of +the transformed matrix can be rigorously proved; see Theorem 3.3 for a precise statement. We remark that +the same idea works even if the SNR Λ is not a constant multiple of an identity matrix, and also a similar +result holds for the additive model of spiked rectangular matrix (Theorem 3.4). +For the multiplicative model of the form Y = (I + UΛU T )1/2X =: (I + UΓU T )X, with Λ = 2Γ + Γ2, +the analysis is significantly more involved due to the following reason: Applying a function q entrywise to +√ +NY , we find that +q( +√ +NYij) = q +�√ +NXij + +√ +N +� +ℓ +uiΓuT +ℓ Xℓj +� +≈ q( +√ +NXij) + +√ +Nq′( +√ +NXij) +� +ℓ +uiΓuT +ℓ Xℓj +≈ +√ +N +� +q( +√ +NXij) +√ +N ++ E[q′( +√ +NXij)] +� +ℓ +uiΓuT +ℓ Xℓj +� +, +and the transformed matrix is of the form UΓ′U T X +Q, which is not a spiked rectangular matrix anymore. +Note that Q depends on X entrywise and thus it cannot be considered as an additive model, either. +In Theorem 3.5, we prove the effective change of the SNR and the BBP-type transition for the multi- +plicative model. The proof of Theorem 3.5 is based on a generalized version of the BBP transition that +works with the matrix of the form UΓU T X +Q. We remark that the strategy for the proof, based on recent +development of random matrix theory, can also be applied to prove a BBP-type transition for other models. +As in the rank-1 case in [34], it is notable that the optimal entrywise transform for the multiplicative +model is different from the one for the additive model. For the spiked Wigner matrix, the optimal transforms +are given by −g′/g for the off-diagonal entries (and −g′ +d/gd for the diagonal entries), where g (and gd for +the diagonal entries) is the density functions of them; the optimal transform for the additive model is also +4 + +given by −g′/g. However, for the multiplicative model, the optimal transform is a linear combination of +the function −g′/g and the identity mapping. Heuristically, it is due to that the effective SNRs depend not +only on Γ′ but also on the correlation between X and Q; the former is maximized when the transform is +−g′/g while the latter is maximized when the transform is the identity mapping. We also remark that the +effective SNRs after the optimal entrywise transform is larger in the additive model, which suggests that the +detection problem is fundamentally harder for the multiplicative model. +With the BBP-type transition for the largest eigenvalues of the transformed matrices, it is also possible +to improve the performance of several statistical inferences [13, 35, 46]. One of the consequences is that the +corresponding eigenspace is adjacent to its true spike U in the sense of direction of arrival (DoA) [23]. In +other words, we can not only reliably estimate the number of spikes by parallel analysis (PA) [27], but also +approximately recover the true spikes and the corresponding SNRs. +For the subcritical case where it is impossible to reliably detect the signal by the improved PCA, we +propose algorithms for weak detection, based on the central limit theorem (CLT) of the LSS, Theorems 5.2, +5.3, 5.5, and 5.6, analogous to the ones introduced in [22]. More precisely, assuming the SNRs are uniform +i.e., Λ = λI, we propose an algorithm for a hypothesis test between +Hk1 : k = k1, +Hk2 : k = k2 +(1.11) +for non-negative integers k1 < k2. While it may seem obvious, it has not been even known in the simple +case k1 = 0 whether the detection becomes easier as k2 increases. Our test in Algorithm 2 verifies the claim +since the error of the proposed test is an increasing function of (k2 − k1) as in Theorem 4.2. As in [22], the +proposed tests are universal, and the various quantities in it can be estimated from the observed data. The +test can further be improved by applying the same entrywise transformation we used for the PCA (Algorithm +3) if the data matrix is of additive type (spiked Wigner matrix or rectangular matrix with spiked mean), +and it also can be adapted to the rank detection problem where we need to estimate the rank k of the signal +without knowing the candidates k1 and k2 a priori (Algorithm 4). +The main mathematical achievement of the second part is the CLT for the LSS of spiked random matrices +with general ranks. +For a rank-1 spiked Wigner matrix, the CLT was first proved for a special spike +1 +√ +N (1, 1, . . . , 1)T in [9] and later extended for a general rank-1 spike by comparison with the special case +[22]. However, the proof in [9] is not readily extended to the spiked Wigner matrices with higher ranks and the +spiked rectangular matrices. In this paper, we overcome the difficulty by introducing a direct interpolation +between the spiked random matrix and the corresponding pure noise matrix and tracking the change of the +LSS. Furthermore, we will prove that the proposed entrywise transformation for the data matrix of additive +type also effectively changes the SNR, and that the LSS of the transformed matrix is also asymptotically +Gaussian; this result was proved previously only for rank-1 spiked Wigner matrices in [22]. Thus, the error +from the proposed test decreases after the transformation as for spiked Wigner matrices in [22]. +Related works +Spiked random matrix models were first introduced by Johnstone [31]. The model can be applied to various +problems such as community detection [1] and submatrix localization [19]. The transition of the largest +eigenvalue was proved by Baik, Ben Arous, and P´ech´e [7] for spiked complex Wishart matrices and generalized +by Benaych-Georges and Nadakuditi [14, 15]. For more results from random matrix theory about the extreme +eigenvalues and the corresponding eigenvectors of spiked random matrices, we refer to [18] and references +therein. +The improved PCA based on the entrywise transformation was considered for rank-1 spiked Wigner +matrices in [42, 50], where the transformation is chosen to maximize the effective SNR of the transformed +5 + +matrix. Detection problems for rank-1 spiked Wigner matrices were also considered, where the analysis is +typically easier due to its symmetry and canonical connection with spin glass models. For more results on +the rank-1 spiked Wigner matrices, we refer to [44, 50, 29, 22] and references therein. +The testing problem for rank-1 spiked Wishart matrices with the spherical prior was considered by +Onatski, Moreira, and Hallin [47, 48], where they proved the optimal error of the hypothesis test. It is later +extended to the case where the entries of the spikes are i.i.d. with bounded support (i.i.d. prior) by El +Alaoui and Jordan [28]. See also [32, 44, 24, 41, 43, 12] for more about detection limits in statistical learning +theory. +Models with sparse or generative structure of the spike have extensively studied in the past literature. +Various statistical and algorithmic methods are applicable to the case where SNR is smaller than the spectral +threshold. In particular, it can be seen that the sparsity of the spikes and the dimension of the latent vector +constituting the generative spike prior actually serve to lower the threshold for the SNR to which several +algorithms are applicable; see [5, 20] and references therein. +Organization of the paper +The rest of the paper is organized as follows: In Section 2, we introduce the precise definitions of models +and relevant previous consequences. In Section 3, we state our results on the improved PCA. In Section 4, +we propose algorithms for LSS-based tests and a test for rank estimation, and analyze their performance. +In Section 5, we state general results on the CLT for the LSS. We conclude the paper in Section 6 with +the summary of our works and future research directions. In Appendix A, we consider examples of spiked +random matrices and provide results from numerical experiments. +In Appendices B and C, we provide +technical details of the proofs. +2 +Preliminaries +In this section, we introduce the precise definition of the models and previous results for the spiked random +matrices. +2.1 +Definitions of models +The noise matrices are defined as follows: +Definition 2.1 (Wigner matrix). An N ×N symmetric matrix W = (Wij) is a (real) Wigner matrix if Wij +(i, j = 1, 2, . . . , N) are independent real random variables such that +• For all i < j, NE[W 2 +ij] = 1, N +3 +2 E[W 3 +ij] = w3, and N 2E[W 4 +ij] = w4 for some w3, w4 ∈ R. +• For all i, NE[W 2 +ii] = w2 for some constant w2 ≥ 0. +• For any positive integer p, there exists Cp, independent of N, such that N +p +2 E[W p +ij] ≤ Cp for all i ≤ j. +Definition 2.2 (Random rectangular matrix). An M × N matrix X = (Xij) is a (real) random rectangular +matrix if Xij (1 ≤ i ≤ M, 1 ≤ j ≤ N) are independent real random variables such that +• For all i, j, E[Xij] = 0, NE[X2 +ij] = 1, N +3 +2 E[X3 +ij] = w3, and N 2E[X4 +ij] = w4 for some constants w3, w4. +• For any positive integer p, there exists Cp, independent of N, such that N +p +2 E[Xp +ij] ≤ Cp for all i, j. +The spiked random matrices are defined as follows: +6 + +Definition 2.3 (Spiked Wigner matrix). An N × N matrix M = UΛ1/2U T + W is a spiked Wigner matrix +with the SNR (matrix) Λ if W is a Wigner matrix and the spike U = [u(1), u(2), . . . , u(k)] ∈ RN×k with +U T U = Ik. +Definition 2.4 (Spiked rectangular matrix - additive model). An M ×N random matrix Y = UΛ1/2V T +X +is a rectangular matrix with spiked mean U, V and the SNR (matrix) Λ if X is a random rectangular matrix +and the spikes U = [u(1), u(2), . . . , u(k)] ∈ RM×k, V = [v(1), v(2), . . . , v(k)] ∈ RN×k with U T U = V T V = +Ik. +Definition 2.5 (Spiked rectangular matrix - multiplicative model). An M × N random matrix Y = (I + +UΛU T )1/2X is a rectangular matrix with spiked covariance U and the SNR (matrix) Λ if X is a rectangular +matrix and U = [u(1), u(2), . . . , u(k)] ∈ RM×k with U T U = Ik. +We assume throughout the paper that the SNR matrix Λ is a k × k diagonal matrices with Λii = λi and +λ1 ≥ λ2 ≥ . . . λk ≥ 0, and M +N → d0 ∈ (0, ∞) as M, N → ∞. +2.2 +Principal component analysis +Here are the results for principal components of spiked models in the context of random matrix theory. +Spiked Wigner matrix +Let M be the spiked Wigner matrix. +The empirical spectral measure of M converges to the Wigner’s +semicircle law µsc, i.e., if we denote by µ1 ≥ µ2 ≥ · · · ≥ µN the eigenvalues of M, then +1 +N +N +� +i=1 +δµi(x)dx → dµsc(x) +(2.1) +weakly in probability as N → ∞, where +dµsc(x) = +√ +4 − x2 +2π +1(−2,2)(x)dx. +(2.2) +The k largest eigenvalue has the following (almost sure) limit: for 1 ≤ i ≤ k +• If λi > 1, then µi → √λi + +1 +√λi . +• If λi < 1, then µi → 2. +Sample covariance matrix +Let S = Y Y T be the sample covariance matrix (Gram matrix) derived from a spiked rectangular matrix +Y . The empirical spectral measure of S converges to the Marchenko–Pastur law µMP , i.e., if we denote by +µ1 ≥ µ2 ≥ · · · ≥ µM the eigenvalues of S, then +1 +M +M +� +i=1 +δµi(x)dx → dµMP (x) +(2.3) +weakly in probability as M, N → ∞, where for M ≤ N +dµMP (x) = +� +(x − d−)(d+ − x) +2πd0x +1(d−,d+)(x)dx, +(2.4) +7 + +with d± = (1 ± √d0)2. The k largest eigenvalue has the following (almost sure) limit: for 1 ≤ i ≤ k +• If λi > √d0, then µi → (1 + λi)(1 + d0 +λi ). +• If λi < √d0, then µi → d+ = (1 + √d0)2. +This in particular shows that the detection can be reliably done by PCA if λ > √d0. We remark that the +results above hold for both the additive model and the multiplicative model. +2.3 +Linear spectral statistics +We introduce the central limit theorems for null models. +Spiked Wigner matrix +The proof of the Gaussian convergence of the LR in [8, 10] is based on the recent study of linear spectral +statistics, defined as +LY (f) = +N +� +i=1 +f(µi) +(2.5) +for a function f, where µ1 ≥ µ2 ≥ . . . µN are the eigenvalues of M. As the Wigner’s semicircle law in (2.1) +suggests, it is required to consider the fluctuation of the LSS about +N +� 2 +−2 +f(x) dµsc(x). +The CLT for the LSS is the statement +� +LM(f) − N +� 2 +−2 +f(x) dµsc(x) +� +⇒ N(mM(f), VM(f)), +(2.6) +where the right-hand side is the Gaussian random variable with the mean mM(f) and the variance VM(f). +The CLT was proved for the null case (λ = 0). We will show that the CLT also holds under the alternative +and the mean mM(f) depends on λ while the variance VM(f) does not. +Spiked rectangular matrices +The LSS for the spiked rectangular matrices defined as +LY (f) = +M +� +i=1 +f(µi) +(2.7) +for a function f, where µ1 ≥ µ2 ≥ . . . µM are the eigenvalues of S = Y Y T . As the Marchenko–Pastur law in +(2.3) suggests, it is required to consider the fluctuation of the LSS about +M +� d+ +d− +f(x) dµMP (x). +The CLT for the LSS is the statement +� +LY (f) − M +� d+ +d− +f(x) dµMP (x) +� +⇒ N(mY (f), VY (f)), +(2.8) +8 + +where the right-hand side is the Gaussian random variable with the mean mY (f) and the variance VY (f). +The CLT was proved for the null case (λ = 0). We will show that the CLT also holds under the alternative +and the mean mY (f) depends on λ while the variance VY (f) does not. +3 +Main result I - Improved PCA +In this section, we state our first main results on the improvement of PCA by entrywise transformations and +provide the results from numerical experiments. +3.1 +Improved PCA +We introduce the following assumptions for the spike and the noise. +Assumption 3.1. For the spike U (and also V in the additive model), we assume, for φ ≤ 1/2, +1. the spikes are φ-localized with high probability, i.e. ∥U∥∞, ∥V ∥∞ ≺ N −φ +2. the spike matrix is φ-orthonormal with high probability, i.e. ∥U T U − Ik∥F , ∥V T V − Ik∥F ≺ N −φ, +and so the spikes are sampled from Stiefel manifold of orthonormal k-frames in RM or RN with high +probability. +For the noise, let P be the distribution of the normalized entries +√ +NWij(i ̸= j) in 2.1 and +√ +NXij in +2.2. Further, for the spiked Wigner matrices, let Pd be the distribution of the normalized diagonal entries +√ +NWii in 2.1. We assume the following: +1. The density functions g and gd of P and Pd, respectively, are smooth, positive everywhere, and sym- +metric (about 0). +2. For any fixed (N-independent) D, the D-th moments of P and Pd are finite. +3. The functions h = −g′/g, hd = −g′ +d/gd and their all derivatives are polynomially bounded in the sense +that |h(ℓ)(w)|, |h(ℓ) +d (w)| ≤ Cℓ|w|Cℓ for some constant Cℓ depending only on ℓ. +The first condition on the prior implies that the spike is not necessarily delocalized, i.e., some entries of +the signal can be significantly larger than N −1/2. The key examples of the prior are as follows: +Example 3.2. We can consider the following examples of the spike prior: +1. the spherical prior, where u(ℓ) (and v(ℓ)) are i.i.d. drawn uniformly from the unit sphere, or +2. the i.i.d. prior, where the entries u1(ℓ), . . . , uM(ℓ) (respectively, v1(ℓ), . . . , vN(ℓ)) are i.i.d. random +variables from the probability measures µℓ (respectively, νℓ) with mean zero and variance M −1 (respec- +tively N −1) such that for any integer p > 2 +E|ui(ℓ)|p, E|vj(ℓ)|p ≤ +Cp +M 1+(p−2)φ +for some (N-independent) constants Cp > 0 and φ ≤ 1 +2, uniformly on i, j and ℓ. +We remark that for the spike Wigner matrices, due to normalization, the variance of the i.i.d. prior µℓ for +ui(ℓ) is N −1. +9 + +Spiked Wigner matrix +Given a spiked Wigner matrix M, we consider a family of the entrywise transformations +hα(x) = −g′(x) +g(x) + αx, +hd(x) = −g′ +d(x)/gd(x) +(3.1) +for α ∈ R. We also consider the transformed matrix � +M whose entries are +� +Mij = +1 +� +FgN h0( +√ +NMij)(i ̸= j), +� +Mii = +� +w2 +Fg,dN hd +�� +N +w2 +Mii +� +, +(3.2) +where the Fisher information Fg and Fg,d of g and gd are given by +Fg = +� ∞ +−∞ +(g′(x))2 +g(x) +dx, +Fg,d = +� ∞ +−∞ +g′ +d(x)2 +gd(x) dx. +Note that Fg ≥ 1 where the equality holds only if g is the standard Gaussian. +Then following theorem asserts that the effective SNRs of the transformed matrix for PCA are λℓFg, +which generalizes Theorem 4.8 in [50]. +Theorem 3.3. Let M be a spiked Wigner matrix in Definition 2.3 satisfying Assumption 3.1 with φ > 1/4. +Let � +M be the transformed matrix obtained as in (3.2) and (�µℓ, �u(ℓ)) the pair of ℓ-th largest eigenvalue and +the corresponding eigenvector of � +M. Then, almost surely, for 1 ≤ ℓ ≤ k +• If λℓ > +1 +Fg , then �µℓ → +� +λℓFg + +1 +√ +λℓFg and |�u(ℓ)T u(ℓ)|2 → 1 − +1 +λℓFg , +• If λℓ < +1 +Fg , then �µℓ → 2 and |�u(ℓ)T u(ℓ)|2 → 0. +For the proof, we adapt the strategy in [50], where the key observation is that the transformed matrix is +approximately equal to another spiked Winger matrix. See Appendix B.2 for the detail of the proof. +We remark that h0 is the optimal (up to constant factor) among all entrywise transformations. See +Appendix B.5.1 for the proof of it. +Spiked rectangular matrices +For a spiked rectangular matrix Y , we consider the family of the entrywise transformations hα(x) defined in +(3.1) and transformed matrices �Y (α) whose entries are +�Y (α) +ij += +1 +� +(α2 + 2α + Fg)N +hα( +√ +NYij). +(3.3) +Note that +For the additive model, we again show that the effective SNRs of the transformed matrix for PCA are +{λℓFg}ℓ. +Theorem 3.4. Let Y be a spiked rectangular matrix in Definition 2.4 satisfying Assumption 3.1 with φ > +1/4. Let �Y ≡ �Y (0) be the transformed matrix obtained as in (3.3) with α = 0 and (�µℓ, �u(ℓ)) the pair of ℓ-th +largest eigenvalue and the corresponding eigenvector of �Y �Y T . Then, almost surely, for 1 ≤ ℓ ≤ k +• If λℓ > +√d0 +Fg , then �µℓ → (1 + λℓFg)(1 + +d0 +λℓFg ) and |�u(ℓ)T u(ℓ)|2 → 1 − +d0(1+λℓFg) +λℓFg(λℓFg+d0). +• If λℓ < +√d0 +Fg , then �µℓ → d+ = (1 + √d0)2 and |�u(ℓ)T u(ℓ)|2 → 0. +10 + +From Theorem 3.4, if λℓ > +√d0 +Fg , we immediately see that the signal in the additive model can be reliably +detected by the transformed PCA. Thus, the detection threshold in the PCA is lowered when the noise is +non-Gaussian. We also remark that h0 is the optimal entrywise transformation (up to constant factor) as in +the Wigner case; see Appendix B.5.2. +For the proof, we adapt the strategy in [34], where the key observation is again that the transformed +matrix is approximately equal to another spiked rectangular matrix. See Appendix B.3 for the detail of the +proof. +For the multiplicative model, we have the following result. +Theorem 3.5. Let Y be a spiked rectangular matrix in Definition 2.5 satisfying Assumption 3.1 with φ > +1/4. Let �Y ≡ �Y (αg,ℓ) be the transformed matrix obtained as in (3.3) with +αg,ℓ := +−γℓFg + +� +4Fg + 4γℓFg + γ2 +ℓ F 2g +2(1 + γℓ) +and (�µℓ, �u(ℓ)) the pair of ℓ-th largest eigenvalue and the corresponding eigenvector of �Y �Y T . Then, almost +surely, +• If (λg)ℓ > √d0, then �µℓ → (1 + (λg)ℓ)(1 + +d0 +(λg)ℓ ) and +|�u(ℓ)T u(ℓ)|2 → 1 − +(λg)ℓ + d0 +(λg)ℓ · ((λg)ℓ + 1), +• If (λg)ℓ < √d0, then �µℓ → d+ = (1 + √d0)2 and |�u(ℓ)T u(ℓ)|2 → 0. +where +(λg)ℓ := γℓ + γ2 +ℓ Fg +2 ++ +γℓ +� +4Fg + 4γℓFg + γ2 +ℓ F 2g +2 +. +Note that +(λg)ℓ ≥ γℓ + γ2 +ℓ Fg +2 ++ +γℓ +� +4 + 4γℓFg + γ2 +ℓ F 2g +2 += 2γℓ + γ2 +ℓ Fg ≥ 2γℓ + γ2 +ℓ = λℓ, +and the inequality is strict if Fg > 1, i.e., g is not Gaussian. +Note that unlike the additive model, we cannot determine αg without prior knowledge on the SNR. +Nevertheless, we can apply the transformation h√ +Fg or h0, which effectively increase all SNRs simultaneously; +see Appendix B.5. +From Theorem 3.5, if (λg)ℓ > √d0, the signal can be reliably detected by the transformed PCA and the +detection threshold in the PCA is lowered if the noise is non-Gaussian. We also remark that hαg,ℓ is the +optimal entrywise transformation (up to constant factor) for the ℓ-th largest eigenvalue; see Appendix B.5. +We finish this section with an outline of the proof of Theorem 3.5. We begin by justifying that the +transformed matrix �Y is approximately of the form (Q+U�Γ +1 +2 U T X), where �Γ = diag(�γ1, · · · , �γk). Then, the +largest eigenvalue of �Y �Y T can be approximated by the largest eigenvalue of (Q+U�Γ +1 +2 U T X)T (Q+U�Γ +1 +2 U T X) +for which we consider an identity +(Q + U�Γ +1 +2 U T X)T (Q + U�Γ +1 +2 U T X) − zI = (QT Q − zI)(I + L(z)), +11 + +where +L(z) = G(z)(XT U�Γ +1 +2 U T Q + QT U�Γ +1 +2 U T X + XT U�ΓU T X), +G(z) = (QT Q − zI)−1. +If z is an eigenvalue of (Q + U�Γ +1 +2 U T X)T (Q + U�Γ +1 +2 U T X) but not of QT Q, the determinant of (I + L(z)) +must be 0 and hence −1 is an eigenvalue of L(z). Since the rank of L(z) is at most 2k, we can find that the +eigenvector of L(z) is a linear combination of vectors G(z)QT u(ℓ) and G(z)XT u(ℓ). Further, by using the +facts in Example 3.2, we can observe that a linear combination of vectors G(z)QT u(ℓ) and G(z)XT u(ℓ) be +a possible candidate for the ℓ-th eigenvector of L(z), and so of �Y T �Y i.e., for some aℓ, bℓ, +L(z)(aℓG(z)QT u(ℓ) + bℓG(z)XT u(ℓ)) = −(aℓG(z)QT u(ℓ) + bℓG(z)XT u(ℓ)). +(3.4) +From the definition of L(z), +L(z) · G(z)XT U = G(z)XT U�Γ +1 +2 (U T QG(z)XT U) + G(z)QT U�Γ +1 +2 (U T XG(z)XT U) ++ G(z)XT U�Γ(U T XG(z)XT U), +and a similar equation holds for L(z) · G(z)QT U. It suggests that if U T QG(z)XT U and U T XG(z)XT U are +concentrated around diagonal matrices where the entries are deterministic functions of z, then the left side +of (3.4) can be well-approximated by a (deterministic) linear combination of G(z)QT u(ℓ) and G(z)XT u(ℓ). +We can then find the location of the largest eigenvalue in terms of a deterministic function of z and conclude +the proof by optimizing the function q. +The concentration of random matrices U T QG(z)XT U and U T XG(z)XT U is the biggest technical chal- +lenge in the proof, mainly due to the dependence between the matrices Q and X. We prove it by applying +the technique of linearization in conjunction with resolvent identities and also several recent results from +random matrix theory, most notably the local Marchenko–Pastur law. +Once we find out the coefficients aℓ and bℓ in (3.4), the eigenvector localization is an easy corollary +since the vector aℓG(z)QT u(ℓ) + bℓG(z)XT u(ℓ) must be a right singular vector of �Y with the corresponding +singular value +� +(1 + (λg)ℓ)(1 + +d0 +(λg)ℓ ). In this paper, we will not go into further detail on this part. +The detailed proof of Theorem 3.5 can be found in Appendix B.4. +4 +Main Result II - Weak Detection +4.1 +Signal detection in rank-1 spiked models +We begin by recalling the LSS-based detection algorithms for rank-1 spiked rectangular matrices in [34]. +Suppose that our goal is to detect the presence of the signal by the hypothesis test between H0 : λ = 0 and +H1 : λ = ω where the SNR ω for the alternative hypothesis H1 is known. The key observation is that the +variances of the limiting Gaussian distributions of the LSS in (2.7) do not depend on the SNR while the means +do. If we denote by VY (f) the common variance, and mY (f)|H0 and mY (f)|H1 the means, respectively, our +goal is to find a function that maximizes the relative difference between the limiting distributions of the LSS +under H0 and under H1, i.e., +����� +mY (f)|H1 − mY (f)|H0 +� +VY (f) +����� . +(4.1) +12 + +Algorithm 1 Hypothesis test for a rank-1 spiked rectangular matrix +Input: data Yij, parameters w4, ω +Lω ← test statistic in (4.3) +mω ← critical value in (4.7) +if Lω ≤ mω then +Accept H0 +else +Reject H0 +end if +As we will see in Theorem 5.5, the optimal function f is of the form C1φω + C2 for some constants C1 and +C2, where +φω(x) = ω +d0 +� +2 +w4 − 1 − 1 +� +x − log +�� +1 + d0 +ω +� +(1 + ω) − x +� +. +(4.2) +The test statistic we use is thus defined as +Lω = +M +� +i=1 +φω(µi) − M +� d+ +d− +φω(x) dµMP (x) += − log det +�� +1 + d0 +ω +� +(1 + ω)I − Y Y T +� ++ ω +d0 +� +2 +w4 − 1 − 1 +� +(Tr Y Y T − M) ++ M +� ω +d0 +− log +� ω +d0 +� +− 1 − d0 +d0 +log(1 + ω) +� +. +(4.3) +Theorem 8 in [34] asserts that Lω converges to a Gaussian, +Lω ⇒ N(m(λ), V0). +(4.4) +Here, the mean of the limiting Gaussian distribution is given by +m(λ) = −1 +2 log +� +1 − ω2 +d0 +� ++ ω2 +2d0 +(w4 − 3) − log +� +1 − λ2 +d0 +� ++ λ2 +d0 +� +2 +w4 − 1 − 1 +� +(4.5) +with λ = 0 under H0 and λ = ω under H1, and the variance +V0 = −2 log +� +1 − ω2 +d0 +� ++ 2ω2 +d0 +� +2 +w4 − 1 − 1 +� +. +(4.6) +Based on the asymptotic normality of Lω, we can construct a test in which we compute the test statistic +Lω and compare it with the average of m(0) and m(ω), i.e., +mω := m(0) + m(ω) +2 += − log +� +1 − ω2 +d0 +� ++ ω2 +2d0 +� +2 +w4 − 1 + w4 − 4 +� +. +(4.7) +See Algorithm 1 for the detail. +The limiting error of the proposed test, Algorithm 1, is given by +err(ω) = P(Lω > mω|H0) + P(Lω ≤ mω|H1) → erfc +�√V0 +4 +√ +2 +� +, +(4.8) +13 + +where V0 is the variance in (4.6) and erfc(·) is the complementary error function. If the noise X is Gaussian, +w4 = 3 and the limiting error in (4.8) is +erfc +�√V0 +4 +√ +2 +� += erfc +� +1 +4 +� +− log +� +1 − ω2 +d0 +�� +, +and it coincides with the error of the LR test; see Section 2.2 of [34]. It shows that our test is optimal with +the Gaussian noise. +4.2 +Signal detection in rank-k spiked models +When the rank of the spike is larger than 1, we first consider a simple case where the data is given as a +spiked Wigner matrix and our goal is to construct an LSS-based algorithm for a hypothesis test between +H0 : Λ = 0 and Hk : Λ = ωIk, where the rank k of the spike for the alternative hypothesis is known. Our +starting point is the following test statistic, which was considered for the rank-1 spiked Wigner matrix in +[22]: +Lω = − log det +� +(1 + ω)I − √ωM +� ++ ωN +2 ++ √ω +� 2 +w2 +− 1 +� +Tr M + ω +� +1 +w4 − 1 − 1 +2 +� +(Tr M 2 − N). +(4.9) +If there is no signal present, Lω ⇒ N(m0, V0), where +m0 = −1 +2 log(1 − ω) + +�w2 − 1 +w4 − 1 − 1 +2 +� +ω + (w4 − 3)ω2 +4 +, +(4.10) +V0 = −2 log(1 − ω) + +� 4 +w2 +− 2 +� +ω + +� +2 +w4 − 1 − 1 +� +ω2. +(4.11) +For a rank-k spiked Wigner matrix, we can consider the same Lω as in (4.9) and prove that it also +converges to a Gaussian with the same variance V0 but an altered mean mk. The following is the precise +statement for the limiting distribution of Lω. +Theorem 4.1. Let M be a rank-k spiked Wigner matrix with a spike U as in Definition 2.3 with Λ = ωIk +for some nonnegative integer k. Then, +Lω ⇒ N(mk, V0) , +(4.12) +where the variance V0 is as in (4.11) and the mean mk is given by +mk = m0 + k +� +− log(1 − ω) + +� 2 +w2 +− 1 +� +ω + +� +1 +w4 − 1 − 1 +2 +� +ω2 +� += m0 + kV0 +2 . +(4.13) +Proof. Theorem 4.1 directly follows from Theorem 5.2 in Section 5. +Since the mean of Lω depends on the rank of the spike, we can construct a hypothesis test between Hk1 +and Hk2 in (1.11) based on Theorems 4.1 and 4.4. In this test, for a given spiked Wigner matrix M, we +compute Lω and compare it with the critical value m(k1+k2)/2, +m(k1+k2)/2 := mk1 + mk2 +2 +. +(4.14) +14 + +Algorithm 2 Hypothesis test for a spiked Wigner matrix +Data: Mij, parameters w2, w4, λ +Lω ← test statistic in (4.9), +m(k1+k2)/2 ← critical value in (4.14) with (4.13) +if Lω ≤ m(k1+k2)/2 then +Accept H1 +else +Accept H2 +end if +See Algorithm 2 for the detail. +In Theorems 5.2 and 5.5, we prove that the proposed test in Algorithm 2 is optimal among all CLT-based +tests, in the sense that the error is minimized with the test statistic Lω also for spiked random matrices. +Theorem 4.2. The error of the test, err(ω) = P(Lω > mω|H0)+P(Lω ≤ mω|H1), in algorithm 2 converges +to +erfc +� +k2 − k1 +4 +� +V0 +2 +� +. +Proof. Theorem 4.2 is a direct consequence of Theorems 4.1 and 4.4. (See also Section 3 of [29] and the +proof of Theorem 2 of [22].) +Remark 4.3. When w4 = 3, we find that the error err(ω) converges to +erfc +� +k2 − k1 +4 +� +− log(1 − ω) + +� 2 +w2 +− 1 +� +ω +� +. +(4.15) +The optimal error for the weak detection, achieved by the LR test, coincides with the limiting error in (4.15) +when the noise is Gaussian and the SNR ω is sufficiently small; see [33]. Thus, our proposed test is optimal +in this case. +The test in Algorithm 2 can be readily extended to the spiked rectangular matrices by replacing the +test statistic in (4.9) with the following one, which was introduced in [34] for the rank-1 spiked rectangular +matrices. +Lω = − log det +�� +1 + d0 +ω +� +(1 + ω)I − Y Y T +� ++ ω +d0 +� +2 +w4 − 1 − 1 +� +(Tr Y Y T − M) ++ M +� ω +d0 +− log +� ω +d0 +� +− 1 − d0 +d0 +log(1 + ω) +� +. +(4.16) +We have the following results for the asymptotic normality of Gaussian fluctuation of Lω: +Theorem 4.4. Let Y be a spiked rectangular matrix in Definition 2.4 or 2.5 with Λ = ωIk for some +nonnegative integer k and λ ∈ (0, √d0) and w4 > 1. Then, for any spikes with U T U = V T V = Ik, +Lω ⇒ N(mk, V0), +(4.17) +where the mean and the variance are given by +mk = m0 + k +� +− log +� +1 − ω2 +d0 +� ++ ω2 +d0 +� +2 +w4 − 1 − 1 +�� +(4.18) +15 + +and +V0 = −2 log +� +1 − ω2 +d0 +� ++ 2ω2 +d0 +� +2 +w4 − 1 − 1 +� +(4.19) +where +m0 = −1 +2 log +� +1 − ω2 +d0 +� ++ ω2 +2d0 +(w4 − 3). +(4.20) +Theorem 4.4 directly follows from the general CLT result in Theorems 5.5. See Appendix C.4 for the +detailed computation for the mean and the variance. +With Theorem 4.4, we find that Algorithm 2 is available for the weak detection of the signal in the spiked +rectangular matrices with the following change: +• Data matrix is Yij (instead of Mij). +• Test statistic Lω is defined by (4.16) (instead of (4.9)). +• Critical value m(k1+k2)/2 is obtained by (4.14) with (4.20) (instead of (4.13)). +The limiting error of the test in this case is again erfc +� +k2−k1 +4 +� +V0 +2 +� +as in Theorem 4.2, where V0 is +defined by (4.19). +4.3 +Test with entrywise transformation for spiked matrices of additive type +The entrywise transform we applied with the PCA in Section 3.1 can also be adapted to be used together +with the proposed test in Algorithm 2; see also [22] where the same idea was applied for the rank-1 spiked +Wigner matrix. Recall the transformation defined in (3.1) and the transformed matrix � +M in (3.2). We +consider a test statistic +�Lω := − log det +� +(1 + ωFg)I − +� +ωFg � +M +� ++ ωFg +2 N ++ √ω +� +2 +� +Fg,d +w2 +− +� +Fg +� +Tr � +M + λ +� GH +� +w4 − 1 − Fg +2 +� +(Tr � +M 2 − N), +(4.21) +where +GH = +1 +2Fg +� ∞ +−∞ +g′(w)2g′′(w) +g(w)2 +dw, +� +w4 = +1 +(Fg)2 +� ∞ +−∞ +(g′(w))4 +(g(w))3 dw. +We then have the following CLT result for �Lω that generalizes the results in [22]. +Theorem 4.5. Assume the conditions in Theorem 4.1, satisfying Assumption 3.1 with φ > 3/8. If λFg < 1, +�Lω ⇒ N( �mk, �V0), +(4.22) +where the mean and the variance are given by +�mk = −1 +2 log(1 − ωFg) + +�(w2 − 1)GH +�w4 − 1 +− Fg +2 +� +ω + �w4 − 3 +4 +(ωFg)2 ++ k +� +− log(1 − ωFg) + +�2Fg,d +w2 +− Fg +� +ω + +� (GH)2 +�w4 − 1 − (Fg)2 +2 +� +ω2 +� +, +(4.23) +�V0 = −2 log(1 − ωFg) + +�4Fg,d +w2 +− 2Fg +� +ω + +�2(GH)2 +�w4 − 1 − (Fg)2 +� +ω2. +(4.24) +16 + +Algorithm 3 Hypothesis test for a spiked Wigner matrix with entrywise transformation +Data: Mij, parameters w2, w4, λ, densities g, gd +� +M ← transformed matrix in (3.2), +�Lω ← test statistic in (4.21), +�m(k1+k2)/2 ← critical value in (4.25) +with (4.23) +if �Lω ≤ �m(k1+k2)/2 then +Accept H1 +else +Accept H2 +end if +Proof. Theorem 4.5 directly follows from Theorem 5.3 in Section 5. +Based on Theorem 4.5, we can adapt the test in Algorithm 2 to construct a test that utilizes the entrywise +transformation. In this test, we compute �LΛ and compare it with the critical value +�m(k1+k2)/2 := ( �mk1 + �mk2)/2. +(4.25) +See Algorithm 3 for the detail. The limiting error of the test is given as follows. +Theorem 4.6. The error of the test in Algorithm 3 converges to +erfc +� +�k2 − k1 +4 +� +�V0 +2 +� +� . +Proof. Theorem 4.6 is a direct consequence of Theorem 5.6. +We also propose an analogous test can for the additive model of the spiked rectangular matrices as follows. +Recall the transformed matrix �Y ≡ �Y (0) in (3.3). Define the test statistic �Lω by +�Lω = − log det +�� +1 + d0 +ωFg +� +(1 + ωFg)I − �Y �Y T +� ++ 2ω +d0 +� GH +�w4 − 1 − Fg +2 +� +(Tr �Y �Y T − M) ++ M +�ωFg +d0 +− log +�ωFg +d0 +� +− 1 − d0 +d0 +log(1 + ωFg) +� +. +(4.26) +We then have the following CLT for the test statistic. +Theorem 4.7. Assume the conditions in Theorem 4.4, satisfying Assumption 3.1 with φ > 3/8. If λ < +√d0/Fg, +�Lω ⇒ N( �mk, �V0), +(4.27) +where the mean and the variance are given by +�m0 = −1 +2 log +� +1 − ω2(Fg)2 +d0 +� ++ ω2(Fg)2 +2d0 +( �w4 − 3) +(4.28) +�mk = �m0 + k +� +− log +� +1 − ω2(Fg)2 +d0 +� ++ 2ω2 +d0 +� (GH)2 +�w4 − 1 − (Fg)2 +2 +�� +(4.29) +and +�V0 = 4ω2 +d0 +� (GH)2 +�w4 − 1 − (Fg)2 +2 +� +− 2 log +� +1 − ω2(Fg)2 +d0 +� +. +(4.30) +17 + +With Theorem 4.7, we can adjust Algorithm 2 for the weak detection of the signal in the additive model +of spiked rectangular matrices, where we make the following change: +• Data matrix is Yij (instead of Mij). +• Transformed matrix is �Y (instead of � +M), defined by (3.3) with α = 0. +• Test statistic �Lω is defined by (4.26) (instead of (4.21)). +• Critical value m(k1+k2)/2 is obtained by (4.25) with (4.29) (instead of (4.23)). +In Appendix A, we consider several examples of spiked Wigner matrices and spiked rectangular matrices, +where we compare the errors from numerical simulations and the theoretical errors of the proposed algorithms. +We find that the numerical errors of the proposed tests closely match the corresponding theoretical errors +and the error from Algorithm 3 is lower than that of Algorithm 2. +4.4 +Rank estimation +The test in Algorithm 2 requires prior knowledge about k1 and k2, the possible ranks of the planted spike. +In this section, we adapt the idea of the proposed tests in Algorithm 2 to estimate the rank of the signal +when there is no prior information on the rank k. Recall that the test statistic Lω defined in (4.9) does not +depend on the rank of the matrix. As proved in Theorem 4.1, the test statistic Lω converges to a Gaussian +random variable with mean mk and the variance V0, where mk is equi-distributed with respect to k and V0 +does not depend on k. It is then natural to set the best candidate for k, which we call κ, be the minimizer +of the distance |Lω − mk|. This procedure is equivalent to find the nearest nonnegative integer of the value +κ′ := 2(Lω − m0) +V0 +(4.31) +rounding half down. +We describe the procedure in Algorithm 4; for example, its probability of error for spiked Wigner matrix +converges to +P(k = 0) · P +� +Z > +√V0 +4 +� ++ +∞ +� +i=1 +P(k = i) · P +� +|Z| > +√V0 +4 +� += +� +1 − P(k = 0) +2 +� +· erfc +� +1 +4 +� +V0 +2 +� +, +(4.32) +where Z is a standard Gaussian random variable. Note that it depends only on P(k = 0). +The error can be lowered if the range of k is known a priori. See Appendix A. It is also possible to +improve Algorithm 4 by pre-transforming the data matrix entrywise as in Section 4.3. We omit the detail. +5 +Central Limit Theorems +In this section, we collect our results on general CLTs for the LSS of spiked random matrices. To precisely +define the statements, we introduce the Chebyshev polynomials of the first kind. +Definition 5.1 (Chebyshev polynomial). The n-th Chebyshev polynomial (of the first kind) Tn is a degree +n polynomial defined by T0(x) = 1, T1(x) = x, and +Tn+1(x) = 2xTn(x) − Tn−1(x). +18 + +Algorithm 4 Rank estimation +Data: Mij (or Yij), parameters w2, w4, λ +Lω ← test statistic in (4.9) or (4.16), +m0 ← mean in (4.10) or (4.20), +m1 ← mean in (4.13) or (4.18) +with k = 1 +κ′ ← value in (4.31) +if Lω ≤ (m0 + m1)/2 then +Set κ = 0 +else +Set κ = ⌈κ′ − 0.5⌉ +end if +We first state a CLT for the LSS of spiked Wigner matrices. Recall that we denote by µ1 ≥ µ2 ≥ · · · ≥ µN +the eigenvalues of a spiked Wigner matrix M. +Theorem 5.2. Assume the conditions in Theorem 4.1. Suppose that a function f is analytic on an open +interval containing [−2, 2]. Then, +� N +� +i=1 +f(µi) − N +� 2 +−2 +√ +4 − z2 +2π +f(z) dz +� +⇒ N (mk(f), V0(f)) . +The mean and the variance of the limiting Gaussian distribution are given by +mk(f) = 1 +4 (f(2) + f(−2)) − 1 +2τ0(f) + (w2 − 2)τ2(f) + (w4 − 3)τ4(f) + k +∞ +� +ℓ=1 +√ +ωℓτℓ(f), +V0(f) = (w2 − 2)τ1(f)2 + 2(w4 − 3)τ2(f)2 + 2 +∞ +� +ℓ=1 +ℓτℓ(f)2 , +where we let +τℓ(f) = 1 +π +� 2 +−2 +Tℓ +�x +2 +� +f(x) +√ +4 − x2 dx. +Furthermore, for mk, m0, and V0 defined in Theorem 4.1, +����� +mk(f) − m0(f) +� +V0(f) +����� ≤ +���� +mk − m0 +√V0 +���� +The equality holds if and only if f(x) = C1φω(x) + C2 for some constants C1 and C2 where +φω(x) := log +� +1 +1 − √ωx + ω +� ++ √ω +� 2 +w2 +− 1 +� +x + ω +� +1 +w4 − 1 − 1 +2 +� +x2. +We will give a proof of Theorem 5.2 in Appendix C. With the entrywise transformation in Section 4.3, +we have the following changes in Theorem 5.2. Recall that �µ1 ≥ �µ2 ≥ · · · ≥ �µN are the eigenvalues of the +transformed matrix � +M. +Theorem 5.3. Assume the conditions in Theorem 5.2, satisfying Assumption 3.1 with φ > 3/8. If λFg < 1, +� N +� +i=1 +f(�µi) − N +� 2 +−2 +√ +4 − z2 +2π +f(z) dz +� +⇒ N( �mk(f), �V0(f)) . +19 + +The mean and the variance of the limiting Gaussian distribution are given by +�mk(f) = 1 +4 (f(2) + f(−2)) − 1 +2τ0(f) + k +� +ωFg,dτ1(f) + (w2 − 2 + kωGH)τ2(f) ++ (� +w4 − 3)τ4(f) + k +∞ +� +ℓ=3 +� +(ωFg)ℓτℓ(f), +(5.1) +�V0(f) = (w2 − 2)τ1(f)2 + 2(� +w4 − 3)τ2(f)2 + 2 +∞ +� +ℓ=1 +ℓτℓ(f)2. +Furthermore, for �mk, �m0, and �V0 defined in Theorem 4.1, +������ +�mk2(f) − �mk1(f) +� +�V0(f) +������ +≤ +������ +�mk2 − �mk1 +� +�V0 +������ +The equality holds if and only if f(x) = C1 �φω(x) + C2 for some constants C1 and C2 with the function +�φω(x) := log +� +1 +1 − +� +ωFgx + ωFg +� ++ +� +2 +� +Fg,d +w2 +− +� +Fg +� +x + ω +� GH +�w4 − 1 − Fg +2 +� +x2. +We will also prove Theorem 5.3 in Appendix C. +Remark 5.4. For a general case where the spike Λ = diag(ω1, · · · , ωk) with possibly distinct ωi’s, we can +prove the CLT and the transformed CLT, analogous to Theorems 5.2 and 5.3, respectively, where the means +of the limiting Gaussians are given by +mM(f) = 1 +4 (f(2) + f(−2)) − 1 +2τ0(f) + (w2 − 2)τ2(f) + (w4 − 3)τ4(f) ++ +k +� +s=1 +∞ +� +ℓ=1 +� +ωℓsτℓ(f), +�mM(f) = 1 +4 (f(2) + f(−2)) − 1 +2τ0(f) + (w2 − 2)τ2(f) + (� +w4 − 3)τ4(f) ++ +k +� +s=1 +� +ωsFg,dτ1(f) + ωsGHτ2(f) + +k +� +s=1 +∞ +� +ℓ=3 +� +(ωsFg)ℓτℓ(f), +and the variances are equal to V0(f) in Theorem 5.2 and �V0(f) in Theorem 5.3, respectively. Adapting the +proposed tests in Algorithms 2 and 3, it is possible to construct hypothesis tests for the weak detection in this +case. +The next result is the CLT for the LSS of spiked rectangular matrices Y , where we denote by µ1 ≥ µ2 ≥ +· · · ≥ µM the eigenvalues of Y Y T . +Theorem 5.5. Assume the conditions in Theorem 4.4. Suppose that a function f is analytic on an open +set containing an interval [d−, d+]. Then, +� M +� +i=1 +f(µi) − M +� d+ +d− +� +(x − d−)(d+ − x) +2πd0x +f(x) dx +� +⇒ N(mk(f), V0(f)). +(5.2) +20 + +The mean and the variance of the limiting Gaussian distribution are given by +mk(f) = +�f(2) + �f(−2) +4 +− τ0( �f) +2 ++ (w4 − 3)τ2( �f) + k +∞ +� +ℓ=1 +� ω +√d0 +�ℓ +τℓ( �f) +and +V0(f) = 2 +∞ +� +ℓ=1 +ℓτℓ( �f)2 + (w4 − 3)τ1( �f)2, +where we let �f(x) = f(√d0x + 1 + d0). +Furthermore, for mk, m0, and V0 defined in Theorem 4.4, +����� +mk2(f) − mk1(f) +� +V0(f) +����� ≤ +���� +mk2 − mk1 +√V0 +���� +The equality holds if and only if f(x) = C1φω(x) + C2 for some constants C1 and C2 with the function +φω(x) = ω +d0 +� +2 +w4 − 1 − 1 +� +x − log +�� +1 + d0 +ω +� +(1 + ω) − x +� +. +Lastly, we state the pre-transformed CLT for the LSS of the additive model of spiked rectangular matrices. +We let �Y be the transformed matrix and �µ1 ≥ �µ2 ≥ · · · ≥ �µN the eigenvalues of �Y �Y T . +Theorem 5.6. Assume the conditions in Theorem 5.5, satisfying Assumption 3.1 with φ > 3/8. If λ < +√d0/Fg, +� M +� +i=1 +f(�µi) − M +� d+ +d− +f(x)ρMP,d0(dx) +� +⇒ N( �mk(f), �V0(f)). +(5.3) +The mean and the variance of the limiting Gaussian distribution are given by +�mk(f) = +�f(2) + �f(−2) +4 +− 1 +2τ0( �f) + kω +√d0 +(GH − Fg)τ1( �f) + (� +w4 − 3)τ2( �f) ++ k +∞ +� +ℓ=1 +�ωFg +√d0 +�ℓ +τℓ( �f) +(5.4) +and +�V0(f) = 2 +∞ +� +ℓ=1 +ℓτℓ( �f)2 + (� +w4 − 3)τ1( �f)2. +(5.5) +where �f(x) = f(√d0x + 1 + d0). +Furthermore, for �mk, �m0, and �V0 defined in Theorem 4.7, The equality holds if and only if f(x) = +C1 �φω(x) + C2 for some constants C1 and C2 with the function +�φω(x) = 2λ +d0 +� GH +�w4 − 1 − Fg +2 +� +x − log +�� d0 +ωFg ++ 1 +� +(ωFg + 1) − x +� +. +Remark 5.7. As in Remark 5.4, for a general case with Λ = diag(ω1, · · · , ωk), the CLT and the transformed +21 + +CLT hold with the adjusted means +mY (f) = +�f(2) + �f(−2) +4 ++ τ0( �f) +2 ++ (w4 − 3)τ2( �f) + +k +� +s=1 +∞ +� +ℓ=1 +� ωs +√d0 +�ℓ +τℓ( �f), +�mY (f) = +�f(2) + �f(−2) +4 +− 1 +2τ0( �f) + (� +w4 − 3)τ2( �f) + +k +� +s=1 +ωs +√d0 +(GH − Fg)τ1( �f) ++ +k +� +s=1 +∞ +� +ℓ=1 +�ωsFg +√d0 +�ℓ +τℓ( �f), +where the variances are given V0(f), �V0(f), respectively. Further, the corresponding optimal functions and +test statistic can be calculated by following the same procedure in [34]. +6 +Conclusion and Future Works +In this paper, we considered the detection problems of the spiked random model with general ranks. First, +we prove the sub-optimality of the PCA for the non-Gaussian noise. Further, we proposed a hypothesis test +based on the central limit theorem for the linear spectral statistics of the data matrix and introduced a test +for rank estimation that do not require any prior information on the rank of the signal. It was shown that +the error of the proposed hypothesis test matches the error of the likelihood ratio test in case the noise is +Gaussian and the signal-to-noise ratio is small. With the knowledge on the density of the noise, the test was +further improved by applying an entrywise transformation. +We believe that the hypothesis test with the entrywise transformed matrix proposed in this paper can +be extended to the multiplicative model of spiked rectangular matrix. This will be discussed in our future +works. +Acknowledgments +The work of J. H. Jung and J. O. Lee was partially supported by National Research Foundation of Korea +under grant number NRF-2019R1A5A1028324. +The work of H. W. Chung was partially supported by +National Research Foundation of Korea under grant number 2017R1E1A1A01076340 and by the Ministry +of Science and ICT, Korea, under an ITRC Program, IITP-2019-2018-0-01402. +References +[1] E. Abbe. Community detection and stochastic block models: recent developments. The Journal of +Machine Learning Research, 18(1):6446–6531, 2017. +[2] O. H. Ajanki, L. Erd˝os, and T. Kr¨uger. Universality for general wigner-type matrices. Probab. Theory +and Related Fields, 169(3):667–727, 2017. +[3] J. Alt. Singularities of the density of states of random gram matrices. Electron. Commun. Probab., +22:1–13, 2017. +[4] J. Alt, L. Erd˝os, and T. Kr¨uger. Local law for random gram matrices. Electron. J. Probab., 22:1–41, +2017. +22 + +[5] B. Aubin, B. Loureiro, A. Maillard, F. Krzakala, and L. Zdeborov´a. The spiked matrix model with +generative priors. Advances in Neural Information Processing Systems, 32, 2019. +[6] Z. D. Bai and J. Yao. On the convergence of the spectral empirical process of Wigner matrices. Bernoulli, +11(6):1059–1092, 2005. +[7] J. Baik, G. B. Arous, S. P´ech´e, et al. Phase transition of the largest eigenvalue for nonnull complex +sample covariance matrices. Ann. Probab., 33(5):1643–1697, 2005. +[8] J. Baik and J. O. Lee. Fluctuations of the free energy of the spherical Sherrington-Kirkpatrick model. +J. Stat. Phys., 165(2):185–224, 2016. +[9] J. Baik and J. O. Lee. Fluctuations of the free energy of the spherical Sherrington-Kirkpatrick model +with ferromagnetic interaction. Ann. Henri Poincar´e, 18(6):1867–1917, 2017. +[10] J. Baik and J. O. Lee. Free energy of bipartite spherical Sherrington-Kirkpatrick model. Ann. Inst. +Henri Poincar´e Probab. Stat., 56(4):2897–2934, 2020. +[11] J. Baik, J. O. Lee, and H. Wu. Ferromagnetic to paramagnetic transition in spherical spin glass. J. +Stat. Phys., 173(5):1484–1522, 2018. +[12] D. Banerjee and Z. Ma. Optimal hypothesis testing for stochastic block models with growing degrees. +arXiv:1705.05305, 2017. +[13] Z. Bao, X. Ding, J. Wang, and K. Wang. +Statistical inference for principal components of spiked +covariance matrices. The Annals of Statistics, 50(2):1144–1169, 2022. +[14] F. Benaych-Georges and R. R. Nadakuditi. The eigenvalues and eigenvectors of finite, low rank pertur- +bations of large random matrices. Adv. Math., 227(1):494–521, 2011. +[15] F. Benaych-Georges and R. R. Nadakuditi. The singular values and vectors of low rank perturbations +of large rectangular random matrices. J. Multivar. Anal., 111:120–135, 2012. +[16] P. J. Bickel and P. Sarkar. Hypothesis testing for automated community detection in networks. J. R. +Stat. Soc., B: Stat. Methodol., 78(1):253–273, 2016. +[17] A. Bloemendal, L. Erd˝os, A. Knowles, H.-T. Yau, and J. Yin. Isotropic local laws for sample covariance +and generalized Wigner matrices. Electron. J. Probab., 19:no. 33, 53, 2014. +[18] A. Bloemendal, A. Knowles, H.-T. Yau, and J. Yin. On the principal components of sample covariance +matrices. Prob. theory and related fields, 164(1-2):459–552, 2016. +[19] C. Butucea, Y. I. Ingster, et al. Detection of a sparse submatrix of a high-dimensional noisy matrix. +Bernoulli, 19(5B):2652–2688, 2013. +[20] T. Cai, Z. Ma, and Y. Wu. Optimal estimation and rank detection for sparse spiked covariance matrices. +Probability theory and related fields, 161(3):781–815, 2015. +[21] H. W. Chung, J. Lee, and J. O. Lee. Asymptotic normality of log likelihood ratio and fundamental limit +of the weak detection for spiked wigner matrices. arXiv preprint arXiv:2203.00821, 2022. +[22] H. W. Chung and J. O. Lee. Weak detection of signal in the spiked wigner model. In International +Conference on Machine Learning, pages 1233–1241, 2019. +23 + +[23] R. Couillet. Robust spiked random matrices and a robust g-music estimator. Journal of Multivariate +Analysis, 140:139–161, 2015. +[24] M. Dia, N. Macris, F. Krzakala, T. Lesieur, L. Zdeborov´a, et al. Mutual information for symmetric +rank-one matrix estimation: A proof of the replica formula. Advances in Neural Information Processing +Systems, 29, 2016. +[25] X. Ding. +High dimensional deformed rectangular matrices with applications in matrix denoising. +Bernoulli, 26(1):387–417, 2020. +[26] X. Ding and F. Yang. Tracy-widom distribution for heterogeneous gram matrices with applications in +signal detection. IEEE Transactions on Information Theory, 2022. +[27] E. Dobriban. Permutation methods for factor analysis and pca. The Annals of Statistics, 48(5):2824– +2847, 2020. +[28] A. El Alaoui and M. I. Jordan. Detection limits in the high-dimensional spiked rectangular model. In +Conference On Learning Theory, pages 410–438, 2018. +[29] A. El Alaoui, F. Krzakala, and M. Jordan. Fundamental limits of detection in the spiked Wigner model. +Ann. Stat, 48(2):863–885, 2020. +[30] L. Erd˝os, A. Knowles, H.-T. Yau, and J. Yin. Spectral statistics of erdos-R´enyi graphs I: Local semicircle +law. Ann. Probab., 41(3B):2279–2375, 2013. +[31] I. M. Johnstone. On the distribution of the largest eigenvalue in principal components analysis. Ann. +Stat, pages 295–327, 2001. +[32] I. M. Johnstone and A. Onatski. Testing in high-dimensional spiked models. Ann. Stat, 48(3):1231–1254, +2020. +[33] J. H. Jung, H. W. Chung, and J. O. Lee. Weak detection in the spiked wigner model with general rank. +arXiv:2001.05676, 2020. +[34] J. H. Jung, H. W. Chung, and J. O. Lee. Detection of signal in the spiked rectangular models. In +International Conference on Machine Learning, pages 5158–5167. PMLR, 2021. +[35] Z. T. Ke, Y. Ma, and X. Lin. Estimation of the number of spiked eigenvalues in a covariance matrix by +bulk eigenvalue matching analysis. Journal of the American Statistical Association, pages 1–19, 2021. +[36] A. Knowles and J. Yin. The isotropic semicircle law and deformation of Wigner matrices. Comm. Pure +Appl. Math., 66(11):1663–1750, 2013. +[37] A. Knowles and J. Yin. Anisotropic local laws for random matrices. Probab. Theory Related Fields, +169(1-2):257–352, 2017. +[38] J. O. Lee and K. Schnelli. Tracy-Widom distribution for the largest eigenvalue of real sample covariance +matrices with general population. Ann. Appl. Probab., 26(6):3786–3839, 2016. +[39] J. O. Lee and K. Schnelli. Local law and Tracy-Widom limit for sparse random matrices. Probab. +Theory Related Fields, 171(1-2):543–616, 2018. +[40] J. Lei. A goodness-of-fit test for stochastic block models. Ann. Stat., 44(1):401–424, 2016. +24 + +[41] M. Lelarge and L. Miolane. Fundamental limits of symmetric low-rank matrix estimation. Probab. +Theory Related Fields, 173(3-4):859–929, 2019. +[42] T. Lesieur, F. Krzakala, and L. Zdeborov´a. Mmse of probabilistic low-rank matrix estimation: Univer- +sality with respect to the output channel. In 2015 53rd Annual Allerton Conference on Communication, +Control, and Computing (Allerton), pages 680–687, 2015. +[43] L. Miolane. Fundamental limits of low-rank matrix estimation: the non-symmetric case. arXiv preprint +arXiv:1702.00473, 2017. +[44] A. Montanari, D. Reichman, and O. Zeitouni. On the limitation of spectral methods: From the gaussian +hidden clique problem to rank-one perturbations of gaussian tensors. In Advances in Neural Information +Processing Systems, pages 217–225, 2015. +[45] F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, editors. NIST handbook of mathematical +functions. U.S. Department of Commerce, National Institute of Standards and Technology, Washington, +DC; Cambridge University Press, Cambridge, 2010. +[46] A. Onatski. Testing hypotheses about the number of factors in large factor models. Econometrica, +77(5):1447–1479, 2009. +[47] A. Onatski, M. J. Moreira, and M. Hallin. Asymptotic power of sphericity tests for high-dimensional +data. Ann. Stat, 41(3):1204–1231, 2013. +[48] A. Onatski, M. J. Moreira, and M. Hallin. Signal detection in high dimension: The multispiked case. +Ann. Stat, 42(1):225–254, 2014. +[49] D. Passemier and J.-F. Yao. On determining the number of spikes in a high-dimensional spiked popu- +lation model. Random Matrices: Theory and Applications, 1(01):1150002, 2012. +[50] A. Perry, A. S. Wein, A. S. Bandeira, and A. Moitra. Optimality and sub-optimality of PCA I: Spiked +random matrix models. Ann. Stat, 46(5):2416–2451, 2018. +A +Examples and Simulations +In Appendix A, we consider specific examples of spiked random matrices under various settings. We first +demonstrate with an example the change of the threshold by the improved PCA in Section 3. We then +provide the details of the proposed tests in Algorithms 2 and 3 with different examples, and the test for +rank estimation in Algorithm 4 for these and compute the theoretical errors. We also perform the numerical +simulation for the proposed tests and compare the numerical errors with the theoretical errors. +A.1 +Spiked Wigner matrix +A.1.1 +Improved PCA with Entrywise Transformation +Our first example is a spiked Wigner matrix with non-Gaussian noise to which we apply the entrywise +transformation for the improved PCA. We let the density function of the noise be a bimodal distribution +with unit variance, defined as +g(x) = gd(x) = +1 +√ +2π +� +e−2(x− +√ +3/2)2 + e−2(x+ +√ +3/2)2� +, +(A.1) +25 + +which is the density function of a random variable +1 +2N + +√ +3 +2 R, +where N is a standard Gaussian random variable and R is a Rademacher random variable, independent to +each other. +We sample Zij = Zji independently from the density g and let Wij = Zij/ +√ +N. +We let u(ℓ) = +(u1(ℓ), u2(ℓ), . . . , uN(ℓ))T , where +√ +Nui(ℓ)’s are i.i.d. Rademacher random variables for i = 1, 2, . . . , N and +ℓ = 1, 2, 3. The data matrix M = UΛ1/2U T , where U = [u(1), u(2), u(3)] and Λ = diag(λ, λ, λ, 0, 0, . . . , 0). +The size of the data matrix is set to be N = 4000. The BBP-transition predicts that the largest eigenvalue +of M pops up from the bulk of the spectrum if λ > 1. +With the entrywise transformation defined in (3.2), we obtain a transformed matrix +� +Mij = +1 +� +FgN h( +√ +NMij) +(A.2) +where +h(x) = −g′(x) +g(x) = +2 +�√ +3 − e4 +√ +3x( +√ +3−2x) + 2x +� +1 + e4 +√ +3x +(A.3) +and Fg = +� ∞ +−∞ +(g′(x))2 +g(x) dx ≈ 2.50810. From Theorem 3.3, it is expected that the largest eigenvalue of � +M +separates from other eigenvalues if λ > +1 +Fg ≈ 0.3987. +In the numerical experiment, we set +λℓ = +ℓ + +1 +Fg +ℓ + 1 +(A.4) +for ℓ = 1, 2, 3, and we compare the spectrum of the matrices M and � +M. In Figure 1, we find three isolated +eigenvalues in the spectrum of � +M (right), which are absent in that of M (left). +-2 +-1.5 +-1 +-0.5 +0 +0.5 +1 +1.5 +2 +0 +5 +10 +15 +20 +25 +30 +-2 +-1.5 +-1 +-0.5 +0 +0.5 +1 +1.5 +2 +0 +5 +10 +15 +20 +25 +30 +Figure 1: The spectrum of the data matrix (N = 4000) with bimodal noise, before (left) and after (right) +the entrywise transformation. Three eigenvalues pop up from the bulk of the spectrum after the entrywise +transformation. +26 + +A.1.2 +Spiked Gaussian Wigner matrix +We consider the weak detection problem with the simplest case of the spiked Gaussian Wigner matrix where +w2 = 2 (i.e., W is a GOE matrix) and the signal u(m) = (u1(m), u2(m), . . . , uN(m)) where +√ +Nui(m)’s are +i.i.d. Rademacher random variable. Note that the parameters w2 = 2 and w4 = 3. +In the numerical simulation done in Matlab, we generated 10,000 independent samples of the 256 × 256 +data matrix M, where we fix k1 = 1 (under H1) and vary k2 from 2 to 5 (under Hk2), with the SNR λ +varying from 0 to 0.7. To apply Algorithm 2, we compute +Lλ = − log det +� +(1 + λ)I − +√ +λM +� ++ λN +2 . +(A.5) +We accept H1 if +Lλ ≤ mk1 + mk2 +2 += −k2 + 2 +2 +log(1 − λ) +and reject H1 otherwise. The (theoretical) limiting error of the test is +erfc +�k2 − 1 +4 +� +− log(1 − λ) +� +. +(A.6) +In Figure 2, we compare the error from the numerical simulation and the theoretical error of the proposed +algorithm, which show that the numerical errors of the test closely match the theoretical errors. +Figure 2: The errors from the simulation with Algorithm 2 (solid) versus the limiting errors (A.6) (dashed) +for the setting in Section A.1.2 with k2 = 2, 3, 4, 5. +A.1.3 +Spiked Wigner matrix +We next consider a spiked Wigner matrix with non-Gaussian noise, where the density function of the noise +matrix is given by +g(x) = gd(x) = +1 +2 cosh(πx/2) = +1 +eπx/2 + e−πx/2 . +(A.7) +27 + +0.9 +0.8 ++ Type II error +0.7 +0.6 +0.5 +k.=2 +(,=2 (limiting) +≥ 0.4 +k,=3 +k,=3 (limiting) +0.3 +=4 (limiting) +0.2 +k,=5 (limiting) +0.1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +rWe sample Zij = Zji from the density g and let Wij = Zij/ +√ +N. +We again let the signal u(m) = +(u1(m), u2(m), . . . , uN(m)) where +√ +Nui(m)’s are i.i.d. Rademacher random variable. Note that the pa- +rameters w2 = 1 and w4 = 5. We again perform the numerical simulation 10,000 samples of the 256 × 256 +data matrix M with the SNR λ varying from 0 to 0.6, where we fix k1 = 1 (under H1) and k2 = 3 (under +H2). +In Algorithm 2, we compute +Lλ = − log det +� +(1 + λ)I − +√ +λM +� ++ λN +2 ++ +√ +λ Tr M − λ +4 (Tr M 2 − N). +(A.8) +We accept H1 if +Lλ ≤ mk1 + mk2 +2 += −k2 + 2 +2 +log(1 − λ) + k2λ +2 +− (k2 − 3)λ2 +8 +and accept H2 otherwise. The (theoretical) limiting error of the test is +erfc +� +k2 − 1 +4 +� +− log(1 − λ) + λ − λ2 +4 +� +. +(A.9) +We can further improve the test by introducing the entrywise transformation given by +h(x) = −g′(x) +g(x) = π +2 tanh πx +2 . +The Fisher information Fg = π2 +8 , which is larger than 1. We thus construct a transformed matrix � +M by +� +Mij = 2 +√ +2 +π +√ +N +h( +√ +NMij) = +� +2 +N tanh +� +π +√ +N +2 +Mij +� +. +If λ > +1 +Fg = +8 +π2 ≈ 0.8106, we can apply PCA for strong detection of the signal. If λ < +8 +π2 , applying Algorithm +3, we compute +�Lλ = − log det +�� +1 + π2λ +8 +� +I − +� +π2λ +8 +� +M +� ++ π2λN +16 ++ π +√ +λ +2 +√ +2 Tr � +M + π2λ +16 (Tr � +M 2 − N). +(Here, Fg = Fg,d = π2 +8 , GH = π2 +16 , and �w4 = 3 +2.) We accept H1 if +�Lλ ≤ −k2 + 2 +2 +log +� +1 − π2λ +8 +� ++ k2π2λ +16 +− 3π4λ2 +512 +and accept H2 otherwise. The limiting error with entrywise transformation is +erfc +� +k2 − 1 +4 +� +− log +� +1 − π2λ +8 +� ++ π2λ +8 +� +. +(A.10) +Since erfc(·) is a decreasing function and π2 +8 > 1, it is immediate to see that the limiting error in (A.10) is +strictly smaller than the limiting error in (A.9). +In Figure 3, we plot the result of the simulation with k2 = 3, which shows that the numerical error from +Algorithm 3 is smaller than that of Algorithm 2; both errors closely match theoretical errors in (A.10) and +(A.9). +28 + +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +Type I + Type II error +Alg.2 k2=3 +Alg.2 k2=3(limiting) +Alg.3 k2=3 +Alg.3 k2=3(limiting) +Figure 3: The errors from the simulation with Algorithm 2 (blue) and with Algorithm 3 (yellow), respectively, +versus the limiting errors (A.9) of Algorithm 2 (red) and (A.10) of Algorithm 3 (purple), respectively, for +the setting in Section A.1.3. +A.1.4 +Rank Estimation +We again consider the example in Section A.1.2 and apply Algorithm 4 to estimate the rank of the signal. +We again perform the numerical simulation 20,000 samples of the 256 × 256 data matrix M with the SNR +λ varying 0.025 to 0.6 and choose the rank of the signal k uniformly from 0 to 4. Since we know that the +range of the rank k is [0, 4], the (theoretical) limiting error in (4.32) changes to +P(k = 0) · P +� +Z > +√V0 +4 +� ++ +3 +� +i=1 +P(k = i) · P +� +|Z| > +√V0 +4 +� ++ P(k = 4) · P +� +Z > +√V0 +4 +� += +� +1 − P(k = 0) + P(k = 4) +2 +� +× erfc +� +1 +4 +� +− log(1 − λ) + +� 2 +w2 +− 1 +� +λ + +� +1 +w4 − 1 − 1 +2 +� +λ2 +� +. +We compute the same test statistic +Lλ = − log det +� +(1 + λ)I − +√ +λM +� ++ λN +2 +(A.11) +and find the nearest nonnegative integer of the value +− +Lλ +log(1 − λ) − 1 +2, +(A.12) +rounding half down. Since P(k = 0) = P(k = 4) = 0.2, the limiting error of the estimation is +� +1 − P(k = 0) + P(k = 4) +2 +� +· erfc +�1 +4 +� +− log(1 − λ) +� += 0.8 · erfc +�1 +4 +� +− log(1 − λ) +� +. +(A.13) +29 + +The result of the simulation can be found in Figure 4, where we compare the error from the estimation +(Algorithm 4) and the theoretical error in (A.13). We can see that the error from the numerical simulation +matches closely the theoretical error. +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.6 +0.62 +0.64 +0.66 +0.68 +0.7 +0.72 +0.74 +0.76 +Probability of error +Alg.4 +limiting error +Figure 4: The errors from the simulation with Algorithm 4 (solid) versus the limiting error (A.13) (dashed) +for the setting in Section A.1.4. +A.2 +Spiked rectangular matrices +In this section, we check the performance of the improved PCA and the pre-transformed LSS-based tests for +spiked rectangular matrices. +A.2.1 +Improved PCA with Entrywise Transformation +Additive model +We consider the data with the non-Gaussian noise whose density function is given by the bimodal dis- +tribution in (A.1). +We sample Zij independently from the density g and let Xij = Zij/ +√ +N. +We let +u(ℓ) = (u1(ℓ), u2(ℓ), . . . , uM(ℓ))T and v(ℓ) = (v1(ℓ), v2(ℓ), . . . , vN(ℓ))T , where +√ +Mui(ℓ)’s and +√ +Nvj(ℓ)’s are +i.i.d. Rademacher random variables for i = 1, 2, . . . , M, j = 1, 2, . . . , N and ℓ = 1, 2, 3. When we apply the +entrywise transformation, defined in (3.3), with α = 0 to the rank-3 spiked mean data matrix, we get +�Yij = +1 +� +FgN h( +√ +NYij) +(A.14) +where +h(x) = −g′(x) +g(x) = +2 +�√ +3 − e4 +√ +3x( +√ +3−2x) + 2x +� +1 + e4 +√ +3x +(A.15) +and Fg = +� ∞ +−∞ +(g′(x))2 +g(x) dx ≈ 2.50810. The size of the data matrix is set to be M = 2000, N = 4000, and the +ratio d0 = M/N = 0.5. +30 + +Theoretically, the threshold for the BBP-transition of the largest eigenvalue is √d0 ≈ 0.7071 with the +vanilla PCA, whereas the threshold is lowered to +√d0 +Fg +≈ 0.2819 with the improved PCA as predicted by +Theorem 3.4. +For ℓ = 1, 2, 3, we set the SNRs +λℓ = +ℓ√d0 + +√d0 +Fg +ℓ + 1 +(A.16) +to observe the transitions of the largest eigenvalue after the transformation. In Figure 5, we compare the +spectrum of the sample covariance matrices, Y Y T (left) and �Y �Y T (right). As in the spiked Wigner case in +Section A.1.1, we again find three outlier eigenvalues only in the spectrum of �Y �Y T (right), which are absent +in that of Y Y T (left). +0 +1 +2 +3 +0 +5 +10 +15 +20 +25 +30 +35 +40 +0 +1 +2 +3 +0 +5 +10 +15 +20 +25 +30 +35 +40 +Figure 5: The spectrum of the sample covariance matrix (M = 2000, N = 4000) with bimodal noise, before +(left) and after (right) the entrywise transformation. Three eigenvalues pop up from the bulk of the spectrum +after the entrywise transformation. +Multiplicative model +In the spiked covariance model, to clearly observe the outlier in our simulation setting, a distribution with +a larger Fisher information value should be used. Thus, we let the density function ga of the noise be the +generalized version of the bimodal distribution with unit variance in (A.1), defined as +ga(x) = +1 +2 +� +2(1 − a2)π +� +e +− (x−a)2 +2(1−a2) + e +− (x+a)2 +2(1−a2) +� +, +which ga is the density function of a random variable +� +1 − a2N + aR. +31 + +We sample Zij independently from the density ga and let Xij = Zij/ +√ +N. We let u(ℓ) = (u1(ℓ), u2(ℓ), . . . , uM(ℓ))T +and v(ℓ) = (v1(ℓ), v2(ℓ), . . . , vN(ℓ))T , where +√ +Mui(ℓ)’s and +√ +Nvj(ℓ)’s are i.i.d. Rademacher random vari- +ables for i = 1, 2, . . . , M, j = 1, 2, . . . , N and ℓ = 1, 2, 3. When we apply the entrywise transformation, +defined in (3.3) to the rank-3 spiked covariance data matrix, we get +�Yij = +1 +� +(α2 + 2α + Fg)N +ha,α( +√ +NYij) +(A.17) +where +ha,α(x) = −g′ +a(x) +ga(x) + αx = +� +(x − a)e +2ax +1−a2 + (x + a) +� +(1 − a2)(1 + e +2ax +1−a2 ) ++ αx +(A.18) +and Fg = +� ∞ +−∞ +(g′ +a(x))2 +ga(x) dx ≈ 5.15583, when a = +√ +21/5. The size of the data matrix is set to be M = 4000, +N = 8000. We also use α = +� +Fg, and the ratio d0 = M/N = 0.5. The threshold for the BBP-transition +of the largest eigenvalue is √d0 ≈ 0.7071 for the vanilla PCA, whereas the threshold changes to λg,ℓ = +(1+√ +Fg) +2 +· (2γℓ + +� +Fgγ2 +ℓ ) = √d0 for the transformed PCA. (See Theorem 3.5.) For ℓ = 1, 2, 3, we set the +SNRs +λℓ = +ℓ√d0 + +2√d0 +1+√ +Fg +ℓ + 1 +(A.19) +to observe the transitions of the largest eigenvalue after the transformation. +We obtain a result analogous to the additive model. See Figure 6. +0 +1 +2 +3 +0 +10 +20 +30 +40 +50 +60 +70 +80 +0 +1 +2 +3 +0 +10 +20 +30 +40 +50 +60 +70 +80 +Figure 6: The spectrum of the sample covariance matrix (M = 4000, N = 8000) with bimodal noise, before +(left) and after (right) the entrywise transformation. Three eigenvalues pop up from the bulk of the spectrum +after the entrywise transformation. +32 + +A.2.2 +Hypothesis Testing with pre-transformed LSS estimator +We now consider an (additive) spiked rectangular matrix with the non-Gaussian noise whose density function +is given by (A.7). We let the signal u = (u1, u2, . . . , uM)T and v = (v1, v2, . . . , vN)T , where +√ +Mui’s and +√ +Nvj’s are i.i.d. Rademacher random variables for i = 1, 2, . . . , M and j = 1, 2, . . . , N. Let the data matrix +Y = +√ +λuvT + X. +Recall that w4 = 5, Fg = π2 +8 , GH = π2 +16 , and �w4 = 3 +2. The LSS estimators are given by +Lω = − log det +�� +1 + d0 +ω +� +(1 + ω)I − Y Y T +� +− ω +2d0 +(Tr Y Y T − M) ++ M +� ω +d0 +− log +� ω +d0 +� +− 1 − d0 +d0 +log(1 + ω) +� +, +(A.20) +and +�Lω = − log det +�� +1 + 8d0 +ωπ2 +� +(1 + ω π2 +8 )I − �Y �Y T +� ++ π2ω +8d0 +(Tr �Y �Y T − M) ++ M +�ωπ2 +8d0 +− log +�ωπ2 +8d0 +� +− 1 − d0 +d0 +log +� +1 + ω π2 +8 +�� +. +(A.21) +With critical values mω = − log +� +1 − ω2 +d0 +� ++ 3ω2 +4d0 and �mω = − log +� +1 − ω2π4 +64d0 +� +− 3π4ω2 +256d0, the errors are +erfc +� +1 +4 +� +− log +� +1 − ω2 +d0 +� +− ω2 +2d0 +� +and +erfc +� +1 +4 +� +− log +� +1 − π4ω2 +64d0 +�� +. +In Figure 7, we plot empirical average (after 1,000 Monte Carlo simulations) of the error of the proposed +test and the theoretical (limiting) error, varying the SNR ω from 0 to 0.5, with M = 256 and N = 512. It +can be checked that the error of the proposed test closely matches the theoretical error. +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +0.35 +0.4 +0.45 +0.5 +0.65 +0.7 +0.75 +0.8 +0.85 +0.9 +0.95 +1 +simulation +limiting error +simulation-transformed +limiting error-tansformed +Figure 7: The error from the simulation (solid) and the theoretical limiting error (dashed), respectively. +33 + +B +Proof of Theorems for improved PCA +In this section, we rigorously prove Theorems 3.4 and 3.5 in Section 3, which are about the detection threshold +of the improved PCA. +B.1 +Preliminaries +We first introduce the following notions, which provide a simple way of making precise statements regarding +the bound up to small powers of N that holds with probability higher than 1 − N −D for all D > 0. +Definition B.1 (Overwhelming probability). We say that an event (or family of events) Ω holds with +overwhelming probability if for all (large) D > 0 we have P(Ω) ≤ N −D for any sufficiently large N. +Definition B.2 (Stochastic domination). Let +ξ = +� +ξ(N)(u) : N ∈ N, u ∈ U (N)� +, +ζ = +� +ζ(N)(u) : N ∈ N, u ∈ U (N)� +be two families of random variables, where U (N) is a possibly N-dependent parameter set. We say that ξ is +stochastically dominated by ζ, uniformly in u, if for all (small) ϵ > 0 and (large) D > 0 +sup +u∈U (N) P +� +|ξ(N)(u)| > N ϵζ(N)(u) +� +≤ N −D +for any sufficiently large N ≥ N0(ε, D). Throughout this appendix, the stochastic domination will always be +uniform in all parameters, including matrix indices and the spectral parameter z. +We write ξ ≺ ζ or ξ = O≺(ζ), if ξ is stochastically dominated by ζ, uniformly in u. +For a Wigner matrix W, we will use the following result for the resolvents, which is called an isotropic +local semicircle law. +Lemma B.3 (Isotropic local semicircle law). Suppose that z ∈ R outside an open interval containing [−2, 2]. +Let ssc(z) be the Stieltjes transform of the Marchenko–Pastur law, which is also given by +ssc(z) = −z + +√ +z2 − 4 +2 +. +(B.1) +Then, +⟨u(ℓ1), (W − zI)−1u(ℓ2)⟩ = ssc(z)⟨u(ℓ1), u(ℓ2)⟩ + O≺(N − 1 +2 ) +See Theorem 2.3 of [36] (also Lemma 7.7 of [22]) for the proof of Lemma B.3. +Further, for a rectangular matrix X, we will use the following analogous result for the resolvents, which +is called an isotropic Marchenko–Pastur law. +Lemma B.4 (Isotropic local Marchenko–Pastur law). Suppose that z ∈ R outside an open interval containing +[d−, d+]. Let s(z) be the Stieltjes transform of the Marchenko–Pastur law, which is also given by +s(z) = (1 − d0 − z) + +� +(1 − d0 − z)2 − 4d0z +2d0z +. +(B.2) +Then, +⟨v(ℓ1), (XT X − zI)−1v(ℓ2)⟩ = − +� +1 +zs(z) + 1 +� +⟨v(ℓ1), v(ℓ2)⟩ + O≺(N − 1 +2 ) +34 + +and +⟨XT u(ℓ1), (XT X − zI)−1XT u(ℓ2)⟩ = (zs(z) + 1)⟨u(ℓ1), u(ℓ2)⟩ + O≺(N − 1 +2 ). +See Theorem 2.5 of [17] (also Lemma 3.7 of [18]) for the proof of Lemma B.4. +The following concentration inequality will be frequently used in the proof, which is sometimes called the +large deviation estimate in random matrix theory. +Lemma B.5 (Large deviation estimate). Let +� +ξ(N) +i +� +and +� +ζ(N) +i +� +be independent families of random variables +and +� +a(N) +ij +� +and +� +b(N) +i +� +be deterministic; here N ∈ N and i, j = 1, . . . , N. +Suppose that complex-valued +random variables ξ(N) +i +and ζ(N) +i +are independent and satisfy for all p ≥ 2 that +Eξ = 0 , +E|ξ|p ≤ +Cp +NBp−2 +(B.3) +for some B ≤ N 1/2 and some (N-independent) constant Cp. Then we have the bounds +� +i +biξi ≺ +� 1 +N +� +i +|bi|2 +�1/2 ++ maxi |bi| +B +, +(B.4) +� +i,j +aijξiζj ≺ +� 1 +N 2 +� +i̸=j +|aij|2 +�1/2 ++ maxi̸=j |aij| +B ++ maxi |aii| +B2 +, +(B.5) +� +i̸=j +aijξiξj ≺ +� 1 +N 2 +� +i̸=j +|aij|2 +�1/2 ++ maxi̸=j |aij| +B +. +(B.6) +If the coefficients a(N) +ij +and b(N) +i +depend on an additional parameter u, then all of these estimates are uniform +in u, i.e. N0 = N0(ε, D) in the definition of ≺ depends not on u but only on the constant C from (B.3). +If B = N 1/2, the bounds can further be simplified to +� +i +biξi ≺ +� 1 +N +� +i +|bi|2 +�1/2 +, +� +i,j +aijξiζj ≺ +� 1 +N 2 +� +i,j +|aij|2 +�1/2 +, +� +i̸=j +aijξiξj ≺ +� 1 +N 2 +� +i̸=j +|aij|2 +�1/2 +. +(B.7) +Proof. These estimates are an immediate consequence of Lemma 3.8 in [30]. +Finally, we recall that, for our prior, |⟨u(ℓ1), u(ℓ2)⟩ − δℓ1ℓ2|, |⟨v(ℓ1), v(ℓ2)⟩ − δℓ1ℓ2| ≺ N −φ. +B.2 +Proof of Theorem 3.3 +We first prove the behavior of the k largest eigenvalues described in Section 2.2, which we will call the BBP +result, in our setting, following the strategy of [14, 15]. +M − zI = W + UΛ1/2U T − zI += (W − zI)(I + (W − zI)−1(UΛ1/2U T )). +(B.8) +Thus, if z is an eigenvalue of M but not of W, then it satisfies +det(I + (W − zI)−1UΛ1/2U T ) = 0, +which also implies that −1 is an eigenvalue of +T ≡ T(z) := (W − zI)−1UΛ1/2U T . +35 + +We then see that +T(W − zI)−1u(ℓ) = (W − zI)−1UΛ1/2U T (W − zI)−1u(ℓ) += +� +λℓ⟨u(ℓ), (W − zI)−1u(ℓ)⟩(W − zI)−1u(ℓ) + O≺(N −φ). +i.e., (W − zI)−1u(ℓ) must be a eigenvector for T with the corresponding eigenvalue −1. Thus, by Lemma +B.3, +� +λℓssc(z) = −1 + O≺(N −φ). +It is elementary to check that the solution of the above equation is z = √λℓ + +1 +√λℓ + O≺(N −φ) if and only +if λℓ > 1. +We now turn to the proof of Theorem 3.3. For the spike ∥U∥∞ ≺ N −φ, suppose that a function q and +its all derivatives are polynomially bounded in the sense of Assumption 3.1. Following the proof of Theorem +4.8 in [50], we have the following local linear estimation of q( +√ +NMij) by +q( +√ +NMij) = q( +√ +NWij) + +√ +λNuiuT +j E[q′( +√ +NWij)] + Rij, +where the error Rij is negligible. Set +Mq := E[q′( +√ +NWij)], +Vq := E[q( +√ +NWij)2], +�λ := λM 2 +q /Vq, +and +Qij := +1 +� +NVq +q( +√ +NWij). +Then the spectrum of the transformed matrix is determined by the matrix Q + U ˆΛ1/2U T . Since Q is also +Wigner matrix with NE[Q2 +ij] = 1, by repeating the same process above, we get the result. +B.3 +Proof of Theorem 3.4 +We first prove the behavior of the k largest eigenvalues described in Section 2.2, which we will again call the +BBP result, in our setting, following the strategy of [14, 15]. Note that the k largest eigenvalue of Y Y T is +equal to the k largest eigenvalues of Y T Y . Consider the identity +Y T Y − zI = (X + UΛ1/2V T )T (X + UΛ1/2V T ) − zI = (XT X − zI)T(z) +(B.9) +where +T ≡ T(z) := (XT X − zI)−1(XT UΛ1/2V T + V Λ1/2U T X + V Λ1/2U T UΛ1/2V T ). +Thus, if z is an eigenvalue of Y Y T but not of XXT , then it satisfies +det(T(z)) = 0, +which also implies that −1 is an eigenvalue of T(z). +36 + +Note that since ∥X∥, ∥(XT X − zI)−1∥ ≺ 1, from Lemma B.5, +⟨b, (XT X − zI)−1XT a⟩ = +� +i,j +� +(XT X − zI)−1XT � +ij biaj +≺ +� +� 1 +N 2 +� +i̸=j +��� +� +(XT X − zI)−1XT � +ij +��� +2 +� +� +1/2 ++ N −φ max +i,j +��� +� +(XT X − zI)−1XT � +ij +��� +≺ +� 1 +N ∥(XT X − zI)−1XT ∥2 +�1/2 ++ N −φ∥(XT X − zI)−1XT ∥ ≺ N −φ. +Then the matrix T satisfies +T · (XT X − zI)−1XT u(ℓ) += (XT X − zI)−1XT UΛ1/2(V T (XT X − zI)−1XT u(ℓ)) ++ (XT X − zI)−1V Λ1/2(U T X(XT X − zI)−1XT u(ℓ)) ++ (XT X − zI)−1V Λ1/2(U T U)Λ1/2(V T (XT X − zI)−1XT u(ℓ)) += +� +λℓ⟨u(ℓ), X(XT X − zI)−1XT u(ℓ)⟩(XT X − zI)−1v(ℓ) + θ1(ℓ) +and +T · (XT X − zI)−1v(ℓ) += (XT X − zI)−1XT UΛ1/2(V T (XT X − zI)−1v(ℓ)) ++ (XT X − zI)−1V Λ1/2(U T X(XT X − zI)−1v(ℓ)) ++ (XT X − zI)−1V Λ1/2(U T U)Λ1/2(V T (XT X − zI)−1v(ℓ)) += +� +λℓ⟨v(ℓ), (XT X − zI)−1v(ℓ)⟩(XT X − zI)−1XT u(ℓ) ++ λℓ⟨v(ℓ), (XT X − zI)−1v(ℓ)⟩(XT X − zI)−1v(ℓ) + θ2(ℓ) +where ∥θ1(ℓ)∥, ∥θ2(ℓ)∥ = O≺(N −φ) since ∥U T U − I∥F ≺ N −φ. +In particular, k extremal eigenvectors of T are a linear combination of (XT X−zI)−1XT u(ℓ) and (XT X− +zI)−1v(ℓ). +Suppose that aℓ(XT X−zI)−1XT u(ℓ)+bℓ(XT X−zI)−1v(ℓ) is an eigenvector of T with the corresponding +eigenvalue −1. Thus, from Lemma B.4, +− +� +aℓ(XT X − zI)−1XT u(ℓ) + bℓ(XT X − zI)−1v(ℓ) +� += T +� +aℓ(XT X − zI)−1XT u(ℓ) + bℓ(XT X − zI)−1v(ℓ) +� += −bℓ +� +λℓ +� +1 +zs(z) + 1 +� +(XT X − zI)−1XT u(ℓ) ++ aℓ +� +λℓ(zs(z) + 1)(XT X − zI)−1v(ℓ) − bℓλℓ +� +1 +zs(z) + 1 +� +(XT X − zI)−1v(ℓ) + �θ(ℓ) +(B.10) +for some �θ(ℓ), which is a linear combination of (XT X −zI)−1XT u(ℓ) and (XT X −zI)−1v(ℓ) with ∥�θ(ℓ)∥ = +O≺(N −φ). +Since U, V , and X are independent, (XT X − zI)−1XT u(ℓ) and (XT X − zI)−1v(ℓ) are linearly inde- +37 + +pendent with overwhelming probability. Thus, from (B.10), +−aℓ = −bℓ +� +λℓ +� +1 +zs(z) + 1 +� ++ O≺(N −φ), +−bℓ = aℓ +� +λℓ(zs(z) + 1) − bℓλℓ +� +1 +zs(z) + 1 +� ++ O≺(N −φ). +It is then elementary to check that +λℓ(zs(z) + 1) + 1 = O≺(N −φ), +which has the solution +z = (1 + λℓ) +� +1 + d0 +λℓ +� ++ O≺(N −φ) +if and only if λℓ > √d0. This proves the BBP result in our setting. +We now turn to the proof of Theorem 3.4. To simplify the exposition, we focus on the case that SNRs are +the same i.e., Λ = λI. For the spike prior in Assumption 3.1, suppose that a function q and its all derivatives +are polynomially bounded in the sense of Assumption 3.1. Following the proof of Theorem 4.8 in [50], we +define the error term from the local linear estimation of q( +√ +NYij) by +q( +√ +NYij) = q( +√ +NXij) + +√ +λNuivT +j q′( +√ +NXij) + Rij +where +Rij = 1 +2q′′( +√ +NXij + eij)λN(uivT +j )2 +for some |eij| ≤ | +√ +λNuivT +j |. The Frobenius norm of R is bounded as +∥R∥2 +F = Tr RT R = λ2N 2 +4 +M +� +i=1 +N +� +j=1 +(uivT +j )4q′′( +√ +NXij + eij)2 +≤ λ2N 2−4φ +4 +M +� +i=1 +N +� +j=1 +(uivT +j )2q′′( +√ +NXij + eij)2. +Since q′′ is polynomially bounded, q′′( +√ +NXij + eij) is uniformly bounded by an N-independent constant. +Thus, with overwhelming probability, +∥R∥2 ≤ ∥R∥2 +F ≤ Cλ2N 2−4φ. +Next, we approximate q( +√ +NXij) by its mean. Let +Eij = q′( +√ +NXij) − E[q′( +√ +NXij)], +∆ij = +√ +λNuivT +j Eij. +Then, ∥∆∥ ≺ N +1 +2 −2φ∥E∥ and, since the entries of matrix E are i.i.d., centered and with finite moments, its +norm ∥E∥ = O( +√ +N) with overwhelming probability. (See, e.g., [18].) Thus, ∥∆∥ = O≺(N 1−2φ). +Set +Mq := E[q′( +√ +NXij)], +Vq := E[q( +√ +NXij)2], +�λ := λM 2 +q /Vq, +38 + +and +Qij := +1 +� +NVq +q( +√ +NXij). +We have proved so far that the difference of the largest eigenvalue of Q + �λ +1 +2 UV T and that of the matrix +� +1 +� +NVq +q( +√ +NYij) +� +is O≺(N +1 +2 −2φ), which is o(1) with overwhelming probability for φ > 1 +4. It is directly applicable to the case +that Λ in our model with �Λℓℓ := λℓM 2 +q /Vq since the above process does not require any information of the +SNRs. The BBP result holds the matrix Q+U �Λ +1 +2 V T , which is another (additive) spiked rectangular matrix. +This shows that the BBP result also holds for �Y with SNR matrix �Λ := +M 2 +q +Vq Λ. This proves Theorem 3.4. +B.4 +Proof of Theorem 3.5 +Recall that the spike prior satisfies the technical conditions in Assumption 3.1 with φ > 1/4. For the sake +of brevity, we assume that Λ = λI. As in the additive case, we further assume that a function q and its all +derivatives are polynomially bounded and consider the local linear approximation of q( +√ +NYij), +q( +√ +NYij) = q( +√ +NXij) + γ +√ +NE[q′( +√ +NXij)] +� +ℓ +uiuT +ℓ Xℓj + Rij + γ∆ij, +(B.11) +where +Rij = 1 +2q′′�√ +NXij + θγ +� +ℓ +uiuT +ℓ +√ +NXℓj +� � +γ +� +ℓ +uiuT +ℓ +√ +NXℓj +�2 +for some θ ∈ [−1, 1] and +∆ij = +√ +NEij +� +ℓ +uiuT +ℓ Xℓj, +Eij = q′( +√ +NXij) − E[q′( +√ +NXij)]. +For any unit vectors a = (a1, a2, . . . , aM) and b = (b1, b2, . . . , bN), +aT ∆b = +� +s +� +i,j +aiui(s)Eijbj +�� +ℓ +uℓ(s) +√ +NXℓj +� += +� +s +� +i,j +aiui(s)2bjEij +√ +NXij + +� +s +� +i,j +aiui(s)Eijbj +� +�� +ℓ̸=i +uℓ(s) +√ +NXℓj +� +� +From the concentration inequalities such as Lemma B.5, +� +ℓ +uiuT +ℓ +√ +NXℓj = +� +s +� +ℓ +ui(s)uℓ(s)T √ +NXℓj ≺ +� +s +|ui(s)| +�� +ℓ +uℓ(s)2 +�1/2 +≺ N −φ. +(B.12) +Recall that ∥E∥ = O( +√ +N) with overwhelming probability. +Note that, by Assumption 3.1, the density +function q have to be an odd function. Further, since q is an odd function (hence xq′(x) is an odd function +39 + +of x), the norm of the matrix whose (i, j)-entry is Eij +√ +NXij is also O( +√ +N). Thus, +aT ∆b ≺ N −2φ + N +1 +2 −φ, +which shows that ∥∆∥ ≺ N +1 +2 −φ. Moreover, since q′′ is polynomially bounded, following the proof of Theorem +3.4 with (B.12), +∥R∥2 ≤ ∥R∥2 +F ≤ CN 2−4φ. +Thus, as in the additive case, the error terms Rij and ∆ij in (B.11) are negligible when finding the limit of +the extreme eigenvalues of the transformed matrix. +Set +Mq := E[q′( +√ +NXij)], +Vq := E[q( +√ +NXij)2], +Eq = E[ +√ +NXijq( +√ +NXij)], +�γ := γMq/ +� +Vq, +and +Qij := +1 +� +NVq +q( +√ +NXij). +With the approximation (B.11), we now focus on the largest eigenvalue of +(Q + �γUU T X)T (Q + �γUU T X). +Note that the assumption on the polynomial boundedness of q implies that the matrix Q is also a rectangular +matrix satisfying the assumptions in Definition 2.2. +Let G(z) and G(z) be the resolvents +G ≡ G(z) := (QQT − zI)−1, +G ≡ G(z) := (QT Q − zI)−1 +for z ∈ R outside an open interval containing [d−, d+]. We note that the following identities hold for G(z) +and G(z): +G(z)Q = QG(z), +QT G(z)Q = I + zG(z). +(B.13) +As in the proof of Theorem 3.4, we consider +(Q + �γUU T X)T (Q + �γUU T X) − zI += (QT Q − zI)(I + (QT Q − zI)−1(�γXT UU T Q + �γQT UU T X + �γ2XT UU T UU T X)). +(B.14) +Let +L ≡ L(z) = G(z)(�γXT UU T Q + �γQT UU T X + �γ2XT UU T UU T X), +Then, as in the proof of Theorem 3.4, if z is an eigenvalue of (Q + �γUU T X)T (Q + �γUU T X) (but not of +QT Q), −1 is an eigenvalue of L(z). Again, the rank of L is at most 2k, with +L · GQT U = �γGXT UU T QGQT U + �γGQT UU T XGQT U + �γ2GXT UU T UU T XGQT U, +L · GXT U = �γGXT UU T QGXT U + �γGQT UU T XGXT U + �γ2GXT UU T UU T XGXT U, +(B.15) +and an eigenvector of L is a linear combination of GQT u(ℓ) and GXT u(ℓ) for 1 ≤ ℓ ≤ k. +In the simplest case where Q is the identity mapping, Q = X, hence the rank of L is k, and the eigenvalue +equation (B.15) is simplified to +L · GQT U = �γGQT U(U T QGQT U) + �γGQT U(U T QGQT U) + �γ2GQT U(U T UU T QGQT U). +(B.16) +40 + +In this case, GQT u(ℓ) are eigenvectors of L corresponding to the eigenvalue −1, i.e., L · GQT u(ℓ) = +−GQT u(ℓ). +The right side of (B.16) can be approximated as follows, which is a direct consequence of +the isotropic local Marchenko–Pastur law (e.g., Theorem 2.5 of [17]). +With the isotropic local Marchenko–Pastur law, (B.16) can be approximated by a deterministic vector +equation on z (and s(z)), and the location of the k largest eigenvalues can be proved by solving the equation. +In a general case where Q is not a multiple of X and the vectors GQT u(ℓ) and GXT u(ℓ) are linearly +independent, however, the eigenvalue equation (B.15) contains other matrices U T QGQT U, U T QGXT U, +and U T XGQT U, which cannot be estimated by Lemma B.4. +For these matrices, we use the following +lemma. +Lemma B.6. Suppose that the assumptions in Lemma B.4 hold. Then, +⟨u(ℓ1), XGQT u(ℓ2)⟩ = ⟨u(ℓ1), QGXT u(ℓ2)⟩ = +� +Eq +� +Vq +(zs(z) + 1) +� +δℓ1ℓ2 + O≺(N −φ) +and +⟨u(ℓ1), XGXT u(ℓ2)⟩ = +� +E2 +q +Vq +zs(z) +� +d0s(z) + d0 − 1 +z +�2 ++ d0s(z) + d0 − 1 +z +� +δℓ1ℓ2 + O≺(N −φ). +We defer the proof to Appendix B.6. +With Lemma B.6, we are ready to finish the proof. From the definition of s(z) in Lemma B.4, we notice +that +s(z) = +1 +1 − d0 − d0zs(z) − z , +(B.17) +or +z +� +d0s(z) + d0 − 1 +z +� += − 1 +s(z) − z. +(B.18) +Set σ(z) := zs(z) + 1. By applying Lemmas B.4 and B.6 to (B.16), for 1 ≤ ℓ ≤ k +L · GQT u(ℓ) = �γ⟨u(ℓ), QGQT u(ℓ)⟩ · GXT u(ℓ) + �γ⟨u(ℓ), XGQT u(ℓ)⟩ · GQT u(ℓ) ++ ∥u(ℓ)∥2�γ2⟨u(ℓ), XGQT u(ℓ)⟩ · GXT u(ℓ) += �γσ(z)GXT u(ℓ) + �γσ(z) Eq +� +Vq +GQT u(ℓ) + �γ2 Eq +� +Vq +σ(z)GXT u(ℓ) + θ1(ℓ) , +(B.19) +and +L · GXT u(ℓ) = �γ⟨u(ℓ), QGXT u(ℓ)⟩ · GXT u(ℓ) + �γ⟨u(ℓ), XGXT u(ℓ)⟩ · GQT u(ℓ) ++ ∥u(ℓ)∥2�γ2⟨u(ℓ), XGXT u(ℓ)⟩ · GXT u(ℓ) += �γσ(z) Eq +� +Vq +GXT u(ℓ) + �γ +�� +σ(z) + +σ(z) +σ(z) − 1 +� E2 +q +Vq +− +σ(z) +σ(z) − 1 +� +GQT u(ℓ) ++ �γ2 +�� +σ(z) + +σ(z) +σ(z) − 1 +� E2 +q +Vq +− +σ(z) +σ(z) − 1 +� +GXT u(ℓ) + θ2(ℓ) , +(B.20) +for some θ1(ℓ), θ2(ℓ), which are linear combinations of GQT u(ℓ) and GXT u(ℓ), with ∥θ1(ℓ)∥, ∥θ2(ℓ)∥ = +O≺(N −φ). +Suppose that aℓGQT u(ℓ)+bℓGXT u(ℓ) is an eigenvector of L with the corresponding eigenvalue −1. From +41 + +(B.19), (B.20), and the linear independence between GQT u(ℓ) and GXT u(ℓ), we find the relation +−aℓ = aℓ�γσ(z) Eq +� +Vq ++ bℓ�γσ(z)2 +σ(z) − 1 · E2 +q +Vq +− bℓ�γσ(z) +σ(z) − 1 + O(N −φ), +−bℓ = aℓ�γσ(z) + aℓ�γ2σ(z) Eq +� +Vq ++ bℓ�γσ(z) Eq +� +Vq ++ bℓ�γ2σ(z)2 +σ(z) − 1 · E2 +q +Vq +− bℓ�γ2σ(z) +σ(z) − 1 + O(N −φ). +We then find that +bℓ +aℓ +� +1 + �γσ(z) Eq +� +Vq +� ++ �γσ(z) − �γ = O(N −φ) +and +aℓ +� +1 + �γσ(z) Eq +� +Vq +� += bℓ +� +�γσ(z) +σ(z) − 1 +� +1 − σ(z) · E2 +q +Vq +�� ++ O(N −φ), +which implies that +1 + 2�γσ(z) Eq +� +Vq ++ �γ2σ(z) = 1 + +� +2γMqEq + γ2M 2 +q +Vq +� +σ(z) = O(N −φ). +(B.21) +From the explicit formula for s, it is not hard to check that (B.21) holds if and only if +λq := 2γMqEq + γ2M 2 +q +Vq +> +� +d0 +and +z = (1 + λq) +� +1 + d0 +λq +� ++ O(N −φ). +(B.22) +We see that it is valid for general Λ in our model, since the above process also does not require any information +of the SNRs as in the additive case. Now, the desired theorem follows from the direct computation for the +case q = hαg; see also Appendix B.5.2. +B.5 +Optimal entrywise transformation +B.5.1 +Additive model +Recall that +E[q′( +√ +NWij)] = E[q′( +√ +NXij)] = Mq, +E[q( +√ +NWij)2] = E[q( +√ +NXij)2] = Vq. +Following the proof of Theorem 3.4 in Appendix B.3, it is not hard to see that the effective SNR is maximized +by optimizing M 2 +q /Vq. Such an optimization problem was already considered in [50] for the spiked Wigner +matrix. For the sake of completeness, we solve this problem by using the calculus of variations. Recall the +density of random variables +√ +NWij and +√ +NXij is g. +To optimize q, we need to maximize +�� ∞ +−∞ +q′(x)g(x)dx +�2 +/ +�� ∞ +−∞ +q(x)2g(x)dx +� += +�� ∞ +−∞ +q(x)g′(x)dx +�2 +/ +�� ∞ +−∞ +q(x)2g(x)dx +� +. +(B.23) +Putting (q + εη) in place of q in (B.23) and differentiating with respect to ε, we find that the optimal q +42 + +satisfies +�� ∞ +−∞ +η(x)g′(x)dx +� �� ∞ +−∞ +q(x)2g(x)dx +� += +�� ∞ +−∞ +q(x)η(x)g(x)dx +� �� ∞ +−∞ +q(x)g′(x)dx +� +(B.24) +for any η. It is then easy to check that q = −Cg′/g is the only solution of (B.24). Since the value in (B.23) +does not change if we replace q by Cq, and the effective SNR is increased with the entrywise transform −g′/g +is the optimal entrywise transformation for PCA. +B.5.2 +Multiplicative model +As we can see from the proof of Theorem 3.5 in Appendix B.4, we need to maximize +2 +�� ∞ +−∞ xq(x)g(x)dx +� �� ∞ +−∞ q′(x)g(x)dx +� ++ γ +�� ∞ +−∞ q′(x)g(x)dx +�2 +�� ∞ +−∞ q(x)2g(x)dx +� += +−2 +�� ∞ +−∞ xq(x)g(x)dx +� �� ∞ +−∞ q(x)g′(x)dx +� ++ γ +�� ∞ +−∞ q(x)g′(x)dx +�2 +�� ∞ +−∞ q(x)2g(x)dx +� +. +(B.25) +Putting (q + εη) in place of q in (B.23) and differentiating with respect to ε, we find that the optimal q +satisfies +− 2 +�� +xηg +� �� +qg′ +� �� +q2g +� +− 2 +�� +xqg +� �� +ηg′ +� �� +q2g +� ++ 2γ +�� +ηg′ +� �� +qg′ +� �� +q2g +� ++ 4 +�� +qηg +� �� +xqg +� �� +qg′ +� +− 2γ +�� +qg′ +�2 �� +qηg +� += 0 +(B.26) +which is written with slight abuse of notation such as +� +xηg = +� ∞ +−∞ xη(x)g(x)dx. Since the equation contains +the terms +� +xηg, +� +ηg′, +� +qηg, +it is natural to consider an ansatz +q(x) = −g′(x) +g(x) + αx +(B.27) +for a constant α. Collecting the terms involving +� +xηg and the terms involving +� +ηg′, we get +2(Fg + α)(Fg + 2α + α2) − 4α(1 + α)(Fg + α) − 2αγ(Fg + α)2 = 0 +and +−2(1 + α)(Fg + 2α + α2) − 2γ(Fg + α)(Fg + 2α + α2) + 4(1 + α)(Fg + α) + 2γ(Fg + α)2 = 0. +We can then check that +α = αg = +−γFg + +� +4Fg + 4γFg + γ2F 2g +2(1 + γ) +, +43 + +and hence (B.26) is satisfied with +q(x) = −g′(x) +g(x) + +−γFg + +� +4Fg + 4γFg + γ2F 2g +2(1 + γ) +x. +The corresponding effective SNR +λhαg ≡ λg = γ + γ2Fg +2 ++ +γ +� +4Fg + 4γFg + γ2F 2g +2 +. +For a general α, when the entrywise transform hα is applied, the effective SNR +λhα = 2γ(1 + α)(Fg + α) + γ2(α + Fg)2 +α2 + 2α + Fg +, +In particular, if α = +� +Fg, +λh√ +Fg = γ(1 + +� +Fg) + γ2 +2 (Fg + +� +Fg) ≥ 2γ + γ2 = λ +where the inequality is strict if Fg > 1. +B.6 +Proof of Lemma B.6 +B.6.1 +Key ingredient: Entrywise local estimates +Recall the definition of the random matrices X and Q. The couple of random matrices (X, Q) is one example +of the following concept for a coupled random matrices: +Definition B.7 (Entrywise correlated random matrices). Suppose that A and B are M × N random rectan- +gular matrices in Definition 2.2 satisfying the following conditions: +• For all 1 ≤ a, b ≤ M and 1 ≤ α, β ≤ N, Aaα and Bbβ are dependent only when a = b and α = β. +• For all a, α, E[Aaα] = E[Baα] = 0, NE[A2 +aα] = wA, NE[B2 +aα] = wB, and NE[AaαBaα] = wAB. +• For any positive integer p, there exists Cp, independent of N, such that +N +p +2 E[Ap +aα], N +p +2 E[Bp +aα] ≤ Cp +for all a, α. +A couple of random matrices (A, B) is called the entrywise correlated. +The key estimates in the proof of Lemma B.6 are the exact bounds on the entries of K := Q(QQT − +zI)−1XT and K := X(QQT − zI)−1XT . We prove the following lemma for the entrywise correlated random +matrices (A, B), which exactly contains the desired result. +Lemma B.8. Let (A, B) be the entrywise correlated random matrices with wB = 1. For z ∈ R outside an +open interval containing [d−, d+], +|(A(BT B − zI)−1AT )ij − (wAs(z) + w2 +ABzs(z)s(z)2)δij| = O≺(N −1/2), +(B.28) +44 + +|(A(BT B − zI)−1BT )ij − (wABs(z) + wABzs(z)s(z)2)δij| = O≺(N −1/2) +(B.29) +and +|(B(BT B − zI)−1AT )ij − (wABs(z) + wABzs(z)s(z)2)δij| = O≺(N −1/2). +(B.30) +Remark B.9. Recall that σ(z) = zs(z) + 1. For (X, Q), since wX = wQ = 1 and wXQ = Eq/ +� +Vq, we have +the following: +For z ∈ R outside an open interval containing [d−, d+], +|Kij − �s(z)δij| = O≺(N −1/2), +|Kij − ˇs(z)δij| = O≺(N −1/2), +(B.31) +where +�s(z) := σ(z) Eq +� +Vq +, +ˇs(z) := zs(z) +� +d0s(z) + d0 − 1 +z +�2 E2 +q +Vq ++ +� +d0s(z) + d0 − 1 +z +� +. +(B.32) +B.6.2 +Linearization +We consider +G ≡ GB(z) = (BBT − zI)−1, +G ≡ GB(z) = (BT B − zI)−1. +In the proof of Lemma B.8, we use the formalism known as the linearization to simplify the computation. +We define an (M + N) × (M + N) symmetric matrix HB by +HB ≡ HB(z) = +�−zIM +B +BT +−IN +� +, +(B.33) +where IM and IN are the identity matrices with size M and N, respectively. +Let RB(z) = HB(z)−1. +(For the invertibility of HB(z), we refer to Section 5.1 in [38].) +By Schur’s +complement formula, +RQ(z) = +� GB(z) +GB(z)B +BT GB(z) +zGB(z) +� +. +(B.34) +Therefore, +Rab(z) = (BBT − zI)−1 +ab = Gab(z), +Rαβ(z) = z(BT B − zI)−1 +α−M,β−M = zGα−M,β−M(z), +(B.35) +and +Rαa(z) = Raα(z) = (GB)a,α−M(z), +(B.36) +where we use lowercase Latin letters a, b, c, . . . for indices from 1 to M and Greek letters α, β, γ, . . . for +indices from (M + 1) to (M + N). We also use uppercase Latin letters A, B, C, . . . for indices from 1 to +(M + N). In the rest of Appendix B, we omit the subscript Q for brevity. +For T ⊂ {1, 2, . . . , M + N}, we define the matrix minor H(T) by +(H(T))AB := 1{A,B /∈T}HAB . +(B.37) +Moreover, for A, B /∈ T we define +R(T) +AB(z) := (H(T))−1 +AB, +(B.38) +In the definitions above, we abbreviate ({A}) by (A); similarly, we write (AB) instead of ({A, B}). +We have the following resolvent (decoupling) identities for the matrix entries of R and R(T), which are +45 + +elementary consequences of Schur’s complement formula; see e.g. Lemma 5.1 of [38]. +Lemma B.10 (Resolvent identities for R). Suppose that z ∈ R is outside an open interval containing +[d−, d+]. +- For a ̸= b, +Rab = −Raa +� +α +HaαR(a) +αb = −Rbb +� +β +R(b) +aβHβb. +- For α ̸= β, +Rαβ = −Rαα +� +a +HαaR(α) +aβ = −Rββ +� +b +R(β) +αb Hbβ. +- For any a and α, +Raα = −Raa +� +β +HaβR(a) +βα = −Rαα +� +b +R(α) +ab Hbα. +- For A, B ̸= C, +RAB = R(C) +AB + RACRCB +RCC +. +Throughout this section, we will frequently use the estimate that all entries of X and Q (and hence all +off-diagonal entries of W) are O≺(N −1/2), which holds since all moments of the entries of +√ +NQ and +√ +NX +are bounded. For the entries of R, we have the following estimates: +Lemma B.11. Let +s(z) = +� +d0s(z) + d0 − 1 +z +� +. +(B.39) +For z ∈ R outside an open interval containing [d−, d+], +|Rij(z) − s(z)δij| , |Rµν(z) − zs(z)δµν| , |Riµ(z)| ≺ N −1/2. +(B.40) +Proof of Lemma B.11. The first two estimates can be checked from Theorem 2.5 (and Remark 2.7) in [17] +with the deterministic unit vectors v = ei and w = ej where ei ∈ RN or RM is a standard basis vector +whose i-th coordinate is 1 and all other coordinates are zero. For the last estimate, we apply Lemma B.10 +to find that +Riµ(z) = −Rii +� +α +HiαR(i) +αµ. +Since Hiα and R(i) +αµ are independent, R(i) +αµ ≺ N −1/2 for α ̸= µ, and R(i) +µµ = Θ(1) with overwhelming probability, +we find from Lemma B.5 that +� +α +HiαR(i) +αµ ≺ +� +1 +N +� +α +|R(i) +αµ|2 +�1/2 +≺ N −1/2. +Proof of Lemma B.8. Throughout this section, for the sake of brevity, we will use the notation +Baα := Ba,(α−M) = Haα, +Aaα := Aa,(α−M). +We begin by estimating the diagonal entry (BGAT )ii. From Schur’s complement formula, (B.35), we can +decompose it into +(BGAT )ii = 1 +z +� +α +HiαRααAiα + 1 +z +� +α̸=β +HiαRαβAiβ. +(B.41) +46 + +From concentration inequalities it is not hard to see that +� +α +BiαAiα = E[BiαAiα] + O≺(N −1/2) = wAB + O≺(N −1/2). +Applying Lemma B.11, we find for the first term in the right side of (B.41) that +1 +z +� +α +HiαRααAiα = wABs(z) + O≺(N −1/2). +(B.42) +We next estimate the second term in the right side of (B.41). We expand it with the resolvent identities +in Lemma B.10 as follows: +� +α̸=β +HiαRαβAiβ = +� +α̸=β +HiαR(i) +αβAiβ + +� +α̸=β +Hiα +RαiRiβ +Rii +Aiβ += +� +α̸=β +HiαR(i) +αβAiβ + +� +α̸=β +Hiα +RαiRiβ +s(z) +Aiβ + O≺(N −1/2). +(B.43) +Here, in the estimate for the second term, we simply counted the power (of N) as it involves two indices for +the sum (hence O(N 2) terms) of Hiα, Rαi, Riβ, Aiβ ≺ N −1/2, hence � +α̸=β HiαRαiRiβAiβ = O≺(1). Applying +Lemma B.5 to the first term in the right side of (B.43), +� +α̸=β +HiαR(i) +αβAiβ ≺ +� +� 1 +N 2 +� +α,β +|R(i) +αβ|2 +� +� +1/2 +≺ N −1/2. +For the second term in the right side of (B.43), we further expand it to find +� +α̸=β +HiαRαiRiβAiβ = +� +α̸=β +HiαRαiRiβAiβ = − +� +α̸=β +Hiα +� +Rii +� +µ +R(i) +αµHµiRiβAiβ +� +Note that +� +µ +R(i) +αµHµi ≺ N −1/2, +as in the proof of Lemma B.11. Since +|Rij − s(z)| ≺ N −1/2, +Riβ = R(α) +iβ + RiαRαβ +Rαα += R(α) +iβ + N −1, +we have +− +� +α̸=β +Hiα +� +Rii +� +µ +R(i) +αµHµiRiβAiβ +� += −s(z) +� +α̸=β +Hiα +�� +µ +R(i) +αµHµiR(α) +iβ Aiβ +� ++ O≺(N −1/2) += −s(z) +� +α̸=β +Hiα +� +� � +µ:µ̸=α +R(i) +αµHµiR(α) +iβ Aiβ +� +� − s(z) +� +α̸=β +(Hiα)2R(i) +ααR(α) +iβ Aiβ + O≺(N −1/2). +(B.44) +47 + +Applying Lemma B.5 again to the first term in the right side of (B.44), +� +α̸=β +Hiα +� +� � +µ:µ̸=α +R(i) +αµHµiR(α) +iβ Aiβ +� +� ≺ +� +� +� 1 +N +� +α +������ +� +β:β̸=α +� +� � +µ:µ̸=α +R(i) +αµHµi +� +� R(α) +iβ Aiβ +������ +2� +� +� +1/2 +≺ +� +� +� 1 +N +� +α +� +� � +β:β̸=α +N −1/2 ���R(α) +iβ Aiβ +��� +� +� +2� +� +� +1/2 +≺ N −1/2. +Similarly, by expanding R(α) +iβ , we find for the second term in the right side of (B.44) that +−s(z) +� +α̸=β +(Hiα)2R(i) +ααR(α) +iβ Aiβ = zs(z)s(z) +� +α̸=β +(Hiα)2R(α) +ii +� +ν:ν̸=α +H(α) +iν R(iα) +νβ Aiβ + O≺(N −1/2) += zs(z)2s(z) +� +α̸=β +(Hiα)2 +� +ν:ν̸=α,β +HiνR(iα) +νβ Aiβ + zs(z)2s(z) +� +α̸=β +(Hiα)2HiβR(iα) +ββ Aiβ + O≺(N −1/2) += z2s(z)2s(z)2 � +α̸=β +(Hiα)2HiβAiβ + O≺(N −1/2), +where we used Lemma B.5 to find +� +ν̸=β:ν,β̸=α +HiνR(iα) +νβ Aiβ ≺ +� +� 1 +N 2 +� +ν̸=β:ν,β̸=α +���R(iα) +νβ +��� +2 +� +� +1/2 +≺ N −1/2. +Thus, since wB = 1, +� +α̸=β +HiαRαiRiβAiβ = z2s(z)2s(z)2 � +α̸=β +(Hiα)2HiβAiβ + O≺(N −1/2) += z2s(z)2s(z)2wAB + O≺(N −1/2), +and putting it back to (B.43) and (B.41), together with (B.42), we conclude that +(AGB)ii = wABs(z) + wABzs(z)s(z)2 + O≺(N −1/2) = wABσ(z) Eq +� +Vq ++ O≺(N −1/2), +(B.45) +where we used the identity zs(z)s(z) = −σ(z). In the same manner, we also find that +(AGA)ii = 1 +z +� +α +AiαRααAiα + zs(z)s(z)2 � +α̸=β +AiαHiαHiβAiβ + O≺(N −1/2) += wAs(z) + w2 +ABzs(z)s(z)2 + O≺(N −1/2). +(B.46) +We next estimate the off-diagonal entry (AGB)ij. We expand it as +(AGB)ij = 1 +z +� +α,β +HiαRαβAjβ = 1 +z +� +α,β +HiαR(i) +αβAjβ + 1 +z +� +α,β +Hiα +RαiRiβ +Rii +Ajβ += 1 +z +� +α,β +HiαR(ij) +αβ Ajβ + 1 +z +� +α,β +Hiα +R(i) +αjR(i) +jβ +R(i) +jj +Ajβ + 1 +z +� +α,β +Hiα +R(j) +αi R(j) +iβ +R(i) +jj +Ajβ + O≺(N −1/2) +(B.47) +48 + +From Lemma B.5, +� +α,β +HiαR(ij) +αβ Ajβ ≺ N −1/2. +We also have +� +α,β +Hiα +R(i) +αjR(i) +jβ +R(i) +jj +Ajβ ≺ +� +� +� 1 +N +� +α +������ +� +β +R(i) +αjR(i) +jβ +R(i) +jj +Ajβ +������ +2� +� +� +1/2 +≺ +� +� +� 1 +N +� +α +������ +� +β +N −3/2 +������ +2� +� +� +1/2 +≺ N −1/2 +and a similar estimate holds for the third term in the right side of (B.47). Thus, +(AGB)ij ≺ N −1/2 +In the same manner, we also find that (AGA)ij ≺ N −1/2. Together with (B.45) and (B.46), this proves +Lemma B.8. +B.6.3 +Isotropic local law +We also assume that wB = 1 and use the same notation in previous section. Then our goal is to prove the +following statement: +Lemma B.12. Let (A, B) be the entrywise correlated random matrices where wB = 1 and x, y are determin- +istic and ℓ2 - normalized vectors in RM. Then, for z ∈ R outside an open interval containing [d−, d+], +⟨x, A(BT B − zI)−1AT y⟩ = (wAs(z) + w2 +ABzs(z)s(z)2)⟨x, y⟩ + O≺(N −1/2). +Proof of Lemma B.12. Note that, due to polarization identity, we suffice to prove for ⟨x, A(BT B−zI)−1AT x⟩. +Recall that we have +(A(BT B − zI)−1AT )ij = (wAs(z) + w2 +ABzs(z)s(z)2)δij + O≺(N −1/2) +and +(A(BT B − zI)−1BT )ij = (B(BT B − zI)−1AT )ij = (wABs(z) + wABzs(z)s(z)2)δij + O≺(N −1/2). +Once the entrywise local law is given, the proof of the isotropic (or anisotropic) type law follows exactly as +in [17]. To be more precisely, we can write +⟨x, A(BT B − zI)−1AT x⟩ = +� +i +xi(AGAT )iixi + +� +i̸=j +xi(AGAT )ijxj. +Then the entrywise local law implies +� +i +xi(AGAT )iixi − (wAs(z) + w2 +ABzs(z)s(z)2)⟨x, x⟩ += +� +i +x2 +i +� +(AGAT )ii − (wAs(z) + w2 +ABzs(z)s(z)2) +� +≺ N −1/2, +49 + +and so the main difficulty is to control the off-diagonal part +ZAB := +� +α,β +� +i̸=j +xiAiαGαβAjβxj = O≺(N −1/2). +For instance, for the sample covariance matrix case +⟨x, BGBT x⟩ = (zs(z) + 1)⟨x, x⟩ + O≺(N −1/2) += (s(z) + zs(z)s(z)2)⟨x, x⟩ + O≺(N −1/2) +was proved in [17] by proving the following bound for higher moments +E|ZB|p ≺ N −p/2 +(B.48) +for any large and even p, where +ZB := +� +i̸=j +xi(BGBT )ijxj = z +� +i̸=j +xiGijxj. +In particular, the (B.48) have proved by using the standard maximal expansion method in [17] and [2], which +only requires the independence between each element, the boundedness of the moment of each entries, and +the entrywise local law. Thus, from the definition of the entrywise correlated random matrices (A, B), it can +be expected that +E|ZAB|p ≺ N −p/2 +also holds for any large and even p, by expanding maximally Gαβ instead of Gab as in (B.47). Then, we can +conclude the proof by using Markov inequality. +To prove such an argument , we only need to check what is changing. First, we express the p-th moment +of ZAB by +E|ZAB|p = E +� +b11̸=b12 +· · · +� +bp1̸=bp2 +� +� +p/2 +� +k=1 +xbk1(AGAT )bk1bk2xbk2 +� +� +� +� +p +� +k=p/2+1 +xbk1(AGAT )bk1bk2xbk2 +� +� . +(B.49) +Let T = {bk1} ∪ {bk2} be the set of indices of x appearing in the fixed summand of the representation of +the p-th moment of ZAB. Then our goal is to decompose the off-diagonal entry of the matrix (AGAT ) into the +two parts by using Lemma B.10, where one consists of the finite number of the maximally expanded term +and the other consists of the terms containing a sufficiently large number of off-diagonal entries. We note +that the latter case is small enough due to the entrywise local laws of off-diagonal entries, and so the leading +order term contained in the formal. +Step 1 : The maximal expansion for the off-diagonal entries of (AGBA). +In our case, the maximally expanded terms (cf. Definition 5.4 of [17]) refer to terms that have one of the +following forms: (AG(T\a,b)AT )ab, (AG(T\a,b)BT )ab, (BG(T\a,b)AT )ab or (BG(T\a,b)BT )ab = z(G(T\a,b))ab, for some +a ̸= b ∈ T. To proceed, we use the following operation successively : +Operation (a) +Let T ⊂ {1, . . . M} be a set of indices. +50 + +• For a ̸= b and c /∈ T , +(AG(T )AT )ab = (AG(T c)AT )ab + (AG(T )BT )ac(BG(T )AT )cb +z(G(T ))cc +• For a ̸= b and c /∈ T , +(AG(T )BT )ab = (AG(T c)BT )ab + (AG(T )BT )ac(BG(T )BT )cb +z(G(T ))cc += (AG(T c)BT )ab + (AG(T )BT )ac(G(T ))cb +(G(T ))cc +• For a ̸= b /∈ T and a, b ̸= c +1 +z (BG(T )BT )ab = (G(T ))ab = (G(T c))ab + (G(T ))ac(G(T ))cb +(G(T ))cc +• For a ̸= b /∈ T +1 +(G(T ))aa += +1 +(G(T b))aa +− +(G(T ))ab(G(T ))ba +(G(T ))aa(G(T b))aa(G(T ))bb +We then observe that the expanded terms contains at most two crossed terms (AG(T\a,b)BT )ab, (BG(T\a,b)AT )ab +and each expansions produce two types of terms, the first one has one more additional upper index, and +the second one at least one more additional off-diagonal entry of BGAT , AGBT or BGBT . Moreover, we also +remark that the denominators are always the diagonal entries of the resolvent G(T ). +It can be seen that the above expansion formulas eventually play the same role as operation (a) in [17]. +Therefore, to obtain the desired decomposition, we only need to iterate the operation (a) until it can no +longer be expanded or contains sufficiently many off-diagonal entries. +Step 2 : The further expansions for the maximally expanded off-diagonal entries +We further expand the maximally expanded term by using the following operations : +Operations (b) (and (c)) +• For a ̸= b ∈ T +(G(T\a,b))ab = z(G(T\a,b))aa(G(T\b))bb(BG(T)BT )ab. +• Furthermore, we use the following type expansion, which is from the above formula, to the terms +(G(T\a,b))aa and (G(T\a,b))bb +(G(T\a,b))aa = (G(T\a))aa + (G(T\a,b))ab(G(T\a,b))ba +(G(T\a,b))bb += (G(T\a))aa + z2(G(T\a,b))aa(G(T\a))aa(G(T\b))bb(BG(T)BT )2 +ab. +Then, this expansion splits such not-maximally expanded term into two parts, one is maximally ex- +panded and the other is a monomial expressed as the product of itself, the diagonal entry, and the +maximally expanded terms. In particular, it can be seen that the number of the off-diagonal entries +included in the latter monomial increases by exactly two. +51 + +• For a ̸= b ∈ T +(AG(T\a,b)AT )ab = (AG(T)AT )ab + z(G(T\b))bb(AG(T)BT )ab(BG(T)AT )bb ++ (AG(T\a,b)BT )aa(BG(T\a,b)AT )ab +z(G(T\a,b))aa += (AG(T)AT )ab + z(G(T\b))bb(AG(T)BT )ab(BG(T)AT )bb ++ z(G(T\a,b))aa(AG(T)BT )aa +� +(BG(T)AT )ab + z(G(T\b))bb(BG(T)BT )ab +� ++ z2(G(T\a,b))aa(G(T\b))bb(AG(T)BT )ab(BG(T)BT )ab× +� +(BG(T)AT )ab + z(G(T\b))bb(BG(T)BT )ab +� +. +since +(AG(T\a,b)BT )aa = −z(G(T\a,b))aa(AG(T\b)BT )aa += −z(G(T\a,b))aa +� +(AG(T)BT )aa + z(G(T\b))bb(AG(T)BT )ab(BG(T)BT )ab +� +and +(BG(T\a,b)AT )ab = −z(G(T\a,b))aa +� +(BG(T)AT )ab + z(G(T\b))bb(BG(T)BT )ab +� +. +The expansion of the first two monomials terminated since every term were maximally expanded. After +this, for any fixed positive integer ℓ, we expand the term which contains the term (G(T\a,b))aa until +the last term is a monomial containing ℓ or more off-diagonal entries by applying the first formula +recursively to the not-maximally expanded diagonal entry (G(T\a,b))aa. +• For a ̸= b ∈ T +(AG(T\a,b)BT )ab = −z(G(T\b))bb(AG(T)BT )ab + (AG(T\a,b)BT )aa(G(T\a,b))ab +(G(T\a,b))aa += −z(G(T\b))bb(AG(T)BT )ab +− z2(G(T\a,b))aa(AG(T)BT )aa(G(T\b))bb(BG(T)BT )ab +− z3(G(T\a,b))aa(G(T\b))2 +bb(AG(T)BT )ab(BG(T)BT )ab(BG(T)BT )ab +Even in this case, we also expand the second and third monomials recursively by applying the first +formula to not-maximally expanded diagonal entry (G(T\a,b))aa. +In particular, we have two observations from the above operations. +• The expansions of the maximally expanded off-diagonal entry consist of the monomials containing only +an odd number of off-diagonal entries: (BG(T)AT )ab, (AG(T)BT )ab and (BG(T)BT )ab. +• The diagonal entries (AG(T)BT )aa = (BG(T)AT )aa for a ∈ T, appear in the expanded term by implement- +ing operation (b) and (c). These terms can be interpreted as a loop of the vertex a in the structure of +the graph considered in [17], since like the maximally expended diagonal entry, these terms are com- +parable to wABs(z) by the entrywise local law. Therefore, similar to the maximally expanded diagonal +entry, terms of such types have no effect on the partial expectation techniques in subsection 5.13 of +[17]. This part will be explained in more detail in the next step. +As with the previous step, from the explanations depicted in each expansion formula, we can see that +the above expansions eventually play the same role as operations (b) and (c) in [17]. +52 + +Step 3 : The further expansions for the maximally expanded diagonal entries +Finally, unless we end up with an expression that includes a sufficiently large numbers of off-diagonal resolvent +entries (such trivial leaves are dealt with separately in Subsection 5.11 of [17]), we need to expand the +maximally expanded diagonal elements (AG(T)BT )aa = (BG(T)AT )aa and (G(T\a))aa for a ∈ T appearing in +the non-trivial leaves (cf. Subsection 5.12 ∼ 14 of [17]), where we need to slightly adjust the proof to the +setting. These terms corresponds to the maximally expanded diagonal G-edge in [17]. +First, for c ∈ T, +1 +(G(T\c) +B +)cc += −z − z(BG(T) +B +BT )cc. +(B.50) +We note that |(G(T))µµ − s(z)| ≺ N −1/2 by following the proof of the entrywise local law. Using (B.50) and +the facts zs(z)s(z) = −(zs(z) + 1) and |s(z)| ≍ 1, we see that +1 +(G(T\c))cc += +1 +s(z) − z +� +(BG(T)BT )cc − s(z) +� +and this implies that +(G(T\c))cc = +ℓ−1 +� +k=0 +(s(z))k+1zk � +(BG(T)BT )cc − s(z) +�k ++ O≺(N −ℓ/2) +for any integer ℓ ≥ 1 since (BG(T)BT )cc − s(z) is O≺(N −1/2), by using Lemma B.5. +Similarly, for a ∈ T, we see that +1 +(AG(T)BT )aa += +1 +wABs(z) − (AG(T) +B +BT )aa − wABs(z) +wABs(z)(AG(T) +B +BT )aa +(B.51) +and so +(AG(T)BT )aa = wABs(z) − (AG(T)BT )aa +wABs(z)−(AG(T)BT )aa +wABs(z) +1 − wABs(z)−(AG(T)BT )aa +wABs(z) +. +By using the estimate +(AG(T)BT )aa − wABs(z) = +� +µ̸=ν +Aaµ(G(T))µνBaν + +� +µ +AaµBaµ +� +(G(T))µµ − s(z) +� ++ s(z) +� +1 +N +� +µ +(NAaµBaµ − wAB) +� +≺ N −1/2, +we have the following series expansion for any integers ℓ ≥ 1, +(AG(T)BT )aa = wABs(z) − (AG(T)BT )aa1(ℓ ≥ 2) +ℓ−1 +� +k=1 +(wABs(z))−k � +wABs(z) − (AG(T)BT )aa +�k ++ O≺(N −ℓ/2) +which corresponds to the term (5.42) in [17]. +This way we end up with an expression where only contains the resolvent terms of the type (AG(T)AT )ab, +(AG(T)BT )ab, (BG(T)AT )ab or (BG(T)BT )ab = (G(T))ab, for some a ̸= b ∈ T. In other words, the x indices +and the indices of the resolvent entries are completely decoupled; only explicit products of entries of (A, B) +53 + +represent the connections between them. +Step 4 : Sketch of the rest of the proof. +Through previous steps, for our case (AGAT ), we observed the modified version of the operations, which are +done for the resolvents G and G in [17]. +After with these modifications, it can be seen that the rest procedures (Step 6 ∼ 8 in [17]) of the proof for +the non-trivial leaves with the stopping rule, which relies on the number of off-diagonal terms (cf. Definition +5.7 of [17]), are also valid for the ZAB. +More precisely, by using the entrywise laws and H¨older’s inequality, the same estimation also holds for +the trivial leave as in Subsection 5.11. Furthermore, the most of the finitely generated non-trivial leaves have +a decay N −p/2 also by applying the same argument in the case of the trivial leaves (Subsection 5.12 in [17]), +and the remaining leading order non-trivial leaves have the same decay by applying the partial expectation +method (Subsection 5.13 in [17]). +We conclude the proof. +Proof of Lemma B.6. From the above version of an isotropic law, we also arrive at the isotropic version of +the entrywise law in Lemma B.8 by taking A = Q + X and B = Q. Then, it is easy to check that +wA = 2 +� +1 + Eq +� +Vq +� +, +wAB = 1 + Eq +� +Vq +, +wB = 1. +Precisely, applying Lemma B.12 directly, we see that +2⟨u, X(QT Q − zI)−1QT u⟩ += ⟨u, X(QT Q − zI)−1QT u⟩ + ⟨u, Q(QT Q − zI)−1XT u⟩ += ⟨u, A(BT B − zI)−1AT u⟩ − ⟨u, B(BT B − zI)−1BT u⟩ − ⟨u, X(BT B − zI)−1XT u⟩ += 2s(z) +� +1 + Eq +� +Vq +� ++ zs(z)s(z)2 +� +1 + Eq +� +Vq +�2 +− s(z) − zs(z)s(z)2 − s(z) − zs(z)s(z)2 E2 +q +Vq += 2 Eq +� +Vq +(s(z) + zs(z)s(z)2) = 2 Eq +� +Vq +(zs(z) + 1) +with O≺(N −φ) error terms, and it exactly matches the entrywise law since correlation wXQ = +Eq +√ +Vq . Thus, +we conclude that the improved PCA via the entrywise transform holds for the spike U s.t. ∥U T U − Ik∥F , +∥U∥∞ ≺ N −φ, where φ > 1/4. +C +Proof of CLTs +In Appendix C, we prove the CLT for the LSS of spiked random matrices. The proof of the CLT for the +LSS is based on the strategy of [6] in which the LSS is first written as a contour integral of the resolvent of +a spiked Wigner matrix. Then, the averaged trace of the resolvent converges to a Gaussian process, which +also implies that the limiting distribution of the LSS is Gaussian. +It is the biggest obstacle in adapting the proof in [6] for spiked matrices that the martingale CLT and +covariance computation are hard to be reproduced with spikes; even with the special choice of rank-1 spike +the proof for the CLT is very tedious as in [9]. In [22], the interpolation between a general rank-1 spike +54 + +and the special rank-1 spiked was introduced to compare the LSS, based on an ansatz that the mean and +the variance of the LSS do not depend on the choice of the spike. In this paper, since we do not have a +reference matrix to be compared with as in the rank-1 case, we introduce a direct interpolation between a +spiked random matrices of general rank and a matrix without any spikes. With the interpolation, we find +the change of the mean in the limiting Gaussian distribution and also prove that its variance is invariant. +C.1 +Proof of CLTs for spiked random matrices +Proof of Theorem 5.2. We adapt the proof of Theorem 5 in [22] with the following change. Instead of inter- +polating the spiked Wigner matrices M with the original signal and with the signal with all 1’s considered in +[9], we directly interpolate M and W and track the change of the mean. Consider the following interpolating +matrix +M(θ) = θ +√ +λUU T + W +and the corresponding eigenvalues {µi(θ)}N +i=1 of M(θ) for θ ∈ [0, 1]. Let Γ be a rectangular contour in the +proof of Theorem 5 in [22]. Applying Cauchy’s integral formula, we have +N +� +i=1 +f(µi(1)) − N +� 2 +−2 +√ +4 − x2 +2π +f(x) dx = − N +2πi +� +Γ +f(z) +� +sN(1, z) − ssc(z) +� +dz +(C.1) +where ssc(z) = −z+ +√ +z2−4 +2 +is the Stieltjes transform of the Wigner semicircle law and sN(θ, z) is the Stieltjes +transform of the empirical spectral distribution (ESD) of M(θ) for θ ∈ [0, 1]. Note that the normalized trace +of the resolvent satisfies +1 +N Tr R(θ, z) = 1 +N +N +� +i=1 +1 +µi(θ) − z = sN(θ, z) +(C.2) +where R(θ, z) is the resolvent corresponding to M(θ), defined as +R(θ, z) := (M(θ) − zI)−1 +(C.3) +for z ∈ C+ and θ ∈ [0, 1]. +The change of the mean in the CLT for W and the CLT for M can be computed by tracking the change +of the corresponding resolvent in (C.3), since (C.1) can be decomposed by +N +� +i=1 +f(µi(1)) − N +� 2 +−2 +√ +4 − x2 +2π +f(x) dx = − 1 +2πi +� +Γ +f(z) +� +Tr R(1, z) − Tr R(0, z) +� +dz +(C.4) +− 1 +2πi +� +Γ +f(z) +� +Tr R(0, z) − Nssc(z) +� +dz +(C.5) +and the fluctuation result of (C.5) is already given in [6]. +Set Γε = {z ∈ C : minw∈Γ |z − w| ≤ ε}. Choose ε so that +min +w∈Γε,x∈[−2,2] |x − w| > 2ε. +55 + +Following the proof of Theorem 5 in [22], on z ∈ Γε +1/2 := Γε ∩ {z ∈ C : |Imz| > N −1/2}, we first find that +∂ +∂θ Tr R(θ, z) = − +k +� +m=1 +√ +λ ∂ +∂z +� +x(m)T R(θ, z)u(m) +� += −k ∂ +∂z +� +√ +λssc(z) +1 + θ +√ +λssc(z) +� ++ O(N − 1 +2 ) += − +k +√ +λs′ +sc(z) +(1 + θ +√ +λssc(z))2 + O(N − 1 +2 ) +(C.6) +with high probability. More precisely, since the elementary resolvent expansion implies +R(0, z) − R(θ, z) = θ +√ +λR(θ, z) +� k +� +ℓ=1 +u(ℓ)u(ℓ)T +� +R(0, z), +(C.7) +we then find that +� +u(m)T R(0, z)u(m) +� += +� +u(m)T R(θ, z)u(m) +� ++ θ +√ +λ +k +� +ℓ=1 +� +u(m)T R(θ, z)u(ℓ) +� � +u(ℓ)T R(0, z)u(m) +� +. +From the rigidity of the eigenvalues, we have a deterministic bound for resolvent +| +� +u(m)T R(θ, z)u(ℓ) +� +| ≤ ∥R(θ, z)∥ ≤ C. +(C.8) +Since columns of spike {u(ℓ)}k +ℓ=1 are orthonormal, the isotropic local law for R(0, z) implies that +� +u(m)T R(0, z)u(ℓ) +� += s(z)δmℓ + O(N −1/2). +(C.9) +uniformly on z ∈ Γε. We then obtain that +� +u(m)T R(0, z)u(m) +� += +� +u(m)T R(θ, z)u(m) +� � +1 + θ +√ +λ +� +u(m)T R(0, z)u(m) +�� ++ O(N − 1 +2 ) +and so +� +u(m)T R(θ, z)u(m) +� += +ssc(z) +1 + θ +√ +λssc(z) ++ O(N − 1 +2 ). +This proves (C.6). +Moreover, on Γε, we easily check that the exactly same argument holds for a finite rank perturbation of +Wigner matrix (e.g. interlacing and rigidity properties). Thus, we conclude that (C.4) is +k +2πi +� +Γ +√ +λs′ +sc(z) +1 + +√ +λssc(z) +f(z)dz + o(1) +with high probability. +Finally, following the computation in the proof of Lemma 4.4 in [9], we then find that the difference +between the LSS of M and the LSS of W is +k +∞ +� +ℓ=1 +√ +λℓτℓ(f). +(C.10) +This proves the desired theorem. +56 + +Proof of Theorem 5.5. The proof of the CLT for the spiked rectangular matrices is quite similar to the case +of spiked Wigner matrix. We first consider the interpolating matrix for the additive model, defined as +Y (θ) = θ +√ +λUV T + X +(C.11) +for θ ∈ [0, 1]. Note that Y (0) = X and Y (1) = Y . Denote by µ1(θ) ≥ µ2(θ) ≥ · · · ≥ µM(θ) the eigenvalues +of Y (θ)Y (θ)T . We also define the resolvent +G(θ, z) = (Y (θ)Y (θ)T − zI)−1, +G(θ, z) = (Y (θ)T Y (θ) − zI)−1 +(C.12) +for z ∈ C. +We choose (N-independent) constants a− < d−, a+ > d+, and v0 ∈ (0, 1) so that the function f is +analytic on the rectangular contour Γ whose vertices are (a− ± iv0) and (a+ ± iv0). With overwhelming +probability, all eigenvalues of Y (θ)Y (θ)T are contained in Γ. Applying Cauchy’s integral formula, we find +that +M +� +i=1 +f(µi(1)) − +M +� +i=1 +f(µi(0)) = − +� 1 +2πi +� +Γ +f(z) (Tr G(1, z) − Tr G(0, z)) dz +� +(C.13) +To estimate the difference Tr G(1, z) − Tr G(0, z), we consider its derivative +∂ +∂θ Tr G(θ, z). Note that +∂Gab(θ) +∂Yij(θ) = −Gai(θ)(Y (θ)T G(θ))jb − (G(θ)Y (θ))ajGib(θ), +dYij(θ) +dθ += +√ +λuivT +j . +(C.14) +Thus, by chain rule +∂ +∂θ Tr G(θ, z) = +M +� +a=1 +M +� +i=1 +N +� +j=1 +∂Yij(θ) +∂θ +∂Gaa(θ) +∂Yij(θ) += − +M +� +a=1 +M +� +i=1 +N +� +j=1 +√ +λuivT +k [Gai(θ)(Y (θ)T G(θ))ja + (G(θ)Y (θ))ajGia(θ)] += −2 +M +� +a=1 +M +� +i=1 +N +� +j=1 +M +� +b=1 +√ +λuivT +j [Ybj(θ)Gba(θ)Gai(θ)] +(C.15) +From the fact +� ∂ +∂z G(θ) +� +bi += (G(θ)2)bi = +� +a +Gba(θ)Gai(θ), +we then find that +∂ +∂θ Tr G(θ, z) = −2 +√ +λ ∂ +∂z +M +� +i=1 +N +� +j=1 +uivT +j (G(θ)Y (θ))ij = −2 +√ +λ ∂ +∂z +k +� +ℓ=1 +⟨u(ℓ), G(θ)Y (θ)v(ℓ)⟩. +(C.16) +It remains to estimate +∂ +∂z⟨u(ℓ), G(θ)Y (θ)v(ℓ)⟩ for 1 ≤ ℓ ≤ k. We suffices to estimate the desired term +for fixed ℓ. From now, we omit ℓ-dependency. Note that +⟨u, G(θ)Y (θ)v⟩ = θ +√ +λ⟨u, G(θ)u⟩ + ⟨u, G(θ)Xv⟩. +57 + +We consider the resolvent expansion +G(0, z) − G(θ, z) = G(θ, z) (H(θ) − H(0)) G(0, z) += G(θ, z) (θ2λuuT + θ +√ +λXvuT + θ +√ +λuvT XT ) G(0, z). +(C.17) +Taking inner products with u and v, we obtain +⟨u, G(0)u⟩ = ⟨u, G(θ)u⟩ + θ2λ⟨u, G(θ)u⟩⟨u, G(0)u⟩ ++ θ +√ +λ⟨u, G(θ)Xv⟩⟨u, G(0)u⟩ + θ +√ +λ⟨u, G(0)Xv⟩⟨u, G(θ)u⟩ +(C.18) +and +⟨u, G(0)Xv⟩ = ⟨u, G(θ)Xv⟩ + θ2λ⟨u, G(θ)Xv⟩⟨u, G(0)Xv⟩ ++ θ +√ +λ⟨u, G(θ)Xv⟩⟨u, G(0)Xv⟩ + θ +√ +λ⟨v, XT G(0)Xv⟩⟨u, G(θ)u⟩, +(C.19) +where we omitted z-dependence for brevity. We then use the following result to control the terms in (C.18) +and (C.19). Recall the definition of s(z) and s(z) in Lemmas B.4 and B.11. Moreover, we consider the same +linearization HX(z) of the matrix X and its inverse RX(z) = HX(z)−1 as in (B.33) and (B.34). +Lemma C.1 (Isotropic local law). For an N-independent constant ε > 0, let Γε be the ε-neighborhood of Γ, +i.e., +Γε = {z ∈ C : min +w∈Γ |z − w| ≤ ε}. +Choose ε small so that the distance between Γε and [d−, d+] is larger than 2ε, i.e., +min +w∈Γε,x∈[d−,d+] |x − w| > 2ε. +(C.20) +Then, for any unit vectors x, y ∈ CM+N independent of X, +|⟨x, (RX(z) − Π(z))y⟩| ≺ N −1/2, +(C.21) +uniformly on z ∈ Γε, where +Π(z) = +�s(z) · IM +0 +0 +zs(z) · IN +� +. +(C.22) +Proof. See Theorems 3.6, 3.7, Corollary 3.9, and Remark 3.10 in [37]. Note that Im s(z), Im s(z) = Θ(η) on +the vertical part of Γε, i.e., the neighborhood of the line segment joining (a++iv0) and (a+−iv0) (respectively +(a− + iv0) and (a− − iv0)). +Set +A := ⟨u, G(0, z)u⟩, +B := ⟨u, G(0, z)Xv⟩, +C := ⟨v, XT G(0, z)Xv⟩. +Recall that +RX(z) = +� G(0, z) +G(0, z)X +XT G(0, z) +zG(0, z) +� +. +(C.23) +Then, as consequences of Lemma C.1 with appropriate choices of the deterministic vectors, +A = s(z) + O≺(N −1/2), +C = ⟨v, zG(0, z)v⟩ + 1 + O(N −1/2) = d0(zs(z) + 1) + O≺(N −1/2), +(C.24) +58 + +and +B = O≺(N −1/2). +We thus have from (C.18) and (C.19) that +⟨u, G(θ)Xv⟩ = −θd0 +√ +λs(z)(zs(z) + 1) +θ2λzs(z) + θ2λ + 1 ++ O≺(N −1/2) +⟨u, G(θ)u⟩ = +s(z) +θ2λzs(z) + θ2λ + 1 + O≺(N −1/2) +(C.25) +and hence +⟨u, G(θ)Y (θ)v⟩ = θ +√ +λ⟨u, G(θ)u⟩ + ⟨u, G(θ)Xv⟩ = +θ +√ +λzs(z) + θ +√ +λ +θ2λzs(z) + θ2λ + 1 + O≺(N −1/2). +(C.26) +Note that this estimate is uniform on θ. Differentiating it with respect to z and plugging it back to (C.16), +we get +∂ +∂θ Tr G(θ, z) = −k +2θλ d +dz(zs(z) + 1) +(θ2λzs(z) + θ2λ + 1)2 + O≺(N −1/2) +and, integrating over θ, we obtain +Tr G(1, z) − Tr G(0, z) = +� 1 +0 +∂ +∂θ Tr G(θ, z)dθ = −k +d +dzλ(zs(z) + 1) +λzs(z) + λ + 1 + O≺(N −1/2). +(C.27) +We now invoke the following relation between the Stieltjes transforms for Marchenko–Pastur law and the +Wigner semicircle law. Let +ssc(z) = −z + +√ +z2 − 4 +2 +be the Stieltjes transform of the Wigner semicircle law and +ϕ(z) = +1 +√d0 +(z − (1 + d0)). +Then +� +d0(zs(z) + 1) = ssc(ϕ(z)). +(C.28) +We thus have +1 +2πi +� +Γ +f(z)λ d +dz(zs(z) + 1) +λzs(z) + λ + 1 dz = +1 +2πi +� +Γ +�f(ϕ(z)) λs′ +sc(ϕ(z))ϕ′(z) +λssc(ϕ(z)) + √d0 +dz += +1 +2πi +� +�Γ +�f(ϕ) +λs′ +sc(ϕ) +λssc(ϕ) + √d0 +dϕ +(C.29) +where we let f(√d0z + 1 + d0) = �f(z) and �Γ = ϕ(Γ). (Note that �Γ contains the interval [−2, 2].) +So far, we have proved that +M +� +i=1 +f(µi(1)) − +M +� +i=1 +f(µi(0)) = +k +2πi +� +�Γ +�f(ϕ) +λs′ +sc(ϕ) +λssc(ϕ) + √d0 +dϕ + O≺(N −1/2). +(C.30) +Since the difference in (C.30) is the sum of a deterministic term and a random term stochastically dominated +59 + +by N −1/2, we can see that the CLT holds for the LSS with the non-null model Y (1). Moreover, the variance +is the same as that of the null model, which is +VY (f) = 2 +∞ +� +ℓ=1 +ℓτℓ( �f)2 + (w4 − 3)τ1( �f)2. +(C.31) +(See, e.g., [10].) +The change of the mean is the first term in the right side of (C.30), which can be computed by following +the proof of Lemma 4.4 in [9]. We obtain +mY (f) = +�f(2) + �f(−2) +4 +− 1 +2τ0( �f) + (w4 − 3)τ2( �f) + k +∞ +� +ℓ=1 +� λ +√d0 +�ℓ +τℓ( �f). +(C.32) +This proves the first part of Theorem 5.2 for the additive model. +For the multiplicative model, we will follow the same strategy as in the additive model. Let +Y (θ) = X + θγUU T X +(C.33) +for θ ∈ [0, 1]. +Note that Y (0) = X and Y (1) = Y . +We denote by µ1(θ) ≥ µ2(θ) ≥ · · · ≥ µM(θ) the +eigenvalues of Y (θ)Y (θ)T , and also let +G(θ, z) = (Y (θ)Y (θ)T − zI)−1, +G(θ, z) = (Y (θ)T Y (θ) − zI)−1 +(C.34) +for z ∈ C. We have the relations +∂Gab(θ) +∂Yij(θ) = −Gai(θ)(Y (θ)T G(θ))jb − (G(θ)Y (θ))ajGib(θ), +∂Yij(θ) +∂θ += γ +M +� +c=1 +uiuT +b Xbj. +(C.35) +Following (C.15)-(C.16), we get +∂ +∂θ Tr G(θ, z) = −γ +M +� +a=1 +M +� +i=1 +N +� +j=1 +M +� +b=1 +uiuT +b Xbj[Gai(θ)(Y (θ)T G(θ))ja + (G(θ)Y (θ))ajGia(θ)] += −2γ +M +� +a=1 +M +� +i=1 +N +� +j=1 +M +� +b=1 +uiuT +b Xbj[(Y (θ)T G(θ))jaGai(θ)] += −2γ ∂ +∂z +M +� +i=1 +N +� +j=1 +M +� +b=1 +uiuT +b Xbj(G(θ)Y (θ))ij += −2γ ∂ +∂z +k +� +ℓ=1 +⟨u(ℓ), G(θ)Y (θ)XT u(ℓ)⟩ = −2γ ∂ +∂z +k +� +ℓ=1 +⟨u(ℓ), G(θ)Y (θ)Y (0)T u(ℓ)⟩. +(C.36) +Moreover, since +Y (0) = X = (I + θγUU T )−1Y (θ) = +� +I − +θγ +1 + θγ UU T +� +Y (θ), +(C.37) +60 + +we have +⟨u(ℓ), G(θ)Y (θ)Y (0)T u(ℓ)⟩ = ⟨u(ℓ), G(θ)Y (θ)Y (θ)T (I + θγUU T )−1u(ℓ)⟩ += ⟨u(ℓ), (I + zG(θ))(I + θγUU T )−1u(ℓ)⟩ += +1 +1 + θγ + +z +1 + θγ ⟨u(ℓ), G(θ)u(ℓ)⟩. +(C.38) +To estimate the term ⟨u(ℓ), G(θ)u(ℓ)⟩, we use the following Anisotropic local law in [37]. +Lemma C.2 (Anisotropic local law). Let Γε be the ε-neighborhood of Γ as in Lemma C.1. Then, for any +unit vectors x, y ∈ CM independent of X, the following estimate holds uniformly on z ∈ Γε : +���� +� +x, +� +G(θ, z) + +� +zI + zs(z)(I + θγUU T )2�−1� +y +����� ≺ N − 1 +2 . +(C.39) +Proof. The proof of Lemma C.2 is the same as that of Lemma C.1. +Now, as in the additive case, we drop the ℓ-dependency. From Lemma C.2, we find that +⟨u, G(θ)u⟩ = − +� +u, +� +zI + zs(z)(I + θγUU T )2�−1 +u +� ++ O(N −1/2) += − +1 +(1 + θγ)2z(1 + s(z)) + O(N −1/2), +(C.40) +and plugging it into (C.38), we obtain +⟨u, G(θ)Y (θ)Y (0)T u⟩ = +1 +1 + θγ − +1 +(1 + θγ)(1 + (1 + θγ)2s(z)) + O(N −1/2). +(C.41) +We thus get +∂ +∂θ Tr G(θ, z) = −2kγ +(1 + θγ)s′(z) +(1 + (1 + θγ)2s(z))2 + O(N −1/2), +(C.42) +and integrating it yields +Tr G(1, z) − Tr G(0, z) = −k +λs′(z) +(1 + s(z))(1 + (1 + λ)s(z)) + O(N −1/2) += −λk d +dz(zs(z) + 1) +λzs(z) + λ + 1 + O(N −1/2). +(C.43) +Since (C.43) coincides with (C.27), the rest of the proof is exactly the same as in the additive case. This +finishes the proof of the first part of Theorem 5.2. +C.2 +Proof of CLTs for entrywise transformed matrices +Proof of Theorem 5.3. We adapt the proof of Theorem 7 in [22] with the following changes. Let S be the +variance matrix of the transformed matrix � +M. We then find that +Sij = E[� +M 2 +ij] − (E[� +Mij])2 = 1 +N + λ(GH − Fg)(uiuT +j )2 + O(N 1−8φ) +and +Sii = E[� +M 2 +ii] − (E[� +Mii])2 = w2 +N + λ(Gg,d − Fg,d)(uiuT +i )2 + O(N 1−8φ). +61 + +Normalizing and centering each entry of the matrix � +M, we arrive at another Wigner matrix � +W where +� +Wij = +1 +� +NSij +(� +Mij − E� +Mij), +� +Wii = +� w2 +NSii +(� +Mii − E� +Mii). +Interpolating � +W and � +M − E[� +M] by � +W(θ) = (1 − θ)� +W + θ(� +M − E[� +M]), � +W(θ) is a general Wigner-type matrix +with the corresponding quadratic vector equation +− +1 +mi(θ, z) = z + +N +� +j=1 +E[� +Wij(θ)2] · mj(θ, z) +where mi(θ, z)δij is the limiting distribution of the (i, j)-element of the resolvent +R +� +W (θ, z) = (� +W(θ) − zI)−1 +for 0 ≤ θ ≤ 1. Recall the ssc(z) is the Stieltjes transform of the Wigner semicircle law. We also directly +check that mi(θ, z) = ssc(z) + C1(uiuT +i ) + C2N −1 = ssc(z) + O(N −2φ). Moreover, the anisotropic local law +for the general Wigner-type matrix in [2] implies that uniformly on z ∈ Γε +1/2 +(u(m)T R +� +W (θ, z)u(ℓ)) = ssc(z)δmℓ + O(N −1/2). +Following the proof of Lemmas B.2 and B.3 in [22], we check that +• Uniformly on z ∈ Γε +1/2, +Tr R +� +W (1, z) − Tr R +� +W (0, z) = kλ(GH − Fg)s′ +sc(z)ssc(z) + O(N 1N −4φ) +(C.44) +• Uniformly on z ∈ Γε\Γε +1/2, +| Tr R +� +W (1, z) − Tr R +� +W (0, z)| = O(N 1N −2/3). +(C.45) +Compared with the bound shown in [22], we give the following remark: +• The error bound in (C.44) is better. This sharper bound can be obtained by using the fact � +a ua(ℓ)2 = +1 instead of +��� +a ua(ℓ)2�� ≤ N∥u(ℓ)∥2 +∞. +Our next step is to consider � +M = � +W(1) + E[� +M]. Since +� +M = � +W(1) + +� +λFgUU T + diag(d1, · · · , dN) + E +where di = E[� +Mii] − +� +λFg(UU T )ii, we then find that +Tr(� +M − zI)−1 − Tr R +� +W (0, z) += kλ(GH − Fg)s′ +sc(z)ssc(z) − +k +� +λFgs′ +sc(z) +1 + +� +λFgssc(z) − k +√ +λ( +� +Fg,d − +� +Fg)s′ +sc(z) + O(N −1/2) +uniformly on z ∈ Γε +1/2. Thus, we obtain the desired CLT by applying Cauchy’s integral formula as in the +proof of Theorem 5.2. +62 + +Proof of Theorem 5.6. Since the proof of the transformed CLT for the spiked Wigner matrix follows the +proof in [22], we only describe the process briefly. On the other hand, there is no technical reference for the +spiked rectangular matrices. As we mentioned before, our consideration is only the additive case. +We consider the optimal entrywise transformation defined by a function +h(w) := −g′(w) +g(w) . +(C.46) +If λ = 0, it is immediate to see that for all i, j +E[h( +√ +NYij)] = +� ∞ +−∞ +h(w)g(w)dw = − +� ∞ +−∞ +g′(w)dw = 0. +Further, with λ = 0, as shown in Proposition 4.2 of [50], +Fg := E[h( +√ +NYij)2] = +� ∞ +−∞ +h(w)2g(w)dw = +� ∞ +−∞ +g′(w)2 +g(w) dw ≥ 1, +(C.47) +where the equality holds if and only if +√ +NXij is a standard Gaussian (hence h(w) = w). +We define a transformed matrix �Y as follows: the terms of �Y are defined by +�Yij = +1 +� +FgN h( +√ +NYij). +(C.48) +Note that the entries of �Y are independent up to symmetry. Since g is smooth, h is also smooth and all +moments of +√ +N �Yij are O(1). Thus, applying a high-order Markov inequality, it is immediate to find that +�Yij = O(N − 1 +2 ). +C.2.1 +Decomposition of the transformed matrix +We first estimate the mean and the variance of entry by using the comparison method with the pre- +transformed entries. For all i, j, we find that +E[�Yij] = +1 +� +FgN +� ∞ +−∞ +h(w)g +� +w − +√ +NλuivT +j +� +dw += − +1 +� +FgN +� ∞ +−∞ +g′(w) +g(w) +� +g +� +w − +√ +NλuivT +j +� +− g(w) +� +dw. +(C.49) +In the Taylor expansion +g +� +w − +√ +NλuivT +j +� +− g(w) += +4 +� +ℓ=1 +g(ℓ)(w) +ℓ! +� +− +√ +NλuivT +j +�ℓ ++ +g(5) � +w − θ +√ +NλuivT +j +� +5! +� +− +√ +NλuivT +j +�5 +(C.50) +63 + +for some θ ∈ (0, 1). Note that the second term and the fourth term in the summation are even functions. +Since g′/g is an odd function, we find that +E[�Yij] = +1 +� +Fg +√ +λuivT +j +� ∞ +−∞ +g′(w)2 +g(w) dw + C3N +�√ +λuivT +j +�3 ++ O(N 2(uivT +j )5) += +� +λFguivT +j + C3N +�√ +λuivT +j +�3 ++ O(N 2(uivT +j )5) +(C.51) +for some (N-independent) constant C3. Similarly, since +� +g′ +g +�2 +is even, +E[�Y 2 +ij] = +1 +FgN +� ∞ +−∞ +�g′(w) +g(w) +�2 +g +� +w − +√ +NλuivT +j +� +dw += 1 +N + +1 +FgN +� ∞ +−∞ +�g′(w) +g(w) +�2 � +g +� +w − +√ +NλuivT +j +� +− g(w) +� +dw += 1 +N + +1 +2Fg +�√ +λuivT +j +�2 � ∞ +−∞ +g′(w)2g′′(w) +g(w)2 +dw + O(N(uivT +j )4) += 1 +N + λGH(uivT +j )2 + O(N(uivT +j )4). +(C.52) +where +GH = +1 +2Fg +� ∞ +−∞ +g′(w)2g′′(w) +g(w)2 +dw. +The evaluation of the mean and the variance shows that the transformed matrix �Y is not a spiked +rectangular matrix when λ > 0, since the variances of the entries are not identical. +Our strategy is to +approximate �Y as a spiked generalized rectangular Gram matrix for which the variances of the each entries +is 1/N in high-dimensional regime. Let S be the variance matrix of �Y defined as +Sij = E[�Y 2 +ij] − (E[�Yij])2. +(C.53) +From (C.51) and (C.52), +Sij = 1 +N + (GH − Fg) +�√ +λuivT +j +�2 ++ O(N∥U∥4 +∞∥V ∥4 +∞), +(C.54) +which shows that �Y is indeed approximately a spiked generalized Gram matrix. +C.2.2 +CLT for a random Gram matrix +We use the local law for general rectangular Gram matrices in [4]. Consider an another M × N rectangular +matrix A = (Aij) defined by +Aij = +1 +� +NSij +(�Yij − E[�Yij]). +(C.55) +Note that E[Aij] = 0, E[A2 +ij] = 1 +N . Then the matrix A is a usual rectangular matrix. We set +GA(z) = (AAT − zI)−1 +(z ∈ C+). +(C.56) +64 + +Next, we introduce an interpolation for A. For 0 ≤ θ ≤ 1, we define a matrix A(θ) by +Aij(θ) = (1 − θ)Aij + θ(�Yij − E[�Yij]) = +� +1 − θ + θ +� +NSij +� +Aij += +� +1 + θNλ(GH − Fg)(uivT +j )2 +2 ++ O(N 2(uivT +j )4) +� +Aij +(C.57) +Note that A(0) = A and A(1) = �Y − E[�Y ]. For 0 ≤ θ ≤ 1, A(θ) is a random Gram matrix considered in +[4] satisfying the conditions (A)–(D) therein. Moreover, if we let +GA(θ, z) = (A(θ)A(θ)T − zI)−1 +(z ∈ C+) +(C.58) +and Sij(θ) = E[Aij(θ)2], then Theorem 1.7 of [4] asserts that the limiting distribution of GA +ij(z) is si(z)δij, +where si(θ, z) is the unique solution to the system of quadratic vector equations +− +1 +si(θ, z) = z + +N +� +j=1 +Sij(θ) zsj(θ, z) +(C.59) +and +− +1 +sj(θ, z) = z + +M +� +i=1 +Sij(θ) zsi(θ, z) +(C.60) +Remark C.3. Recall that s(z) is the Stieltjes transform of the Marchenko-Pastur measure. We can then +find that si(θ, z) = s(z)+C1(uiuT +i )+C2N −1 = s(z)+O(N −1/2) and sj(θ, z) = s(z)+C1(vjvT +j )+C2N −1 = +s(z) + O(N −1/2); see also Lemma 3.9 of [4]. +For the resolvent GA(θ, z), we will use the following lemma for the random Gram matrix: +Lemma C.4 (Anisotropic local law for random Gram matrix). Let Γε be the ε-neighborhood of Γ as in +Lemma C.1. Then, for any deterministic x = (x1, . . . , xM), y = (y1, . . . , yM) ∈ CM with ∥x∥ = ∥y∥ = 1, +the following estimate holds uniformly on z ∈ Γε ∩ {z ∈ C+ : Im z > N − 1 +2 }: +������ +M +� +i=1 +M +� +j=1 +xiGA +ij(θ, z)yj − +M +� +i=1 +si(θ, z)xiyi +������ += O(N − 1 +2 ). +(C.61) +and, for any deterministic x = (x1, . . . , xN), y = (y1, . . . , yN) ∈ CN with ∥x∥ = ∥y∥ = 1, +������ +N +� +i=1 +N +� +j=1 +xiGA +ij(θ, z)yj − +N +� +i=1 +si(θ, z)xiyi +������ += O(N − 1 +2 ). +(C.62) +Proof of Lemma C.4. Let Ψ(z) = +� +1 +M Im z be the control parameter for the random gram matrix model. We +then note that the bound for the entrywise local law is N −1/2 since Ψ(z) ≺ N −1/2 on Γε ∩ {z ∈ C+ : Im z > +N − 1 +2 }. With the entrywise local law in [4], the proof of the anisotropic law exactly follows the maximal +65 + +expansion argument used in [2, 17] and Lemma B.12. We consider the following decomposition of (C.61): +M +� +i̸=j +xiGA +ij(θ, z)xj + +M +� +i=1 +(GA +ii − si(θ, z))xixi. +(C.63) +From now, we drop A, θ and z-dependencies for brevity and use the linearization matrix HA(θ)(z) ≡ H and +its inverse R. Then, in usual, we suffices to prove that +Z ≡ +� +a̸=b +xaRabxb ≺ N −1/2. +To prove the above high probability bound, we will bound the 2p-moments E[|Z|2p] ≺ N −p/2 by deriving +the maximally expanded form via the resolvent identity in Lemma B.10. +Now, we will check the representation of the maximally expanded diagonal resolvent elements. e.g. R(B\b) +bb +, +b ∈ B. Together with Remark C.3, we then conclude that the standard argument in [2] is valid for our model. +By applying Shur’s complement lemma and (C.59), for b ∈ B, +1 +R(B\b) +bb += −z − +(B) +� +α,β +HbαR(B) +αβ Hβb += +1 +sb(θ, z) + +� +β +Sbβ (zsβ(θ, z)) − +(B) +� +α,β +HbαR(B) +αβ Hβb += +1 +sb(θ, z) − +(B) +� +β +(HbβR(B) +ββ Hβb − Sbβ (zsβ(θ, z))) − +(B) +� +α̸=β +HbαR(B) +αβ Hβb. +(C.64) +Then (C.64) and the analogue representation of R(B\β) +ββ +for β ∈ B replace the (6.2) in [2]. +With linearization H and its inverse R, one useful by-product of the above argument is +⟨x, GA(θ)A(θ)y⟩ = +� +a +� +α +xaRaαyα ≺ N −1/2. +(C.65) +Note that our model satisfies the closeness condition (A3) of Assumption 2.2 in [3] (See also Remark 2.4 +therein). On Γ\Γε +1/2, we use the following results on the rigidity of eigenvalues. +Lemma C.5 (Rigidity of eigenvalues for the random Gram matrix). Denote by µA +1 (θ) ≥ µA +2 (θ) ≥ · · · ≥ +µA +M(θ) the eigenvalues of A(θ)A(θ)T . Let γi be the classical location of the eigenvalues with respect to the +Marchenko-Pastur measure defined by +� ∞ +γi +ρMP,d0(dx) = 1 +M +� +i − 1 +2 +� +(C.66) +for i = 1, 2, . . . , M. Then, +|µA +i (θ) − γi| = O(M −2/3). +(C.67) +Proof. Note that the rigidity of the eigenvalues with an error of at most O(M −2/3) holds for random gram +matrices at the classical location of the eigenvalues with respect to the probability measure ρ from the Stieltjes +transform sρ(z) := +1 +M +� +i si(z), see Lemma 4 in [26]. Moreover, since |si(θ, z) − s(z)|, |sj(θ, z) − s(z)| = +66 + +O(M −2φ) for all i and j, we also have the desired rigidity near the classical location of Marchenko-Pastur +law ρMP,d0. +Remark C.6. In fact, rigorous proofs of the rigidity and anisotropic law are not given in [4, 3]. However, +as in the proof of anisotropic local law for general Wigner-type matrix in [2], the above lemmas may be proved +by using the local laws in [4] and standard methods in [2] (Remark 2.10 in [4] and Remark 2.7 in [3].) +On Γε +1/2, as a simple corollary to Lemma C.4, we obtain +��⟨x, GA(θ, z)y⟩ − s(z)⟨x, y⟩ +�� = O(N − 1 +2 ), +(C.68) +and +��⟨x, GA(θ, z)y⟩ − s(z)⟨x, y⟩ +�� = O(N − 1 +2 ). +(C.69) +We have the following lemma for the difference between Tr GA(0, z) and Tr GA(1, z) on Γε +1/2. +Lemma C.7. Let GA(θ, z) be defined as in Equations (C.57) and (C.58). Then, the following holds uniformly +for z ∈ Γε +1/2: +Tr GA(1, z) − Tr GA(0, z) = −λ(GH − Fg)k ∂ +∂z (zs(z) + 1) + O(N∥U∥2 +∞∥V ∥2 +∞). +(C.70) +We will prove Lemma C.7 later. +From Lemma C.5, we find that +| Tr GA(1, z) − Tr GA(0, z)| = +����� +N +� +i=1 +� +1 +µA +i (1) − z − +1 +µA +i (0) − z +������ = +����� +N +� +i=1 +µA +i (0) − µA +i (1) +(µA +i (1) − z)(µA +i (0) − z) +����� +≤ +����� +N +� +i=1 +|µA +i (0) − γi| + |γi − µA +i (1)| +(µA +i (1) − z)(µA +i (0) − z) +����� = O(N 1/3) +(C.71) +uniformly for z ∈ Γ. Thus, from (C.70) and (C.71), +1 +2πi +� +Γ +f(z) Tr GA(1, z)dz − 1 +2πi +� +Γ +f(z) Tr GA(0, z)dz += +1 +2πi +� +Γε +1/2 +f(z) +� +Tr GA(1, z) − Tr GA(0, z) +� +dz + 1 +2πi +� +Γ\Γε +1/2 +f(z) +� +Tr GA(1, z) − Tr GA(0, z) +� +dz += −λ(GH − Fg)k +2πi +� +Γε +1/2 +f(z) ∂ +∂z (zs(z) + 1)dz + O(N∥U∥2 +∞∥V ∥2 +∞) + O(N −1/6) += −λ(GH − Fg)k +2πi +� +Γ +f(z) ∂ +∂z (zs(z) + 1)dz + o(1). +(C.72) +Furthermore, using the relation (C.28), we have +1 +2πi +� +Γ +f(z) ∂ +∂z (zs(z) + 1)dz = +1 +2πi +� +Γ +f(z) 1 +√d0 +s′ +sc(ϕ(z))ϕ′(z)dz += +1 +2√d0πi +� +�Γ +�f(ϕ)s′ +sc(ϕ)dϕ = +1 +√d0 +τ1( �f). +67 + +C.2.3 +CLT for a random Gram matrix with a spike and small perturbation +Recall that A(1) = �Y − E[�Y ]. Our next step in the approximation is to consider �Y = A(1) + E[�Y ]. Since +E[�Y ] is not a matrix of rank k, we instead consider +B(θ) = A(1) + θ +� +λFgUV T , +GB(θ, z) = (B(θ)B(θ)T − zI)−1 +(C.73) +To prove this part of CLT, we will adapt the strategy for the proof of Theorem 5.5 with Lemmas C.4 and +C.5. We then find that, uniformly for z ∈ Γε +1/2, +Tr GB(1, z) − Tr GB(0, z) = −k +d +dzλFg(zs(z) + 1) +λFgzs(z) + λFg + 1 + O≺(N −φ), +(C.74) +since ∥U T U − Ik∥F , ∥V T V − Ik∥F ≺ N −φ. Using the rigidity (Lemma C.5) and the eigenvalue interlacing +property, we have +Tr GB′(1, z) − Tr GB′(0, z) = O(1) +on Γ\Γ1/2 +and so +1 +2πi +� +Γ +f(z) Tr GB(1, z)dz − 1 +2πi +� +Γ +f(z) Tr GB(0, z)dz += − k +2πi +� +Γ +f(z) λFg d +dz(zs + 1) +λFg(zs + 1) + 1dz + o(1). +(C.75) +The remaining part is to control an effect of small perturbation (E[�Y ] − +� +λFgUV T )ij = CN(uivT +j )3 + +O(N 2(uivT +j )5). First, we let +B′ = B(1) + CN(uivT +j )3, +GB′(z) = (B′(B′)T − zI)−1 +(C.76) +For 1 ≤ ℓ1, ℓ2, ℓ3 ≤ k, we consider vectors u3 and v3 such that +(u3(ℓ1, ℓ2, ℓ3))i = u3 +i (ℓ1, ℓ2, ℓ3) := +√ +Nui(ℓ1)ui(ℓ2)ui(ℓ3) +and +(v3(ℓ1, ℓ2, ℓ3))j = v3 +j (ℓ1, ℓ2, ℓ3) := +√ +Nvj(ℓ1)vj(ℓ2)vj(ℓ3). +We then observe that B′ contains k3 additional small spikes: +B′ = B(1) + C +� +ℓ1,ℓ2,ℓ3 +u3(ℓ1, ℓ2, ℓ3)v3(ℓ1, ℓ2, ℓ3)T +where ∥u3(ℓ1, ℓ2, ℓ3)∥∞, ∥v3(ℓ1, ℓ2, ℓ3)∥∞ ≺ N 1/2−3φ. +In the above point of view, we are able to consider B′ as another spiked Gram matrix model with two +types of spikes u(ℓ)v(ℓ)T and u3(ℓ1, ℓ2, ℓ3)(v3(ℓ1, ℓ2, ℓ3))T . As before, for 0 ≤ θ ≤ 1, let +B′(θ) = A(1) + θ +� +λFg +� +ℓ +u(ℓ)v(ℓ)T + θC +� +ℓ1,ℓ2,ℓ3 +u3(ℓ1, ℓ2, ℓ3)(v3(ℓ1, ℓ2, ℓ3))T +and +GB′(θ, z) = (B′(θ)B′(θ)T − zI)−1. +68 + +Following the proof of Theorem 5.5, we have +∂ +∂θ Tr GB′(θ, z) = −2 +� +λFg +∂ +∂z +� +ℓ +⟨u(ℓ), GB′(θ, z)B′(θ)v(ℓ)⟩ +− 2C ∂ +∂z +� +ℓ1,ℓ2,ℓ3 +⟨u3(ℓ1, ℓ2, ℓ3), GB′(θ, z)B′(θ)v3(ℓ1, ℓ2, ℓ3)⟩, +and it can be observed that the first term of the right-hand side is the leading order term, since ∥u3(ℓ1, ℓ2, ℓ3)∥∞, ∥v3(ℓ1, ℓ2, ℓ3)∥∞ ≺ +N 1/2−3φ < N −φ. Moreover, from the definition of B′(θ), the leading order term of ⟨u(ℓ), GB′(θ, z)B′(θ)v(ℓ)⟩ +is ⟨u(ℓ), GB′(θ, z)B′(0)v(ℓ)⟩+θ +� +λFg⟨u(ℓ), GB′(θ, z)u(ℓ)⟩, since ⟨v(ℓ1), v(ℓ2)⟩ = δℓ1ℓ2+O(N −φ), ⟨v(ℓ), v3(ℓ1, ℓ2, ℓ3)⟩ = +O(N 1/2−2φ) and ⟨v3(ℓ1, ℓ2, ℓ3), v3(ℓ4, ℓ5, ℓ6)⟩ = O(N 1−4φ). Carrying out the remaining procedures presented +in the proof of Theorem 5.5 and collecting the leading order terms, we eventually obtain +Tr GB′(1, z) − Tr GB′(0, z) = −k +d +dzλFg(zs(z) + 1) +λFgzs(z) + λFg + 1 + O≺(N 1/2−2φ) +(C.77) +uniformly for z ∈ Γε +1/2. Here, we last apply Lemma C.4 and (C.65) for B′(0) = A(1). Further, on Γ\Γ1/2, +from the rigidity and the interlacing property of the eigenvalues, +Tr GB′(1, z) − Tr GB′(0, z) = O(1). +(C.78) +Thus, we conclude that +1 +2πi +� +Γ +f(z) Tr GB′(z)dz − 1 +2πi +� +Γ +f(z) Tr GB′(0, z)dz += − k +2πi +� +Γ +f(z) λFg d +dz(zs + 1) +λFg(zs + 1) + 1dz + o(1). +Furthermore, we set Eij = (�Y − B′)ij = O(N 2(uivT +j )5). Then +∥E∥ ≤ ∥E∥F = O(N 2∥U∥4 +∞∥V ∥4 +∞) = o(N −1) +(C.79) +for some φ > 3/8. This implies that +1 +2πi +� +Γ +f(z) Tr G +�Y (z)dz − 1 +2πi +� +Γ +f(z) Tr GB′(1, z)dz = o(1). +Remark C.8. Under the assumption that φ > 3/8, we suffices to consider E[�Yij] up to O(N 2(uivT +j )5) error. +However, (C.77) and (C.78) are valid for any finite approximation of E[�Y ] as presented in (C.51), even for +any φ > 1/4. This means that the condition φ > 3/8 can be improved by considering a higher order expansion +of E[�Y ]. For example, if we consider +E[�Y ] = +� +λFguivT +j + C1N(uivT +j )3 + C2N 2(uivT +j )5 + O(N 3(uivT +j )7), +then it can be checked that the contributions of the second and third terms are negligible, and the error +Eij = O(N 3(uivT +j )7) is also negligible if φ > 1/3, since +∥E∥ ≤ ∥E∥F = O(N 3∥U∥6 +∞∥V ∥6 +∞) ≺ N 3−12φ = o(N −1). +69 + +C.2.4 +Conclusion for the proof of pre-transformed CLT +We are now ready to prove pre-transformed CLT. Denote by �µ1 ≥ �µ2 ≥ · · · ≥ �µN the eigenvalues of �Y �Y T . +Recall that we denoted by µA +1 (0) ≥ µA +2 (0) ≥ · · · ≥ µA +N(0) the eigenvalues of A(0)A(0)T . From Cauchy’s +integral formula, we have +M +� +i=1 +f(�µi) − M +� d+ +d− +f(x) ρMP,d0(dx) += +� M +� +i=1 +f(µA +i (0)) − +� d+ +d− +f(x) ρMP,d0(dx) +� ++ +� M +� +i=1 +f(�µi) − +M +� +i=1 +f(µA +i (0)) +� += +� M +� +i=1 +f(µA +i (0)) − M +� d+ +d− +f(x) ρMP,d0(dx) +� +− +� 1 +2πi +� +Γ +f(z) Tr G +�Y (z)dz − 1 +2πi +� +Γ +f(z) Tr GA(0, z)dz +� +. +(C.80) +Since AA∗ is a usual sample covariance matrix, the first term in the right-hand side converges to a Gaussian +random variable. Further, as computed in (C.52), +E[�Y 4 +ij] =: � +w4 +N 2 + +1 +(NFg)2 +� ∞ +−∞ +�g′(w) +g(w) +�4 � +g +� +w − +√ +NλuivT +j +� +− g(w) +� +dw, +where the first term is the leading term of E[�Y 4 +ij] and hence the leading term of E[A4 +ij] as well. This means +that the difference between �w4 and E[A4 +ij] is negligible in the sense that it has no contribution in the limiting +behavior of the resolvent, which can be checked from standard Green function comparison theorems. (Refer +to [26].) +Thus, the mean and the variance of the limiting Gaussian distribution are given by +mA(f) = +�f(2) + �f(−2) +4 +− 1 +2τ0( �f) + (� +w4 − 3)τ2( �f) +(C.81) +and +VA(f) = 2 +∞ +� +ℓ=1 +ℓτℓ( �f)2 + (� +w4 − 3)τ1( �f)2, +(C.82) +respectively. +For the second term in the right-hand side of (C.80), by (C.75), we obtain that +1 +2πi +� +Γ +f(z) Tr G +�Y (z)dz − 1 +2πi +� +Γ +f(z) Tr GA(0, z)dz += − k +2πi +� +Γ +f(z) λFg d +dz(sz + 1) +λFg(sz + 1) + 1dz + o(1) +(C.83) +with high probability. From (C.80), we thus find that the CLT for the LSS holds, i.e., +� M +� +i=1 +f(�µi) − M +� d+ +d− +f(x) ρMP,d0(dx) +� +→ N(m�Y (f), V�Y (f)), +(C.84) +and the variance V�Y (f) = VA(f) since the second term in (C.80) converges to a deterministic value as +70 + +N → ∞, which corresponds to the change of the mean. In particular, +m�Y (f) − mA(f) = (GH − Fg)λk +2πi +� +Γ +f(z) ∂ +∂z (zs(z) + 1)dz + k +2πi +� +Γ +f(z) λFg d +dz(sz + 1) +λFg(sz + 1) + 1dz. +(C.85) +Following the computation in the proof of Lemma 4.4 in [9] with the relation (C.28), we find that the +right-hand side of (C.85) is given by +k +2πi +� +Γ +f(z)(zs(z) + 1)′ +� +λ(GH − Fg) + +λFg +λFg(zs(z) + 1) + 1 +� +dz += λk +√d0 +(GH − Fg)τ1( �f) + k +∞ +� +ℓ=1 +� λFg +√d0 +�ℓ +τℓ( �f). +(C.86) +(See also Remark 1.7 of [9].) Thus, +m�Y (f) = +�f(2) + �f(−2) +4 +− 1 +2τ0( �f) + λk +√d0 +(GH − Fg)τ1( �f) + (� +w4 − 3)τ2( �f) + k +∞ +� +ℓ=1 +� λFg +√d0 +�ℓ +τℓ( �f) +(C.87) +and +V�Y (f) = 2 +∞ +� +ℓ=1 +ℓτℓ( �f)2 + (� +w4 − 3)τ1( �f)2. +(C.88) +C.3 +Proof of Lemma C.7 +Notational remarks +In the rest of the section, we use C order to denote a constant that is independent of N. Even if the constant +is different from one place to another, we may use the same notation C as long as it does not depend +on N for the convenience of the presentation. Now, we recall the linearization HA(θ)(z) and its inverse +RA(θ, z) = HA(θ)(z)−1. For simplicity, we drop the subscript A and index z of the linearization entries. +Proof of Lemma C.7. To prove the lemma, we consider +∂ +∂θ Tr GA(θ, z) = +� +b +� +a +� +α +∂Aaα(θ) +∂θ +∂Gbb(θ) +∂Aaα(θ) += +� +b +� +a +� +α +∂Haα(θ) +∂θ +∂Rbb(θ) +∂Haα(θ) += − +� +b +� +a +� +α +∂Haα(θ) +∂θ +[Rba(θ)Rαb(θ) + Rbα(θ)Rab(θ)] += −2 +� +a +� +α +∂Haα(θ) +∂θ +(R(θ)2)aα += −2 +� +a +� +α +∂Haα(θ) +∂θ +∂ +∂z Raα(θ), +(C.89) +where we again used that +∂ +∂zGA(θ, z) = GA(θ, z)2. We expand the right-hand side by using the definition of +71 + +A(θ), +Haα(θ) = Aaα(θ) = +� +1 − θ + θ +� +NSaα +� +Aaα(0) = +� +1 − θ + θ +� +NSaα +� +Haα(0), +(C.90) +and so +� +a +� +α +∂Haα(θ) +∂θ +Raα(θ) = +� +a +� +α +� +−1 + +� +NSaα +� +Haα(0)Raα(θ) += +� +a +� +α +−1 + √NSaα +1 − θ + θ√NSaα +Haα(θ)Raα(θ) += Nλ(GH − Fg) +2 +� +a +� +α +(uavT +α)2Haα(θ)Raα(θ) + O(N∥U∥2 +∞∥V ∥2 +∞). +(C.91) +From now, we further drop the θ-dependency for the brevity. +Then +∂ +∂θ Tr GA(θ, z) = −Nλ(GH − Fg) ∂ +∂z +� +a +� +α +(uavT +α)2HaαRaα + O(N∥U∥2 +∞∥V ∥2 +∞). +Here, we used the properties that Haα = Aab(θ) = O(N − 1 +2 ), Rab = GA +ba(θ) = O(N − 1 +2 ) for b ̸= a, Raa = +GA +aa(θ) = O(1), and � +a ua(ℓ1)ua(ℓ2) = δℓ1ℓ2 = � +α vα(ℓ1)vα(ℓ2), which imply +�����N 2 � +a +� +α +(uavT +α)4HaαRaα +����� ≤ N 2∥U∥2 +∞∥V ∥2 +∞ +� +a +� +α +(uavT +α)2|HaαRaα| = O(N∥U∥2 +∞∥V ∥2 +∞). +(C.92) +Together with Remark C.3, from the elementary equality for R and H, we have +� +a +� +α +ua(ℓ)2HaαRaα = +� +a +ua(ℓ)2 +�� +α +HaαRaα +� += +� +a +ua(ℓ)2(1 + zRaa) += 1 + zs(z) + O(N − 1 +2 ), +(C.93) +and +� +a +� +α +vα(ℓ)2HaαRaα = +� +α +vα(ℓ)2 +�� +a +HaαRaα +� += +� +α +vα(ℓ)2(1 + Rαα) += d0(1 + zs(z)) + O(N − 1 +2 ). +(C.94) +72 + +Plugging them into (C.91), we get +4 +λ(GH − Fg) × (C.91) += N +� +ℓ +� +a +� +α +� 1 +N ua(ℓ)2HaαRaα + 1 +M vα(ℓ)2HaαRaα ++ +� +ua(ℓ)2 − 1 +M +� +vα(ℓ)2HaαRaα + ua(ℓ)2 +� +vα(ℓ)2 − 1 +N +� +HaαRaα +� ++ 2N +� +ℓ1̸=ℓ2 +� +a +� +α +ua(ℓ1)ua(ℓ2)vα(ℓ1)vα(ℓ2)HaαRaα + O(N∥U∥2 +∞∥V ∥2 +∞) += N +� +ℓ +� +a +� +α +� � +ua(ℓ)2 − 1 +M +� +vα(ℓ)2HaαRaα + ua(ℓ)2 +� +vα(ℓ)2 − 1 +N +� +HaαRaα +� ++ 2N +� +ℓ1̸=ℓ2 +� +a +� +α +ua(ℓ1)ua(ℓ2)vα(ℓ1)vα(ℓ2)HaαRaα ++ 2k(zs(z) + 1) + O(N∥U∥2 +∞∥V ∥2 +∞). +(C.95) +It remains to estimate the first three terms in (C.95). Set +X1 ≡ X1(θ, z, ℓ) := +� +a +� +α +� +ua(ℓ)2 − 1 +M +� +vα(ℓ)2HaαRaα, +(C.96) +X2 ≡ X2(θ, z, ℓ) := +� +a +� +α +ua(ℓ)2 +� +vα(ℓ)2 − 1 +N +� +HaαRaα +(C.97) +and +X3 ≡ X3(θ, z, ℓ1, ℓ2) := +� +a +� +α +ua(ℓ1)ua(ℓ2)vα(ℓ1)vα(ℓ2)HaαRaα +(ℓ1 ̸= ℓ2). +(C.98) +We notice that |X1|, |X2|, |X3| = O(N −1) on Γ1/2 by a naive power counting as in (C.91) after applying +H¨older inequality once. To obtain a better bound, we use a method based on a recursive moment estimate, +introduced in [39]. We need the following lemma: +Lemma C.9. Let X1, X2 and X3 be as in (C.96), (C.97) and (C.98). Define an event Ωε by +Ωε = +� +a,b,α,β +{|Haα|, |Raα| ≤ N − 1 +2 +ε} ∩ {|Rab − s(z)δab| ≤ N − 1 +2 +ε} ∩ {|Rαβ − zs(z)δαβ| ≤ N − 1 +2 +ε} +Then, for any fixed (large) D and (small) ε, which may depend on D, +E[|X|2D|Ωε] ≤ CN − 1 +2 +ε∥u∥2 +∞E[|X|2D−1|Ωε] + CN −1+4ε∥u∥4 +∞E[|X|2D−2|Ωε] ++ CN −2+10ε∥u∥6 +∞E[|X|2D−3|Ωε] + CN −3+14ε∥u∥8 +∞E[|X|2D−4|Ωε], +(C.99) +where X is X1, X2 and X3. +Since the rank of the signal k is fixed, we suffices to prove the above lemma for fixed ℓ, ℓ1 and ℓ2. We will +prove Lemma C.9 for X1 at the end of this section (the calculation for the X2 and X3 is almost the same). +With Lemma C.9, we are ready to obtain an improved bound for X. First, note that the contribution from +the exceptional event Ωc +ε is negligible i.e., P(Ωc +ε) < N −D2, which can be checked by applying a high-order +73 + +Markov inequality with the moment condition on �Y (See Assumption 3.1). We decompose E[|X|2D] by +E[|X|2D] = E[|X|2D · 1(Ωε)] + E[|X|2D · 1(Ωc +ε)] = E[|X|2D|Ωε] · P(Ωε) + E[|X|2D · 1(Ωc +ε)]. +(C.100) +Then the second term in the right-hand side of (C.100), +E[|X|2D · 1(Ωc +ε)] ≤ +� +E[|X|4D] +� 1 +2 (P(Ωc +ε)) +1 +2 ≤ N − D2 +2 � +E[|X|4D] +� 1 +2 +(C.101) +and by using a trivial bound for the resolvent |Rab(z)| ≤ ∥GA(z)∥ ≤ +1 +Im z +E[|X|4D] ≤ E +�� +a +� +α +|HaαRaα| +�4D +≤ (M 2N)4D +(Im z)4D max +a,b,α E|HaαHbα|4D ≤ CN 14D. +(C.102) +To bound the right-hand side of (C.99), we use Young’s inequality: For any a, b > 0 and p, q > 0 with +1 +p + 1 +q = 1, +ab ≤ ap +p + bq +q . +We then find that the first term has the following upper bound +N − 1 +2 +ε∥u∥2 +∞|X|2D−1 = N +(2D−1)ε +2D +N − 1 +2 +ε∥u∥2 +∞ · N − (2D−1)ε +2D +|X|2D−1 +≤ +1 +2DN (2D−1)ε(N − 1 +2 +ε∥u∥2 +∞)2D + 2D − 1 +2D +N −ε|X|2D. +(C.103) +Applying Young’s inequality for other terms in (C.99), we get +E[|X|2D|Ωε] ≤ CN (2D−1)ε(N − 1 +2 +ε∥u∥2 +∞)2D + CN (D−1)ε(N −1+4ε∥u∥4 +∞)D ++ CN ( 2D +3 −1)ε(N −2+10ε∥u∥6 +∞) +2D +3 + CN ( D +2 −1)ε(N −3+14ε∥u∥8 +∞) +D +2 ++ CN −εE[|X|2D|Ωε]. +(C.104) +Absorbing the last term in the right-hand side to the left-hand side and plugging the estimates (C.101) and +(C.102) into (C.100), we now get +E[|X|2D] ≤ CN (2D−1)ε(N − 1 +2 +ε∥u∥2 +∞)2D + CN (D−1)ε(N −1+4ε∥u∥4 +∞)D ++ CN ( 2D +3 −1)ε(N −2+10ε∥u∥6 +∞) +2D +3 + CN ( D +2 −1)ε(N −3+14ε∥u∥8 +∞) +D +2 + CN − D2 +2 +7D. +(C.105) +From the (2D)-th order Markov inequality, for any fixed ε′ > 0 independent of D, +P +� +|X| ≥ N ε′N − 1 +2 ∥u∥2 +∞ +� +≤ N −2Dε′ +E[|X|2D] +(N − 1 +2 ∥u∥2∞)2D ≤ N −2Dε′N 8Dε. +(C.106) +By choosing ε = 1/D, for sufficiently large D, we find that +|X| = O(N − 1 +2 ∥u∥2 +∞). +(C.107) +74 + +We now return to (C.89) and use (C.95) with the bound (C.107), +M +� +j=1 +N +� +k=1 +∂Ajk(θ) +∂θ +(GA(θ)A(θ))jk = (GH − Fg)λk +2 +(1 + zs(z)) + O(N∥u∥2 +∞∥v∥2 +∞). +(C.108) +To handle the derivative of the right-hand side, we use Cauchy’s integral formula with a rectangular contour, +contained in Γε +1/2, whose perimeter is larger than ε. Then, we get from (C.89) that +∂ +∂θ Tr GA(θ, z) = −λ(GH − Fg) · k ∂ +∂z (1 + zs(z)) + O(N∥u∥2 +∞∥v∥2 +∞). +(C.109) +After integrating over θ from 0 to 1, we conclude that (C.70) holds for a fixed z ∈ Γε +1/2. +At last, we prove Lemma C.9. +Proof of Lemma C.9. As we mentioned above, we consider X = X1 and drop the ℓ-dependency. i.e. +E[|X|2D] = E +�� +a +� +α +� +u2 +a − 1 +M +� +v2 +αHaαRaαXD−1X +D +� +We use the following inequality that generalizes Stein’s lemma (see Proposition 5.2 of [11]): Let Φ be a C2 +function. Fix a (small) ε > 0, which may depend on D. Recall that Ωε is the complement of the exceptional +event on which |Haα| or |Raα| is exceptionally large for some a, α, defined by Ωε by +� +a,b,α,β +{|Haα|, |Raα| ≤ N − 1 +2 +ε} ∩ {|Rab − s(z)δab| ≤ N − 1 +2 +ε} ∩ {|Rαβ − zs(z)δαβ| ≤ N − 1 +2 +ε} +Then, +E[HaαΦ(Haα)|Ωε] = (E[H2 +aα|Ωε] − E[Haα|Ωε]2)E[Φ′(Haα)|Ωε] + ε1, +(C.110) +where the error term ε1 admits the bound +|ε1| ≤ C1E +� +|Haα|3 sup +|t|≤1 +Φ′′(tHaα) +���Ωε +� +(C.111) +for some constant C1. Note that by applying a decomposition (C.100) to E[Haα|Ωε] and E[H2 +aα|Ωε], we see +that +E[Haα|Ωε] − E(E[Haα|Ωε]) = E[Haα|Ωε] = O(N −D0) +(C.112) +and +E[H2 +aα|Ωε] = E(E[H2 +aα|Ωε]) + O(N −D0) = 1 +N + O(∥u∥2 +∞∥v∥2 +∞) + O(N −D0) +(C.113) +for D0 = D2+1 +2 +> 1. The estimate (C.110) follows from the proof of Proposition 5.2 of [11] with p = 1, where +we use the inequality (5.38) therein only up to second to the last line. +In the estimate (C.110), we choose +Φ(Haα) = RaαXD−1X +D +(C.114) +so that +E[|X|2D|Ωε] = +� +a +� +α +� +u2 +a − 1 +M +� +v2 +αE [HaαΦ(Haα)|Ωε] . +(C.115) +75 + +Applying (C.112) and (C.113) to the equation (C.110), +E [HaαΦ(Haα)|Ωε] = E +� +H2 +aα +� +E[Φ′(Haα)|Ωε] + ε1 += E[H2 +aα] +� +−E +� +RaaRααXD−1X +D|Ωε +� +− E +� +R2 +aαXD−1X +D|Ωε +� ++(D − 1)E +� +Raα +∂X +∂Haα +XD−2X +D��Ωε +� ++ DE +� +Raα +∂X +∂Haα +XD−1X +D−1��Ωε +�� ++ ε1, +(C.116) +for sufficiently large D. We plug it into (C.115) and estimate each term. Then the term originated from the +first term in (C.116) can be separated by +� +a +� +α +� +u2 +a − 1 +M +� +v2 +αE[H2 +aα]E +� +RaaRααXD−1X +D|Ωε +� += +� +a +� +α +� +u2 +a − 1 +M +� +v2 +αE[H2 +aα]E +� +(Raa − s)RααXD−1X +D|Ωε +� ++ s +� +a +� +α +� +u2 +a − 1 +M +� +v2 +αE[H2 +aα]E +� +RααXD−1X +D|Ωε +� +. +(C.117) +The first term satisfies that +����� +� +a +� +α +� +u2 +a − 1 +M +� +v2 +αE[H2 +aα]E +� +(Raa − s)RααXD−1X +D|Ωε +������ +≤ CM∥u∥2 +∞N −1N − 1 +2 +εE[|X|2D−1|Ωε] +� +α +v2 +α = CN − 1 +2 +ε∥u∥2 +∞E[|X|2D−1|Ωε] +(C.118) +for some constant C since � +α v2 +α = 1. Using (C.113) and � +a +� +u2 +a − +1 +M +� += 0, we also have +�����s +� +a +� +α +� +u2 +a − 1 +M +� +v2 +αE[H2 +aα|Ωε]E +� +RααXD−1X +D|Ωε +������ +≤ C∥u∥2 +∞∥v∥2 +∞|s| +� +a +� +α +� +u2 +a + 1 +M +� +v2 +αE +� +|RααXD−1X +D||Ωε +� +≤ C∥u∥2 +∞∥v∥2 +∞E[|X|2D−1|Ωε] +(C.119) +for some constant C and large D > 1. For the second term in (C.116), we also have +����� +� +a +� +α +� +u2 +a − 1 +M +� +v2 +αE[H2 +aα|Ωε]E +� +R2 +aαXD−1X +D|Ωε +������ +≤ CN −1∥u∥2 +∞ +����� +� +a +� +α +v2 +αE +� +R2 +aαXD−1X +D|Ωε +������ +≤ CN −1+2ε∥u∥2 +∞E[|X|2D−1|Ωε] +� +α +v2 +α +≤ CN −1+2ε∥u∥2 +∞E[|X|2D−1|Ωε]. +(C.120) +76 + +To estimate the third term and the fourth term in (C.116), we notice that on Ωε +���� +∂X +∂Haα +���� = +������ +− +� +b +� +β +� +u2 +b − 1 +M +� +v2 +βHbβ[RabRαβ + RbαRaβ] + +� +u2 +a − 1 +M +� +v2 +αRaα +������ +≤ CN − 1 +2 +3ε∥u∥2 +∞ +� +α +v2 +α + CN − 1 +2 +ε∥u∥2 +∞∥v∥2 +∞ ≤ CN − 1 +2 +3ε∥u∥2 +∞. +(C.121) +for some constant C. Similarly, we can observe that +���� +∂2X +∂H2aα +���� ≤ CN − 1 +2 +3ε∥u∥2 +∞. +(C.122) +Thus, we also obtain that +����� +� +a +� +α +� +u2 +a − 1 +M +� +v2 +αE[H2 +aα|Ωε]E +� +Raα +∂X +∂Haα +XD−2X +D��Ωε +������ +≤ CN −1+4ε∥u∥4 +∞E[|X|2D−2|Ωε] +(C.123) +and +����� +� +a +� +α +� +u2 +a − 1 +M +� +v2 +αE[H2 +aα|Ωε]E +� +Raα +∂X +∂Haα +XD−1X +D−1��Ωε +������ +≤ CN −1+4ε∥u∥4 +∞E[|X|2D−2|Ωε]. +(C.124) +Hence, from (C.116), (C.120), (C.123), and (C.124), +����� +� +a +� +α +� +u2 +a − 1 +M +� +v2 +αE[H2 +aα|Ωε]E[Φ′(Haα)|Ωε] +����� +≤ CN − 1 +2 +ε∥u∥2 +∞E[|X|2D−1|Ωε] + CN −1+4ε∥u∥4 +∞E[|X|2D−2|Ωε] + ε1. +(C.125) +It remains to estimate |ε1| in (C.111). Proceeding as before, +� +a +� +α +� +u2 +a − 1 +M +� +v2 +αE +� +|Haα|3Φ′′(Haα) +���Ωε +� +≤ CN −1+4ε∥u∥2 +∞E[|X|2D−1|Ωε] + CN −2+7ε∥u∥4 +∞E[|X|2D−2|Ωε] ++ CN −2+10ε∥u∥6 +∞E[|X|2D−3|Ωε]. +(C.126) +Our last goal is to find the bound for the error term ε1. To handle Φ′′(tHaα), we want to compare +Φ′′(Haα) and Φ′′(tHaα) for some |t| < 1. Let GA,t be the resolvent of A where Aaα is replaced by tAaα, +and let Xt be defined as X in (C.96) with the same replacement for Aaα and also GA is replaced by GA,t. +Correspondingly, we also consider the replacement Rt of the linearization R by substituting tHaα into Haα +(also for Hαa). Then, +Rt +AB − RAB = (Rt(H − Ht)R)AB = (1 − t)Rt +AaHaαRαB + (1 − t)Rt +AαHαaRaB. +(C.127) +77 + +and +Xt − X = +� +b +� +β +� +u2 +b − 1 +M +� +v2 +β(Ht +bβRt +bβ − HbβRbβ) += +� +b +� +β +� +u2 +b − 1 +M +� +v2 +βHbβ(Rt +bβ − Rbβ) + (t − 1) +� +u2 +a − 1 +M +� +v2 +α(HaαRt +aα) += (1 − t) +� +b +� +β +� +u2 +b − 1 +M +� +v2 +βHbβRt +baHaαRαβ ++ (1 − t) +� +b +� +β +� +u2 +b − 1 +M +� +v2 +βHbβRt +bαHαaRaβ + (t − 1) +� +u2 +a − 1 +M +� +v2 +αHaαRt +aα. +(C.128) +Thus, on Ωε, +|Xt − X| ≤ CN −1+4ε∥u∥2 +∞. +(C.129) +Using the estimates (C.127) and (C.129), on Ωε, we obtain that +|Φ′′(Haα) − Φ′′(tHaα)| ≤ C|Φ′′(Haα)| + N − 5 +2 +11ε∥u∥6 +∞|X|2D−4 +(C.130) +uniformly on t ∈ (−1, 1). +Combining (C.115) and (C.125) with (C.126), (C.130), and (C.111), we finally get +E[|X|2D|Ωε] ≤ CN − 1 +2 +ε∥u∥2 +∞E[|X|2D−1|Ωε] + CN −1+4ε∥u∥4 +∞E[|X|2D−2|Ωε] ++ CN −2+10ε∥u∥6 +∞E[|X|2D−3|Ωε] + CN −3+14ε∥u∥8 +∞E[|X|2D−4|Ωε]. +(C.131) +This proves the desired lemma for X = X1. +For the cases X = X2 or X = X3, the proofs are almost the same with the following changes: +• For X = X2, we change the role of U and V . In other words, we will use � +a ua(ℓ)2 = 1 and +� +α +� +vα(ℓ)2 − 1 +N +� += 0. Then the recursive bound for E[|X2|2D|Ωε] obtained by putting v instead of +u in the upper bound in (C.99). +• In the same way, we use � +a ua(ℓ1)ua(ℓ2) = δℓ1ℓ2 and � +α |vα(ℓ1)||vα(ℓ2)| ≤ 1 instead of � +a +� +u2 +a − +1 +M +� += +0 and � +a ua(ℓ)2 = 1, respectively. We then obtain the exactly same recursive bound in (C.99) for X3. +C.4 +Computation of the test statistic +In this section, we prove the second part of Theorem 5.5 and also provide the details on the computation of +the test statistic in Theorem 4.2. By performing the same calculations as we will do in this section, we can +obtain optimal functions for the other models, so we omit the details. (Refer to [22, 33, 34].) Recall that +mY (f)|H1 − mY (f)|H0 = +k +� +s=1 +∞ +� +ℓ=1 +� ωs +√d0 +�ℓ +τℓ( �f) +(C.132) +and +VY (f) = 2 +∞ +� +ℓ=2 +ℓτℓ( �f)2 + (w4 − 1)τ1( �f)2. +(C.133) +78 + +Assuming w2 > 0 and w4 > 1, from Cauchy’s inequality and the identity log(1 − λ) = − �∞ +ℓ=1 λℓ/ℓ, +����� +mY (f)|H1 − mY (f)|H0 +� +VY (f) +����� +2 +≤ +k +� +p,q=1 +ωpωq +d0 +� +1 +w4 − 1 − 1 +2 +� +− 1 +2 log +� +1 − ωpωq +d0 +� += +���� +m(Ω) − m(0) +√V0 +���� +2 +, +(C.134) +which proves the first part of the theorem. The equality in (C.134) holds if and only if +√d0(w4 − 1)τ1( �f) +� +s ωs += 2ℓ(√d0)ℓτℓ( �f) +� +s ωℓs +(ℓ = 2, 3, 4, . . . ). +(C.135) +We now find all functions f that satisfy (C.135). Letting 2C be the common value in (C.135), +τ1( �f) = +2C +√d0(w4 − 1) +� +s +ωs, +τℓ( �f) = +C +ℓ(√d0)ℓ +� +s +ωℓ +s +(ℓ = 2, 3, 4, . . . ). +(C.136) +We can expand �f in terms of the Chebyshev polynomials as +�f(x) = +∞ +� +ℓ=0 +CℓTℓ +�x +2 +� +. +(C.137) +The orthogonality relation of the Chebyshev polynomials implies that for ℓ ≥ 1 +τℓ( �f) = Cℓ +π +� 2 +−2 +Tℓ +�x +2 +� +Tℓ +�x +2 +� +dx +√ +4 − x2 = Cℓ +π +� 1 +−1 +Tℓ (y) Tℓ (y) +dy +� +1 − y2 = Cℓ +2 . +(C.138) +Thus, (C.136) holds if and only if +�f(x) = c0 + 2C +� +s +� +2ωs +√d0(w4 − 1)T1 +�x +2 +� ++ +∞ +� +ℓ=2 +1 +ℓ +� ωs +√d0 +�ℓ +Tℓ +�x +2 +�� += c0 + 2C +� +s +� +ωs +√d0 +� +2 +w4 − 1 − 1 +� +T1 +�x +2 +� ++ +∞ +� +ℓ=1 +1 +ℓ +� ωs +√d0 +�ℓ +Tℓ +�x +2 +�� +(C.139) +for some constant c0. We notice that the following identity holds for the Chebyshev polynomials: +∞ +� +ℓ=1 +tℓ +ℓ Tℓ (x) = log +� +1 +√ +1 − 2tx + t2 +� +. +(C.140) +(See, e.g., (18.12.9) of [45].) Since T1(x) = x, we find that (C.139) is equivalent to +�f(x) = c0 + C +� +s +� ωs +√d0 +� +2 +w4 − 1 − 1 +� +x − log +�d0 − ωs +√d0x + ω2 +s +d0 +�� +, +(C.141) +79 + +or +f(x) = c0 + C +� +s +�ωs +d0 +� +2 +w4 − 1 − 1 +� +x − ωs(1 + d0) +d0 +� +2 +w4 − 1 − 1 +�� +− C +� +s +log +�ωs +d0 +�� +1 + d0 +ωs +� +(1 + ωs) − x +�� +. +(C.142) +This concludes the proof of Theorem 5.2 with an optimal function +φΩ(x) = �φΩ(ϕ(x)) +(C.143) +where +�φΩ(x) = c0 + +� +s +� ωs +√d0 +� +2 +w4 − 1 − 1 +� +x − log +�d0 − ωs +√d0x + ω2 +s +d0 +�� +. +(C.144) +Choosing +c0 = +� +s +�(1 + d0) +d0 +� +2 +w4 − 1 − 1 +� +ωs + log(ωs/d0) +� +, +we get (4.2). Further, we can see that +φΩ(x) = +� +s +φωs(x). +(C.145) +From this, we directly obtain that LΩ = � +s Lωs, +mY (φω)|H0 = −1 +2 +� +s +log +� +1 − ω2 +s +d0 +� ++ +1 +2d0 +(w4 − 3) +� +s +ω2 +s, +(C.146) +mY (φω)|H1 = mY (φω)|H0 + +� +p,q +� +− log +� +1 − ωpωq +d0 +� ++ ωpωq +d0 +� +2 +w4 − 1 − 1 +�� +(C.147) +and +VY (φω)|H1 = VY (φω)|H0 = 2 +� +p,q +� +− log +� +1 − ωpωq +d0 +� ++ ωpωq +d0 +� +2 +w4 − 1 − 1 +�� +. +(C.148) +80 + diff --git a/D9E4T4oBgHgl3EQf6Q7Z/content/tmp_files/load_file.txt b/D9E4T4oBgHgl3EQf6Q7Z/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b8eb4c1db5bc73b785c407d253479e263d3ca0a1 --- /dev/null +++ b/D9E4T4oBgHgl3EQf6Q7Z/content/tmp_files/load_file.txt @@ -0,0 +1,2589 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf,len=2588 +page_content='Detection problems in the spiked matrix models Ji Hyung Jung∗, Hye Won Chung†, and Ji Oon Lee‡ January 16, 2023 Abstract We study the statistical decision process of detecting the low-rank signal from various signal-plus- noise type data matrices, known as the spiked random matrix models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We first show that the principal component analysis can be improved by entrywise pre-transforming the data matrix if the noise is non- Gaussian, generalizing the known results for the spiked random matrix models with rank-1 signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' As an intermediate step, we find out sharp phase transition thresholds for the extreme eigenvalues of spiked random matrices, which generalize the Baik-Ben Arous-P´ech´e (BBP) transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We also prove the central limit theorem for the linear spectral statistics for the spiked random matrices and propose a hypothesis test based on it, which does not depend on the distribution of the signal or the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' When the noise is non-Gaussian noise, the test can be improved with an entrywise transformation to the data matrix with additive noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We also introduce an algorithm that estimates the rank of the signal when it is not known a priori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' 1 Introduction One of the most natural approach to ‘signal-plus-noise’ type data is to consider spiked random matrices, which are the low-rank deformation of large random matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Most notable examples of spiked random matrices include spiked Wigner matrix and spiked Wishart matrix, where the signals are given as a low-rank mean matrix (spiked Wigner matrix) and a low-rank perturbation of the identity in its covariance matrix (spiked Wishart matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' In this paper, we focus on the following three types of noisy data matrices, known as spiked random matrices, which generalize spiked Wigner/Wishart matrices: Spiked Wigner matrix: the data matrix is of the form UΛ1/2U T + W, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1) where U = [u(1), u(2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' , u(k)] ∈ RN×k with U T U = Ik, and W is an N × N Wigner matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The signal-to-noise ratio (SNR) Λ = diag(λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' , λk) with λ1 ≥ λ2 ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' λk > 0 for some positive integer k, independent of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' ∗Department of Mathematical Sciences, KAIST, Daejeon, 34141, Korea email: jhjung66@kaist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='kr †School of Electrical Engineering, KAIST, Daejeon, 34141, Korea email: hwchung@kaist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='kr ‡Department of Mathematical Sciences, KAIST, Daejeon, 34141, and School of Mathematics, KIAS, Seoul, 02455, Korea email: jioon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='lee@kaist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='edu 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='05331v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='ST] 12 Jan 2023 Rectangular matrix with spiked mean (additive model): the data matrix is of the form UΛ1/2V T + X, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2) where U = [u(1), u(2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' , u(k)] ∈ RM×k, V = [v(1), v(2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' , v(k)] ∈ RN×k with U T U = V T V = Ik, and X is an M × N random i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' matrix whose entries are centered with variance N −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The SNR Λ is given as in the spiked Wigner matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Rectangular matrix with spiked covariance (multiplicative model): the data matrix is of the form (I + UΛU T )1/2X, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3) where U = [u(1), u(2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' , u(k)] with U T U = Ik and X is an M × N random i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' matrix whose entries are centered with variance N −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The SNR Λ is given as in the spiked Wigner matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Here, Ik is the identity matrix with rank k and we allow the case k = 0 where no signal is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Throughout the paper, for the ease of notation, we denote by W an N × N Wigner matrix, and X an M × N random i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' To describe the detection problems we consider in this paper, we first review the known results for the simplest case of the spiked random matrix models with rank-1 spike, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='e, k = 1 in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2), and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Signal detection problem in rank-1 spiked random matrices: Many problems concerning the signal detection can be answered in the case with Gaussian noise and rank-1 spike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' In this case, the spikes U = u and V = v are vectors, and SNR λ1 = λ, hence the spiked random matrices are of the following forms: √ λuuT + W (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4) √ λuvT + X (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5) (I + λuuT )1/2X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='6) For this case, reliable detection of the signal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=', detection with probability 1 − o(1) as M, N → ∞, is impossible if the SNR λ is below a certain threshold [44, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The threshold is 1 as N → ∞ for spiked Wigner matrices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' for spiked rectangular matrices, with additional assumption M/N → d0 as N → ∞, the threshold is √d0 for a general class of priors [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' On the other hand, the signal can be reliably detected by the principal component analysis (PCA) if the SNR is above the threshold in which case the signal can actually be estimated [24, 41, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' In the subcritical case where the signal is not reliably detectable, it is natural to consider a hypothesis test on the presence of the signal between H0 : λ = 0 and H1 : λ = ω, commonly referred to as the weak detection, which is also known as the sphericity test in the case the spike is drawn from the uniform distribution on the unit sphere, known as the spherical prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' As asserted by Neyman–Pearson lemma, the likelihood ratio (LR) test is optimal in the sense that it minimizes the sum of the Type-I error and the Type-II error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' It was proved for several distributions of the spikes, called priors, that this sum for a spiked Wigner matrix converges to erfc �1 4 � − log (1 − λ) � (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='7) when H is Gaussian Orthogonal Ensemble (GOE), and for a spiked Wishart matrix erfc � 1 4 � − log � 1 − λ2 d0 �� (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='8) 2 when XXT is a Wishart Ensemble;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=', [47, 29, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Here, erfc(·) is the complementary error function defined as erfc(x) = � ∞ x e−t2dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='9) Though optimal, the LR test is not efficient, and it is desirable to construct a test that does not depend on information about the prior, which is typically not known in many practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' In [22], an optimal and universal test for spiked Wigner matrices was proposed, which is based on the linear spectral statistics (LSS) of the data matrix, a linear functional defined as LN(f) = N � i=1 f(µi) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='10) for a given function f, where µ1, · · · µN are the eigenvalues of the data matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The test is extended to spiked rectangular matrices in [34], where the singular values of the data matrix is used instead of the eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' If the noise is non-Gaussian, it is possible to improve the PCA by transforming the data matrix entrywise for spiked Wigner matrices [42, 50] and for spiked rectangular matrices [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' In this improved PCA, the threshold is lowered by a certain factor that depends on the Fisher information of the noise distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Below this threshold, the LSS-based test proposed in [22] for spiked Wigner matrices can also be improved by applying the entrywise transformation for the improved PCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' It is not known whether the reliable detection is impossible below the threshold except for the case of the spiked Wigner matrix with Rademacher prior [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Spiked random matrices with general rank: The more relevant structure for application is that the latent signal contains multiple spikes, or a spike with a higher rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' For such models of spiked random matrices, similar to the cases with rank-1 spikes, it is natural to ask the following questions: What is the spectral threshold for a reliable detection lower than the existing one for Gaussian noise if the noise is non-Gaussian?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Can we design an efficient algorithm to weakly detect the presence of signal (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=', better than a random guess) when a reliable detection is not feasible?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Contrary to the rank-1 spike case, the questions addressed above have never been answered, even for the simplest case of Gaussian noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Furthermore, for the spikes with general rank, we need to consider another important problem of finding the rank of the spike in case it is not known a priori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' While viable solutions to resolve the issue in the context of the community detection were suggested in [40, 16] for any spiked Wigner matrices and [49, 25] for spiked rectangular matrices, these methods are not applicable in the sub-critical case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' To the best of our knowledge, there are no spectral algorithms for estimating the rank of signal in the sub-critical regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We thus aim to the following question as well: Can we design an efficient algorithm to estimate the rank of signal when a reliable detection is not feasible?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Main contributions Our main contributions are mainly divided into three parts as follows: (Strong detection) We prove that the PCA can be improved by an entrywise transformation if the noise is non-Gaussian, under a mild assumption on the distribution (prior) of the spike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' 3 (Weak detection I) We propose a universal test to detect the presence of signal with low computational complexity, based on the linear spectral statistics (LSS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The test does not require any prior information on the signal, and if the noise is Gaussian the error of the proposed test is optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' For the spiked Wigner matrix or the additive model of the spiked rectangular matrix with the non-Gaussian noise, we suggest an improved test via an entrywise transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (Weak detection II) We present an LSS-based test for estimating the rank of a signal when Λ = λI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Heuristically, it is possible to increase the SNR via an entrywise transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Here, we illustrate the main idea of the entrywise transformation for the spiked Wigner matrix of the form M = UΛ1/2U T + W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' If |uiuT j |, |uivT j | ≪ Wij, then by applying a function q entrywise to √ NY , we obtain a transformed matrix whose entries are q( √ NMij) = q( √ NWij + √ NuiΛ1/2uT j ) ≈ q( √ NWij) + √ Nq′( √ NWij)uiΛ1/2uT j , where ui and vi denote i−th row vector of the signal matrix U and V , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' With negligible error, it is possible to approximate the coefficient q′( √ NWij) in the second term in the right side by its expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (See Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3 for the proof) Then, q( √ NMij) = q( √ NWij + √ NuiΛ1/2uT j ) ≈ √ N � q( √ NWij) √ N + E[q′( √ NWij)]uiΛ1/2uT j � and the transformed matrix is approximately of the form U(Λ′)1/2U T + Q after a proper normalization, which becomes another spiked Wigner matrix with different SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' By optimizing the transformation q, we find that the SNR is effectively increased (or equivalently, the threshold √d0 is lowered) in the PCA for the transformed matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The change of the threshold and a BBP-type transition for the largest eigenvalues of the transformed matrix can be rigorously proved;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3 for a precise statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We remark that the same idea works even if the SNR Λ is not a constant multiple of an identity matrix, and also a similar result holds for the additive model of spiked rectangular matrix (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' For the multiplicative model of the form Y = (I + UΛU T )1/2X =: (I + UΓU T )X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' with Λ = 2Γ + Γ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' the analysis is significantly more involved due to the following reason: Applying a function q entrywise to √ NY ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' we find that q( √ NYij) = q �√ NXij + √ N � ℓ uiΓuT ℓ Xℓj � ≈ q( √ NXij) + √ Nq′( √ NXij) � ℓ uiΓuT ℓ Xℓj ≈ √ N � q( √ NXij) √ N + E[q′( √ NXij)] � ℓ uiΓuT ℓ Xℓj � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' and the transformed matrix is of the form UΓ′U T X +Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' which is not a spiked rectangular matrix anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Note that Q depends on X entrywise and thus it cannot be considered as an additive model, either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' In Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5, we prove the effective change of the SNR and the BBP-type transition for the multi- plicative model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5 is based on a generalized version of the BBP transition that works with the matrix of the form UΓU T X +Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We remark that the strategy for the proof, based on recent development of random matrix theory, can also be applied to prove a BBP-type transition for other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' As in the rank-1 case in [34], it is notable that the optimal entrywise transform for the multiplicative model is different from the one for the additive model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' For the spiked Wigner matrix, the optimal transforms are given by −g′/g for the off-diagonal entries (and −g′ d/gd for the diagonal entries), where g (and gd for the diagonal entries) is the density functions of them;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' the optimal transform for the additive model is also 4 given by −g′/g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' However, for the multiplicative model, the optimal transform is a linear combination of the function −g′/g and the identity mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Heuristically, it is due to that the effective SNRs depend not only on Γ′ but also on the correlation between X and Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' the former is maximized when the transform is −g′/g while the latter is maximized when the transform is the identity mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We also remark that the effective SNRs after the optimal entrywise transform is larger in the additive model, which suggests that the detection problem is fundamentally harder for the multiplicative model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' With the BBP-type transition for the largest eigenvalues of the transformed matrices, it is also possible to improve the performance of several statistical inferences [13, 35, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' One of the consequences is that the corresponding eigenspace is adjacent to its true spike U in the sense of direction of arrival (DoA) [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' In other words, we can not only reliably estimate the number of spikes by parallel analysis (PA) [27], but also approximately recover the true spikes and the corresponding SNRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' For the subcritical case where it is impossible to reliably detect the signal by the improved PCA, we propose algorithms for weak detection, based on the central limit theorem (CLT) of the LSS, Theorems 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5, and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='6, analogous to the ones introduced in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' More precisely, assuming the SNRs are uniform i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=', Λ = λI, we propose an algorithm for a hypothesis test between Hk1 : k = k1, Hk2 : k = k2 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='11) for non-negative integers k1 < k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' While it may seem obvious, it has not been even known in the simple case k1 = 0 whether the detection becomes easier as k2 increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Our test in Algorithm 2 verifies the claim since the error of the proposed test is an increasing function of (k2 − k1) as in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' As in [22], the proposed tests are universal, and the various quantities in it can be estimated from the observed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The test can further be improved by applying the same entrywise transformation we used for the PCA (Algorithm 3) if the data matrix is of additive type (spiked Wigner matrix or rectangular matrix with spiked mean), and it also can be adapted to the rank detection problem where we need to estimate the rank k of the signal without knowing the candidates k1 and k2 a priori (Algorithm 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The main mathematical achievement of the second part is the CLT for the LSS of spiked random matrices with general ranks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' For a rank-1 spiked Wigner matrix, the CLT was first proved for a special spike 1 √ N (1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' , 1)T in [9] and later extended for a general rank-1 spike by comparison with the special case [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' However, the proof in [9] is not readily extended to the spiked Wigner matrices with higher ranks and the spiked rectangular matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' In this paper, we overcome the difficulty by introducing a direct interpolation between the spiked random matrix and the corresponding pure noise matrix and tracking the change of the LSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Furthermore, we will prove that the proposed entrywise transformation for the data matrix of additive type also effectively changes the SNR, and that the LSS of the transformed matrix is also asymptotically Gaussian;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' this result was proved previously only for rank-1 spiked Wigner matrices in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Thus, the error from the proposed test decreases after the transformation as for spiked Wigner matrices in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Related works Spiked random matrix models were first introduced by Johnstone [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The model can be applied to various problems such as community detection [1] and submatrix localization [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The transition of the largest eigenvalue was proved by Baik, Ben Arous, and P´ech´e [7] for spiked complex Wishart matrices and generalized by Benaych-Georges and Nadakuditi [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' For more results from random matrix theory about the extreme eigenvalues and the corresponding eigenvectors of spiked random matrices, we refer to [18] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The improved PCA based on the entrywise transformation was considered for rank-1 spiked Wigner matrices in [42, 50], where the transformation is chosen to maximize the effective SNR of the transformed 5 matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Detection problems for rank-1 spiked Wigner matrices were also considered, where the analysis is typically easier due to its symmetry and canonical connection with spin glass models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' For more results on the rank-1 spiked Wigner matrices, we refer to [44, 50, 29, 22] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The testing problem for rank-1 spiked Wishart matrices with the spherical prior was considered by Onatski, Moreira, and Hallin [47, 48], where they proved the optimal error of the hypothesis test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' It is later extended to the case where the entries of the spikes are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' with bounded support (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' prior) by El Alaoui and Jordan [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' See also [32, 44, 24, 41, 43, 12] for more about detection limits in statistical learning theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Models with sparse or generative structure of the spike have extensively studied in the past literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Various statistical and algorithmic methods are applicable to the case where SNR is smaller than the spectral threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' In particular, it can be seen that the sparsity of the spikes and the dimension of the latent vector constituting the generative spike prior actually serve to lower the threshold for the SNR to which several algorithms are applicable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' see [5, 20] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Organization of the paper The rest of the paper is organized as follows: In Section 2, we introduce the precise definitions of models and relevant previous consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' In Section 3, we state our results on the improved PCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' In Section 4, we propose algorithms for LSS-based tests and a test for rank estimation, and analyze their performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' In Section 5, we state general results on the CLT for the LSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We conclude the paper in Section 6 with the summary of our works and future research directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' In Appendix A, we consider examples of spiked random matrices and provide results from numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' In Appendices B and C, we provide technical details of the proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' 2 Preliminaries In this section, we introduce the precise definition of the models and previous results for the spiked random matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1 Definitions of models The noise matrices are defined as follows: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1 (Wigner matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' An N ×N symmetric matrix W = (Wij) is a (real) Wigner matrix if Wij (i, j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' , N) are independent real random variables such that For all i < j, NE[W 2 ij] = 1, N 3 2 E[W 3 ij] = w3, and N 2E[W 4 ij] = w4 for some w3, w4 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' For all i, NE[W 2 ii] = w2 for some constant w2 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' For any positive integer p, there exists Cp, independent of N, such that N p 2 E[W p ij] ≤ Cp for all i ≤ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2 (Random rectangular matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' An M × N matrix X = (Xij) is a (real) random rectangular matrix if Xij (1 ≤ i ≤ M, 1 ≤ j ≤ N) are independent real random variables such that For all i, j, E[Xij] = 0, NE[X2 ij] = 1, N 3 2 E[X3 ij] = w3, and N 2E[X4 ij] = w4 for some constants w3, w4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' For any positive integer p, there exists Cp, independent of N, such that N p 2 E[Xp ij] ≤ Cp for all i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The spiked random matrices are defined as follows: 6 Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3 (Spiked Wigner matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' An N × N matrix M = UΛ1/2U T + W is a spiked Wigner matrix with the SNR (matrix) Λ if W is a Wigner matrix and the spike U = [u(1), u(2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' , u(k)] ∈ RN×k with U T U = Ik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4 (Spiked rectangular matrix - additive model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' An M ×N random matrix Y = UΛ1/2V T +X is a rectangular matrix with spiked mean U, V and the SNR (matrix) Λ if X is a random rectangular matrix and the spikes U = [u(1), u(2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' , u(k)] ∈ RM×k, V = [v(1), v(2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' , v(k)] ∈ RN×k with U T U = V T V = Ik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5 (Spiked rectangular matrix - multiplicative model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' An M × N random matrix Y = (I + UΛU T )1/2X is a rectangular matrix with spiked covariance U and the SNR (matrix) Λ if X is a rectangular matrix and U = [u(1), u(2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' , u(k)] ∈ RM×k with U T U = Ik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We assume throughout the paper that the SNR matrix Λ is a k × k diagonal matrices with Λii = λi and λ1 ≥ λ2 ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' λk ≥ 0, and M N → d0 ∈ (0, ∞) as M, N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2 Principal component analysis Here are the results for principal components of spiked models in the context of random matrix theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Spiked Wigner matrix Let M be the spiked Wigner matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The empirical spectral measure of M converges to the Wigner’s semicircle law µsc, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=', if we denote by µ1 ≥ µ2 ≥ · · · ≥ µN the eigenvalues of M, then 1 N N � i=1 δµi(x)dx → dµsc(x) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1) weakly in probability as N → ∞, where dµsc(x) = √ 4 − x2 2π 1(−2,2)(x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2) The k largest eigenvalue has the following (almost sure) limit: for 1 ≤ i ≤ k If λi > 1, then µi → √λi + 1 √λi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' If λi < 1, then µi → 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Sample covariance matrix Let S = Y Y T be the sample covariance matrix (Gram matrix) derived from a spiked rectangular matrix Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The empirical spectral measure of S converges to the Marchenko–Pastur law µMP , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=', if we denote by µ1 ≥ µ2 ≥ · · · ≥ µM the eigenvalues of S, then 1 M M � i=1 δµi(x)dx → dµMP (x) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3) weakly in probability as M, N → ∞, where for M ≤ N dµMP (x) = � (x − d−)(d+ − x) 2πd0x 1(d−,d+)(x)dx, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4) 7 with d± = (1 ± √d0)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The k largest eigenvalue has the following (almost sure) limit: for 1 ≤ i ≤ k If λi > √d0, then µi → (1 + λi)(1 + d0 λi ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' If λi < √d0, then µi → d+ = (1 + √d0)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' This in particular shows that the detection can be reliably done by PCA if λ > √d0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We remark that the results above hold for both the additive model and the multiplicative model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3 Linear spectral statistics We introduce the central limit theorems for null models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Spiked Wigner matrix The proof of the Gaussian convergence of the LR in [8, 10] is based on the recent study of linear spectral statistics, defined as LY (f) = N � i=1 f(µi) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5) for a function f, where µ1 ≥ µ2 ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' µN are the eigenvalues of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' As the Wigner’s semicircle law in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1) suggests, it is required to consider the fluctuation of the LSS about N � 2 −2 f(x) dµsc(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The CLT for the LSS is the statement � LM(f) − N � 2 −2 f(x) dµsc(x) � ⇒ N(mM(f), VM(f)), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='6) where the right-hand side is the Gaussian random variable with the mean mM(f) and the variance VM(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The CLT was proved for the null case (λ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We will show that the CLT also holds under the alternative and the mean mM(f) depends on λ while the variance VM(f) does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Spiked rectangular matrices The LSS for the spiked rectangular matrices defined as LY (f) = M � i=1 f(µi) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='7) for a function f, where µ1 ≥ µ2 ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' µM are the eigenvalues of S = Y Y T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' As the Marchenko–Pastur law in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3) suggests, it is required to consider the fluctuation of the LSS about M � d+ d− f(x) dµMP (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The CLT for the LSS is the statement � LY (f) − M � d+ d− f(x) dµMP (x) � ⇒ N(mY (f), VY (f)), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='8) 8 where the right-hand side is the Gaussian random variable with the mean mY (f) and the variance VY (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The CLT was proved for the null case (λ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We will show that the CLT also holds under the alternative and the mean mY (f) depends on λ while the variance VY (f) does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' 3 Main result I - Improved PCA In this section, we state our first main results on the improvement of PCA by entrywise transformations and provide the results from numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1 Improved PCA We introduce the following assumptions for the spike and the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' For the spike U (and also V in the additive model), we assume, for φ ≤ 1/2, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' the spikes are φ-localized with high probability, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' ∥U∥∞, ∥V ∥∞ ≺ N −φ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' the spike matrix is φ-orthonormal with high probability, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' ∥U T U − Ik∥F , ∥V T V − Ik∥F ≺ N −φ, and so the spikes are sampled from Stiefel manifold of orthonormal k-frames in RM or RN with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' For the noise, let P be the distribution of the normalized entries √ NWij(i ̸= j) in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1 and √ NXij in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Further, for the spiked Wigner matrices, let Pd be the distribution of the normalized diagonal entries √ NWii in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We assume the following: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The density functions g and gd of P and Pd, respectively, are smooth, positive everywhere, and sym- metric (about 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' For any fixed (N-independent) D, the D-th moments of P and Pd are finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The functions h = −g′/g, hd = −g′ d/gd and their all derivatives are polynomially bounded in the sense that |h(ℓ)(w)|, |h(ℓ) d (w)| ≤ Cℓ|w|Cℓ for some constant Cℓ depending only on ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The first condition on the prior implies that the spike is not necessarily delocalized, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=', some entries of the signal can be significantly larger than N −1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The key examples of the prior are as follows: Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We can consider the following examples of the spike prior: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' the spherical prior, where u(ℓ) (and v(ℓ)) are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' drawn uniformly from the unit sphere, or 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' prior, where the entries u1(ℓ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' , uM(ℓ) (respectively, v1(ℓ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' , vN(ℓ)) are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' random variables from the probability measures µℓ (respectively, νℓ) with mean zero and variance M −1 (respec- tively N −1) such that for any integer p > 2 E|ui(ℓ)|p, E|vj(ℓ)|p ≤ Cp M 1+(p−2)φ for some (N-independent) constants Cp > 0 and φ ≤ 1 2, uniformly on i, j and ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We remark that for the spike Wigner matrices, due to normalization, the variance of the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' prior µℓ for ui(ℓ) is N −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' 9 Spiked Wigner matrix Given a spiked Wigner matrix M, we consider a family of the entrywise transformations hα(x) = −g′(x) g(x) + αx, hd(x) = −g′ d(x)/gd(x) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1) for α ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We also consider the transformed matrix � M whose entries are � Mij = 1 � FgN h0( √ NMij)(i ̸= j), � Mii = � w2 Fg,dN hd �� N w2 Mii � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2) where the Fisher information Fg and Fg,d of g and gd are given by Fg = � ∞ −∞ (g′(x))2 g(x) dx, Fg,d = � ∞ −∞ g′ d(x)2 gd(x) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Note that Fg ≥ 1 where the equality holds only if g is the standard Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Then following theorem asserts that the effective SNRs of the transformed matrix for PCA are λℓFg, which generalizes Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='8 in [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Let M be a spiked Wigner matrix in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3 satisfying Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1 with φ > 1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Let � M be the transformed matrix obtained as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2) and (�µℓ, �u(ℓ)) the pair of ℓ-th largest eigenvalue and the corresponding eigenvector of � M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Then, almost surely, for 1 ≤ ℓ ≤ k If λℓ > 1 Fg , then �µℓ → � λℓFg + 1 √ λℓFg and |�u(ℓ)T u(ℓ)|2 → 1 − 1 λℓFg , If λℓ < 1 Fg , then �µℓ → 2 and |�u(ℓ)T u(ℓ)|2 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' For the proof, we adapt the strategy in [50], where the key observation is that the transformed matrix is approximately equal to another spiked Winger matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' See Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2 for the detail of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We remark that h0 is the optimal (up to constant factor) among all entrywise transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' See Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1 for the proof of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Spiked rectangular matrices For a spiked rectangular matrix Y , we consider the family of the entrywise transformations hα(x) defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1) and transformed matrices �Y (α) whose entries are �Y (α) ij = 1 � (α2 + 2α + Fg)N hα( √ NYij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3) Note that For the additive model, we again show that the effective SNRs of the transformed matrix for PCA are {λℓFg}ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Let Y be a spiked rectangular matrix in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4 satisfying Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1 with φ > 1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Let �Y ≡ �Y (0) be the transformed matrix obtained as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3) with α = 0 and (�µℓ, �u(ℓ)) the pair of ℓ-th largest eigenvalue and the corresponding eigenvector of �Y �Y T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Then, almost surely, for 1 ≤ ℓ ≤ k If λℓ > √d0 Fg , then �µℓ → (1 + λℓFg)(1 + d0 λℓFg ) and |�u(ℓ)T u(ℓ)|2 → 1 − d0(1+λℓFg) λℓFg(λℓFg+d0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' If λℓ < √d0 Fg , then �µℓ → d+ = (1 + √d0)2 and |�u(ℓ)T u(ℓ)|2 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' 10 From Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4, if λℓ > √d0 Fg , we immediately see that the signal in the additive model can be reliably detected by the transformed PCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Thus, the detection threshold in the PCA is lowered when the noise is non-Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We also remark that h0 is the optimal entrywise transformation (up to constant factor) as in the Wigner case;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' see Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' For the proof, we adapt the strategy in [34], where the key observation is again that the transformed matrix is approximately equal to another spiked rectangular matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' See Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3 for the detail of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' For the multiplicative model, we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Let Y be a spiked rectangular matrix in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5 satisfying Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1 with φ > 1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Let �Y ≡ �Y (αg,ℓ) be the transformed matrix obtained as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3) with αg,ℓ := −γℓFg + � 4Fg + 4γℓFg + γ2 ℓ F 2g 2(1 + γℓ) and (�µℓ, �u(ℓ)) the pair of ℓ-th largest eigenvalue and the corresponding eigenvector of �Y �Y T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Then, almost surely, If (λg)ℓ > √d0, then �µℓ → (1 + (λg)ℓ)(1 + d0 (λg)ℓ ) and |�u(ℓ)T u(ℓ)|2 → 1 − (λg)ℓ + d0 (λg)ℓ · ((λg)ℓ + 1), If (λg)ℓ < √d0, then �µℓ → d+ = (1 + √d0)2 and |�u(ℓ)T u(ℓ)|2 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' where (λg)ℓ := γℓ + γ2 ℓ Fg 2 + γℓ � 4Fg + 4γℓFg + γ2 ℓ F 2g 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Note that (λg)ℓ ≥ γℓ + γ2 ℓ Fg 2 + γℓ � 4 + 4γℓFg + γ2 ℓ F 2g 2 = 2γℓ + γ2 ℓ Fg ≥ 2γℓ + γ2 ℓ = λℓ, and the inequality is strict if Fg > 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=', g is not Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Note that unlike the additive model, we cannot determine αg without prior knowledge on the SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Nevertheless, we can apply the transformation h√ Fg or h0, which effectively increase all SNRs simultaneously;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' see Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' From Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5, if (λg)ℓ > √d0, the signal can be reliably detected by the transformed PCA and the detection threshold in the PCA is lowered if the noise is non-Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We also remark that hαg,ℓ is the optimal entrywise transformation (up to constant factor) for the ℓ-th largest eigenvalue;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' see Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We finish this section with an outline of the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We begin by justifying that the transformed matrix �Y is approximately of the form (Q+U�Γ 1 2 U T X), where �Γ = diag(�γ1, · · · , �γk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Then, the largest eigenvalue of �Y �Y T can be approximated by the largest eigenvalue of (Q+U�Γ 1 2 U T X)T (Q+U�Γ 1 2 U T X) for which we consider an identity (Q + U�Γ 1 2 U T X)T (Q + U�Γ 1 2 U T X) − zI = (QT Q − zI)(I + L(z)), 11 where L(z) = G(z)(XT U�Γ 1 2 U T Q + QT U�Γ 1 2 U T X + XT U�ΓU T X), G(z) = (QT Q − zI)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' If z is an eigenvalue of (Q + U�Γ 1 2 U T X)T (Q + U�Γ 1 2 U T X) but not of QT Q, the determinant of (I + L(z)) must be 0 and hence −1 is an eigenvalue of L(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Since the rank of L(z) is at most 2k, we can find that the eigenvector of L(z) is a linear combination of vectors G(z)QT u(ℓ) and G(z)XT u(ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Further, by using the facts in Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2, we can observe that a linear combination of vectors G(z)QT u(ℓ) and G(z)XT u(ℓ) be a possible candidate for the ℓ-th eigenvector of L(z), and so of �Y T �Y i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=', for some aℓ, bℓ, L(z)(aℓG(z)QT u(ℓ) + bℓG(z)XT u(ℓ)) = −(aℓG(z)QT u(ℓ) + bℓG(z)XT u(ℓ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4) From the definition of L(z), L(z) · G(z)XT U = G(z)XT U�Γ 1 2 (U T QG(z)XT U) + G(z)QT U�Γ 1 2 (U T XG(z)XT U) + G(z)XT U�Γ(U T XG(z)XT U), and a similar equation holds for L(z) · G(z)QT U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' It suggests that if U T QG(z)XT U and U T XG(z)XT U are concentrated around diagonal matrices where the entries are deterministic functions of z, then the left side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4) can be well-approximated by a (deterministic) linear combination of G(z)QT u(ℓ) and G(z)XT u(ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We can then find the location of the largest eigenvalue in terms of a deterministic function of z and conclude the proof by optimizing the function q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The concentration of random matrices U T QG(z)XT U and U T XG(z)XT U is the biggest technical chal- lenge in the proof, mainly due to the dependence between the matrices Q and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We prove it by applying the technique of linearization in conjunction with resolvent identities and also several recent results from random matrix theory, most notably the local Marchenko–Pastur law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Once we find out the coefficients aℓ and bℓ in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4), the eigenvector localization is an easy corollary since the vector aℓG(z)QT u(ℓ) + bℓG(z)XT u(ℓ) must be a right singular vector of �Y with the corresponding singular value � (1 + (λg)ℓ)(1 + d0 (λg)ℓ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' In this paper, we will not go into further detail on this part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The detailed proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5 can be found in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' 4 Main Result II - Weak Detection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1 Signal detection in rank-1 spiked models We begin by recalling the LSS-based detection algorithms for rank-1 spiked rectangular matrices in [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Suppose that our goal is to detect the presence of the signal by the hypothesis test between H0 : λ = 0 and H1 : λ = ω where the SNR ω for the alternative hypothesis H1 is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The key observation is that the variances of the limiting Gaussian distributions of the LSS in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='7) do not depend on the SNR while the means do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' If we denote by VY (f) the common variance, and mY (f)|H0 and mY (f)|H1 the means, respectively, our goal is to find a function that maximizes the relative difference between the limiting distributions of the LSS under H0 and under H1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=', ����� mY (f)|H1 − mY (f)|H0 � VY (f) ����� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1) 12 Algorithm 1 Hypothesis test for a rank-1 spiked rectangular matrix Input: data Yij, parameters w4, ω Lω ← test statistic in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3) mω ← critical value in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='7) if Lω ≤ mω then Accept H0 else Reject H0 end if As we will see in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5, the optimal function f is of the form C1φω + C2 for some constants C1 and C2, where φω(x) = ω d0 � 2 w4 − 1 − 1 � x − log �� 1 + d0 ω � (1 + ω) − x � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2) The test statistic we use is thus defined as Lω = M � i=1 φω(µi) − M � d+ d− φω(x) dµMP (x) = − log det �� 1 + d0 ω � (1 + ω)I − Y Y T � + ω d0 � 2 w4 − 1 − 1 � (Tr Y Y T − M) + M � ω d0 − log � ω d0 � − 1 − d0 d0 log(1 + ω) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3) Theorem 8 in [34] asserts that Lω converges to a Gaussian, Lω ⇒ N(m(λ), V0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4) Here, the mean of the limiting Gaussian distribution is given by m(λ) = −1 2 log � 1 − ω2 d0 � + ω2 2d0 (w4 − 3) − log � 1 − λ2 d0 � + λ2 d0 � 2 w4 − 1 − 1 � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5) with λ = 0 under H0 and λ = ω under H1, and the variance V0 = −2 log � 1 − ω2 d0 � + 2ω2 d0 � 2 w4 − 1 − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='6) Based on the asymptotic normality of Lω, we can construct a test in which we compute the test statistic Lω and compare it with the average of m(0) and m(ω), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=', mω := m(0) + m(ω) 2 = − log � 1 − ω2 d0 � + ω2 2d0 � 2 w4 − 1 + w4 − 4 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='7) See Algorithm 1 for the detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The limiting error of the proposed test, Algorithm 1, is given by err(ω) = P(Lω > mω|H0) + P(Lω ≤ mω|H1) → erfc �√V0 4 √ 2 � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='8) 13 where V0 is the variance in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='6) and erfc(·) is the complementary error function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' If the noise X is Gaussian, w4 = 3 and the limiting error in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='8) is erfc �√V0 4 √ 2 � = erfc � 1 4 � − log � 1 − ω2 d0 �� , and it coincides with the error of the LR test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2 of [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' It shows that our test is optimal with the Gaussian noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2 Signal detection in rank-k spiked models When the rank of the spike is larger than 1, we first consider a simple case where the data is given as a spiked Wigner matrix and our goal is to construct an LSS-based algorithm for a hypothesis test between H0 : Λ = 0 and Hk : Λ = ωIk, where the rank k of the spike for the alternative hypothesis is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Our starting point is the following test statistic, which was considered for the rank-1 spiked Wigner matrix in [22]: Lω = − log det � (1 + ω)I − √ωM � + ωN 2 + √ω � 2 w2 − 1 � Tr M + ω � 1 w4 − 1 − 1 2 � (Tr M 2 − N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='9) If there is no signal present, Lω ⇒ N(m0, V0), where m0 = −1 2 log(1 − ω) + �w2 − 1 w4 − 1 − 1 2 � ω + (w4 − 3)ω2 4 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='10) V0 = −2 log(1 − ω) + � 4 w2 − 2 � ω + � 2 w4 − 1 − 1 � ω2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='11) For a rank-k spiked Wigner matrix, we can consider the same Lω as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='9) and prove that it also converges to a Gaussian with the same variance V0 but an altered mean mk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The following is the precise statement for the limiting distribution of Lω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Let M be a rank-k spiked Wigner matrix with a spike U as in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3 with Λ = ωIk for some nonnegative integer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Then, Lω ⇒ N(mk, V0) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='12) where the variance V0 is as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='11) and the mean mk is given by mk = m0 + k � − log(1 − ω) + � 2 w2 − 1 � ω + � 1 w4 − 1 − 1 2 � ω2 � = m0 + kV0 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='13) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1 directly follows from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2 in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Since the mean of Lω depends on the rank of the spike, we can construct a hypothesis test between Hk1 and Hk2 in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='11) based on Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' In this test, for a given spiked Wigner matrix M, we compute Lω and compare it with the critical value m(k1+k2)/2, m(k1+k2)/2 := mk1 + mk2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='14) 14 Algorithm 2 Hypothesis test for a spiked Wigner matrix Data: Mij, parameters w2, w4, λ Lω ← test statistic in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='9), m(k1+k2)/2 ← critical value in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='14) with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='13) if Lω ≤ m(k1+k2)/2 then Accept H1 else Accept H2 end if See Algorithm 2 for the detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' In Theorems 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5, we prove that the proposed test in Algorithm 2 is optimal among all CLT-based tests, in the sense that the error is minimized with the test statistic Lω also for spiked random matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The error of the test, err(ω) = P(Lω > mω|H0)+P(Lω ≤ mω|H1), in algorithm 2 converges to erfc � k2 − k1 4 � V0 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2 is a direct consequence of Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (See also Section 3 of [29] and the proof of Theorem 2 of [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=') Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' When w4 = 3, we find that the error err(ω) converges to erfc � k2 − k1 4 � − log(1 − ω) + � 2 w2 − 1 � ω � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='15) The optimal error for the weak detection, achieved by the LR test, coincides with the limiting error in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='15) when the noise is Gaussian and the SNR ω is sufficiently small;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' see [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Thus, our proposed test is optimal in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The test in Algorithm 2 can be readily extended to the spiked rectangular matrices by replacing the test statistic in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='9) with the following one, which was introduced in [34] for the rank-1 spiked rectangular matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Lω = − log det �� 1 + d0 ω � (1 + ω)I − Y Y T � + ω d0 � 2 w4 − 1 − 1 � (Tr Y Y T − M) + M � ω d0 − log � ω d0 � − 1 − d0 d0 log(1 + ω) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='16) We have the following results for the asymptotic normality of Gaussian fluctuation of Lω: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Let Y be a spiked rectangular matrix in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4 or 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5 with Λ = ωIk for some nonnegative integer k and λ ∈ (0, √d0) and w4 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Then, for any spikes with U T U = V T V = Ik, Lω ⇒ N(mk, V0), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='17) where the mean and the variance are given by mk = m0 + k � − log � 1 − ω2 d0 � + ω2 d0 � 2 w4 − 1 − 1 �� (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='18) 15 and V0 = −2 log � 1 − ω2 d0 � + 2ω2 d0 � 2 w4 − 1 − 1 � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='19) where m0 = −1 2 log � 1 − ω2 d0 � + ω2 2d0 (w4 − 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='20) Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4 directly follows from the general CLT result in Theorems 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' See Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4 for the detailed computation for the mean and the variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' With Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4, we find that Algorithm 2 is available for the weak detection of the signal in the spiked rectangular matrices with the following change: Data matrix is Yij (instead of Mij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Test statistic Lω is defined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='16) (instead of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='9)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Critical value m(k1+k2)/2 is obtained by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='14) with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='20) (instead of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='13)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The limiting error of the test in this case is again erfc � k2−k1 4 � V0 2 � as in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2, where V0 is defined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3 Test with entrywise transformation for spiked matrices of additive type The entrywise transform we applied with the PCA in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1 can also be adapted to be used together with the proposed test in Algorithm 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' see also [22] where the same idea was applied for the rank-1 spiked Wigner matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Recall the transformation defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1) and the transformed matrix � M in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We consider a test statistic �Lω := − log det � (1 + ωFg)I − � ωFg � M � + ωFg 2 N + √ω � 2 � Fg,d w2 − � Fg � Tr � M + λ � GH � w4 − 1 − Fg 2 � (Tr � M 2 − N), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='21) where GH = 1 2Fg � ∞ −∞ g′(w)2g′′(w) g(w)2 dw, � w4 = 1 (Fg)2 � ∞ −∞ (g′(w))4 (g(w))3 dw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We then have the following CLT result for �Lω that generalizes the results in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Assume the conditions in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1, satisfying Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1 with φ > 3/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' If λFg < 1, �Lω ⇒ N( �mk, �V0), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='22) where the mean and the variance are given by �mk = −1 2 log(1 − ωFg) + �(w2 − 1)GH �w4 − 1 − Fg 2 � ω + �w4 − 3 4 (ωFg)2 + k � − log(1 − ωFg) + �2Fg,d w2 − Fg � ω + � (GH)2 �w4 − 1 − (Fg)2 2 � ω2 � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='23) �V0 = −2 log(1 − ωFg) + �4Fg,d w2 − 2Fg � ω + �2(GH)2 �w4 − 1 − (Fg)2 � ω2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='24) 16 Algorithm 3 Hypothesis test for a spiked Wigner matrix with entrywise transformation Data: Mij, parameters w2, w4, λ, densities g, gd � M ← transformed matrix in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2), �Lω ← test statistic in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='21), �m(k1+k2)/2 ← critical value in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='25) with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='23) if �Lω ≤ �m(k1+k2)/2 then Accept H1 else Accept H2 end if Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5 directly follows from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3 in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Based on Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5, we can adapt the test in Algorithm 2 to construct a test that utilizes the entrywise transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' In this test, we compute �LΛ and compare it with the critical value �m(k1+k2)/2 := ( �mk1 + �mk2)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='25) See Algorithm 3 for the detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The limiting error of the test is given as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The error of the test in Algorithm 3 converges to erfc � �k2 − k1 4 � �V0 2 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='6 is a direct consequence of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We also propose an analogous test can for the additive model of the spiked rectangular matrices as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Recall the transformed matrix �Y ≡ �Y (0) in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Define the test statistic �Lω by �Lω = − log det �� 1 + d0 ωFg � (1 + ωFg)I − �Y �Y T � + 2ω d0 � GH �w4 − 1 − Fg 2 � (Tr �Y �Y T − M) + M �ωFg d0 − log �ωFg d0 � − 1 − d0 d0 log(1 + ωFg) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='26) We then have the following CLT for the test statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Assume the conditions in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4, satisfying Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1 with φ > 3/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' If λ < √d0/Fg, �Lω ⇒ N( �mk, �V0), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='27) where the mean and the variance are given by �m0 = −1 2 log � 1 − ω2(Fg)2 d0 � + ω2(Fg)2 2d0 ( �w4 − 3) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='28) �mk = �m0 + k � − log � 1 − ω2(Fg)2 d0 � + 2ω2 d0 � (GH)2 �w4 − 1 − (Fg)2 2 �� (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='29) and �V0 = 4ω2 d0 � (GH)2 �w4 − 1 − (Fg)2 2 � − 2 log � 1 − ω2(Fg)2 d0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='30) 17 With Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='7, we can adjust Algorithm 2 for the weak detection of the signal in the additive model of spiked rectangular matrices, where we make the following change: Data matrix is Yij (instead of Mij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Transformed matrix is �Y (instead of � M), defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3) with α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Test statistic �Lω is defined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='26) (instead of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='21)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Critical value m(k1+k2)/2 is obtained by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='25) with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='29) (instead of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='23)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' In Appendix A, we consider several examples of spiked Wigner matrices and spiked rectangular matrices, where we compare the errors from numerical simulations and the theoretical errors of the proposed algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We find that the numerical errors of the proposed tests closely match the corresponding theoretical errors and the error from Algorithm 3 is lower than that of Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4 Rank estimation The test in Algorithm 2 requires prior knowledge about k1 and k2, the possible ranks of the planted spike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' In this section, we adapt the idea of the proposed tests in Algorithm 2 to estimate the rank of the signal when there is no prior information on the rank k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Recall that the test statistic Lω defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='9) does not depend on the rank of the matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' As proved in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1, the test statistic Lω converges to a Gaussian random variable with mean mk and the variance V0, where mk is equi-distributed with respect to k and V0 does not depend on k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' It is then natural to set the best candidate for k, which we call κ, be the minimizer of the distance |Lω − mk|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' This procedure is equivalent to find the nearest nonnegative integer of the value κ′ := 2(Lω − m0) V0 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='31) rounding half down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We describe the procedure in Algorithm 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' for example, its probability of error for spiked Wigner matrix converges to P(k = 0) · P � Z > √V0 4 � + ∞ � i=1 P(k = i) · P � |Z| > √V0 4 � = � 1 − P(k = 0) 2 � erfc � 1 4 � V0 2 � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='32) where Z is a standard Gaussian random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Note that it depends only on P(k = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The error can be lowered if the range of k is known a priori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' See Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' It is also possible to improve Algorithm 4 by pre-transforming the data matrix entrywise as in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We omit the detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' 5 Central Limit Theorems In this section, we collect our results on general CLTs for the LSS of spiked random matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' To precisely define the statements, we introduce the Chebyshev polynomials of the first kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1 (Chebyshev polynomial).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The n-th Chebyshev polynomial (of the first kind) Tn is a degree n polynomial defined by T0(x) = 1, T1(x) = x, and Tn+1(x) = 2xTn(x) − Tn−1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' 18 Algorithm 4 Rank estimation Data: Mij (or Yij), parameters w2, w4, λ Lω ← test statistic in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='9) or (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='16), m0 ← mean in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='10) or (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='20), m1 ← mean in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='13) or (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='18) with k = 1 κ′ ← value in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='31) if Lω ≤ (m0 + m1)/2 then Set κ = 0 else Set κ = ⌈κ′ − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5⌉ end if We first state a CLT for the LSS of spiked Wigner matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Recall that we denote by µ1 ≥ µ2 ≥ · · · ≥ µN the eigenvalues of a spiked Wigner matrix M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Assume the conditions in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Suppose that a function f is analytic on an open interval containing [−2, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Then, � N � i=1 f(µi) − N � 2 −2 √ 4 − z2 2π f(z) dz � ⇒ N (mk(f), V0(f)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The mean and the variance of the limiting Gaussian distribution are given by mk(f) = 1 4 (f(2) + f(−2)) − 1 2τ0(f) + (w2 − 2)τ2(f) + (w4 − 3)τ4(f) + k ∞ � ℓ=1 √ ωℓτℓ(f), V0(f) = (w2 − 2)τ1(f)2 + 2(w4 − 3)τ2(f)2 + 2 ∞ � ℓ=1 ℓτℓ(f)2 , where we let τℓ(f) = 1 π � 2 −2 Tℓ �x 2 � f(x) √ 4 − x2 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Furthermore, for mk, m0, and V0 defined in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1, ����� mk(f) − m0(f) � V0(f) ����� ≤ ���� mk − m0 √V0 ���� The equality holds if and only if f(x) = C1φω(x) + C2 for some constants C1 and C2 where φω(x) := log � 1 1 − √ωx + ω � + √ω � 2 w2 − 1 � x + ω � 1 w4 − 1 − 1 2 � x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We will give a proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2 in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' With the entrywise transformation in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3, we have the following changes in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Recall that �µ1 ≥ �µ2 ≥ · · · ≥ �µN are the eigenvalues of the transformed matrix � M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Assume the conditions in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2, satisfying Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1 with φ > 3/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' If λFg < 1, � N � i=1 f(�µi) − N � 2 −2 √ 4 − z2 2π f(z) dz � ⇒ N( �mk(f), �V0(f)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' 19 The mean and the variance of the limiting Gaussian distribution are given by �mk(f) = 1 4 (f(2) + f(−2)) − 1 2τ0(f) + k � ωFg,dτ1(f) + (w2 − 2 + kωGH)τ2(f) + (� w4 − 3)τ4(f) + k ∞ � ℓ=3 � (ωFg)ℓτℓ(f), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1) �V0(f) = (w2 − 2)τ1(f)2 + 2(� w4 − 3)τ2(f)2 + 2 ∞ � ℓ=1 ℓτℓ(f)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Furthermore, for �mk, �m0, and �V0 defined in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1, ������ �mk2(f) − �mk1(f) � �V0(f) ������ ≤ ������ �mk2 − �mk1 � �V0 ������ The equality holds if and only if f(x) = C1 �φω(x) + C2 for some constants C1 and C2 with the function �φω(x) := log � 1 1 − � ωFgx + ωFg � + � 2 � Fg,d w2 − � Fg � x + ω � GH �w4 − 1 − Fg 2 � x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We will also prove Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3 in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' For a general case where the spike Λ = diag(ω1, · · · , ωk) with possibly distinct ωi’s, we can prove the CLT and the transformed CLT, analogous to Theorems 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3, respectively, where the means of the limiting Gaussians are given by mM(f) = 1 4 (f(2) + f(−2)) − 1 2τ0(f) + (w2 − 2)τ2(f) + (w4 − 3)τ4(f) + k � s=1 ∞ � ℓ=1 � ωℓsτℓ(f), �mM(f) = 1 4 (f(2) + f(−2)) − 1 2τ0(f) + (w2 − 2)τ2(f) + (� w4 − 3)τ4(f) + k � s=1 � ωsFg,dτ1(f) + ωsGHτ2(f) + k � s=1 ∞ � ℓ=3 � (ωsFg)ℓτℓ(f), and the variances are equal to V0(f) in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2 and �V0(f) in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Adapting the proposed tests in Algorithms 2 and 3, it is possible to construct hypothesis tests for the weak detection in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The next result is the CLT for the LSS of spiked rectangular matrices Y , where we denote by µ1 ≥ µ2 ≥ · · ≥ µM the eigenvalues of Y Y T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Assume the conditions in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Suppose that a function f is analytic on an open set containing an interval [d−, d+].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Then, � M � i=1 f(µi) − M � d+ d− � (x − d−)(d+ − x) 2πd0x f(x) dx � ⇒ N(mk(f), V0(f)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2) 20 The mean and the variance of the limiting Gaussian distribution are given by mk(f) = �f(2) + �f(−2) 4 − τ0( �f) 2 + (w4 − 3)τ2( �f) + k ∞ � ℓ=1 � ω √d0 �ℓ τℓ( �f) and V0(f) = 2 ∞ � ℓ=1 ℓτℓ( �f)2 + (w4 − 3)τ1( �f)2, where we let �f(x) = f(√d0x + 1 + d0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Furthermore, for mk, m0, and V0 defined in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4, ����� mk2(f) − mk1(f) � V0(f) ����� ≤ ���� mk2 − mk1 √V0 ���� The equality holds if and only if f(x) = C1φω(x) + C2 for some constants C1 and C2 with the function φω(x) = ω d0 � 2 w4 − 1 − 1 � x − log �� 1 + d0 ω � (1 + ω) − x � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Lastly, we state the pre-transformed CLT for the LSS of the additive model of spiked rectangular matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We let �Y be the transformed matrix and �µ1 ≥ �µ2 ≥ · · · ≥ �µN the eigenvalues of �Y �Y T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Assume the conditions in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5, satisfying Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1 with φ > 3/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' If λ < √d0/Fg, � M � i=1 f(�µi) − M � d+ d− f(x)ρMP,d0(dx) � ⇒ N( �mk(f), �V0(f)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3) The mean and the variance of the limiting Gaussian distribution are given by �mk(f) = �f(2) + �f(−2) 4 − 1 2τ0( �f) + kω √d0 (GH − Fg)τ1( �f) + (� w4 − 3)τ2( �f) + k ∞ � ℓ=1 �ωFg √d0 �ℓ τℓ( �f) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4) and �V0(f) = 2 ∞ � ℓ=1 ℓτℓ( �f)2 + (� w4 − 3)τ1( �f)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5) where �f(x) = f(√d0x + 1 + d0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Furthermore, for �mk, �m0, and �V0 defined in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='7, The equality holds if and only if f(x) = C1 �φω(x) + C2 for some constants C1 and C2 with the function �φω(x) = 2λ d0 � GH �w4 − 1 − Fg 2 � x − log �� d0 ωFg + 1 � (ωFg + 1) − x � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' As in Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4, for a general case with Λ = diag(ω1, · · · , ωk), the CLT and the transformed 21 CLT hold with the adjusted means mY (f) = �f(2) + �f(−2) 4 + τ0( �f) 2 + (w4 − 3)τ2( �f) + k � s=1 ∞ � ℓ=1 � ωs √d0 �ℓ τℓ( �f), �mY (f) = �f(2) + �f(−2) 4 − 1 2τ0( �f) + (� w4 − 3)τ2( �f) + k � s=1 ωs √d0 (GH − Fg)τ1( �f) + k � s=1 ∞ � ℓ=1 �ωsFg √d0 �ℓ τℓ( �f), where the variances are given V0(f), �V0(f), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Further, the corresponding optimal functions and test statistic can be calculated by following the same procedure in [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' 6 Conclusion and Future Works In this paper, we considered the detection problems of the spiked random model with general ranks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' First, we prove the sub-optimality of the PCA for the non-Gaussian noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Further, we proposed a hypothesis test based on the central limit theorem for the linear spectral statistics of the data matrix and introduced a test for rank estimation that do not require any prior information on the rank of the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' It was shown that the error of the proposed hypothesis test matches the error of the likelihood ratio test in case the noise is Gaussian and the signal-to-noise ratio is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' With the knowledge on the density of the noise, the test was further improved by applying an entrywise transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We believe that the hypothesis test with the entrywise transformed matrix proposed in this paper can be extended to the multiplicative model of spiked rectangular matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' This will be discussed in our future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Acknowledgments The work of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Jung and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Lee was partially supported by National Research Foundation of Korea under grant number NRF-2019R1A5A1028324.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The work of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Chung was partially supported by National Research Foundation of Korea under grant number 2017R1E1A1A01076340 and by the Ministry of Science and ICT, Korea, under an ITRC Program, IITP-2019-2018-0-01402.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' References [1] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Abbe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Community detection and stochastic block models: recent developments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The Journal of Machine Learning Research, 18(1):6446–6531, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' [2] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Ajanki, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Erd˝os, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Kr¨uger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Universality for general wigner-type matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Theory and Related Fields, 169(3):667–727, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' [3] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Singularities of the density of states of random gram matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=', 22:1–13, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' [4] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Alt, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Erd˝os, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Kr¨uger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Local law for random gram matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=', 22:1–41, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' 22 [5] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Aubin, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Loureiro, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Maillard, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Krzakala, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Zdeborov´a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The spiked matrix model with generative priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 32, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' [6] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Bai and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Yao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' On the convergence of the spectral empirical process of Wigner matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Bernoulli, 11(6):1059–1092, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' [7] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Baik, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Arous, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' P´ech´e, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=', 33(5):1643–1697, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' [8] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Baik and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Fluctuations of the free energy of the spherical Sherrington-Kirkpatrick model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=', 165(2):185–224, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' [9] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Baik and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Fluctuations of the free energy of the spherical Sherrington-Kirkpatrick model with ferromagnetic interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Henri Poincar´e, 18(6):1867–1917, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' [10] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Baik and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Free energy of bipartite spherical Sherrington-Kirkpatrick model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Henri Poincar´e Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=', 56(4):2897–2934, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' [11] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Baik, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Lee, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Wu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Ferromagnetic to paramagnetic transition in spherical spin glass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=', 173(5):1484–1522, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' [12] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Banerjee and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Ma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Optimal hypothesis testing for stochastic block models with growing degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' arXiv:1705.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='05305, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' [13] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Bao, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Ding, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Wang, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Statistical inference for principal components of spiked covariance matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The Annals of Statistics, 50(2):1144–1169, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' [14] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Benaych-Georges and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Nadakuditi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The eigenvalues and eigenvectors of finite, low rank pertur- bations of large random matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=', 227(1):494–521, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' [15] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Benaych-Georges and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Nadakuditi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The singular values and vectors of low rank perturbations of large rectangular random matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Multivar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=', 111:120–135, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' [16] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Bickel and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Sarkar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Hypothesis testing for automated community detection in networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=', B: Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=', 78(1):253–273, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' [17] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Bloemendal, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Erd˝os, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Knowles, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Yau, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Yin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Isotropic local laws for sample covariance and generalized Wigner matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=', 19:no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' 33, 53, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' [18] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Bloemendal, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Knowles, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Yau, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Yin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' On the principal components of sample covariance matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' theory and related fields, 164(1-2):459–552, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' [19] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Butucea, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Ingster, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Detection of a sparse submatrix of a high-dimensional noisy matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Bernoulli, 19(5B):2652–2688, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' [20] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Cai, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Ma, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Wu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Optimal estimation and rank detection for sparse spiked covariance matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Probability theory and related fields, 161(3):781–815, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' [21] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Chung, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Lee, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Asymptotic normality of log likelihood ratio and fundamental limit of the weak detection for spiked wigner matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' arXiv preprint arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='00821, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' [22] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Chung and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Weak detection of signal in the spiked wigner model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' In International Conference on Machine Learning, pages 1233–1241, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' 23 [23] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Couillet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Robust spiked random matrices and a robust g-music estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Journal of Multivariate Analysis, 140:139–161, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' [24] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Dia, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Macris, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Krzakala, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Lesieur, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Zdeborov´a, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Mutual information for symmetric rank-one matrix estimation: A proof of the replica formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 29, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' [25] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Ding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' High dimensional deformed rectangular matrices with applications in matrix denoising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Bernoulli, 26(1):387–417, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' [26] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Ding and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Yang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Tracy-widom distribution for heterogeneous gram matrices with applications in signal detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' IEEE Transactions on Information Theory, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' [27] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Dobriban.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Permutation methods for factor analysis and pca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The Annals of Statistics, 48(5):2824– 2847, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' [28] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' El Alaoui and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Jordan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Detection limits in the high-dimensional spiked rectangular model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' In Conference On Learning Theory, pages 410–438, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' [29] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' El Alaoui, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Krzakala, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Jordan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Fundamental limits of detection in the spiked Wigner model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Stat, 48(2):863–885, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' [30] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Erd˝os, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Knowles, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Yau, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Yin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Spectral statistics of erdos-R´enyi graphs I: Local semicircle law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=', 41(3B):2279–2375, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' [31] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Johnstone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' On the distribution of the largest eigenvalue in principal components analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Stat, pages 295–327, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' [32] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Johnstone and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Onatski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Testing in high-dimensional spiked models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Stat, 48(3):1231–1254, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' [33] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Jung, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Chung, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Weak detection in the spiked wigner model with general rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' arXiv:2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='05676, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' [34] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Jung, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Chung, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Detection of signal in the spiked rectangular models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' In International Conference on Machine Learning, pages 5158–5167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' PMLR, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' [35] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Ke, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Ma, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Lin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Estimation of the number of spiked eigenvalues in a covariance matrix by bulk eigenvalue matching analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Journal of the American Statistical Association, pages 1–19, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' [36] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Knowles and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Yin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The isotropic semicircle law and deformation of Wigner matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Pure Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=', 66(11):1663–1750, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' [37] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Knowles and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Yin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Anisotropic local laws for random matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Theory Related Fields, 169(1-2):257–352, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' [38] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Lee and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Schnelli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Tracy-Widom distribution for the largest eigenvalue of real sample covariance matrices with general population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=', 26(6):3786–3839, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' [39] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Lee and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Schnelli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Local law and Tracy-Widom limit for sparse random matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Theory Related Fields, 171(1-2):543–616, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' [40] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Lei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' A goodness-of-fit test for stochastic block models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=', 44(1):401–424, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' 24 [41] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Lelarge and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Miolane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Fundamental limits of symmetric low-rank matrix estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Theory Related Fields, 173(3-4):859–929, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' [42] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Lesieur, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Krzakala, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Zdeborov´a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Mmse of probabilistic low-rank matrix estimation: Univer- sality with respect to the output channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' In 2015 53rd Annual Allerton Conference on Communication, Control, and Computing (Allerton), pages 680–687, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' [43] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Miolane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Fundamental limits of low-rank matrix estimation: the non-symmetric case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' arXiv preprint arXiv:1702.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='00473, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' [44] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Montanari, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Reichman, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Zeitouni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' On the limitation of spectral methods: From the gaussian hidden clique problem to rank-one perturbations of gaussian tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' In Advances in Neural Information Processing Systems, pages 217–225, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' [45] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Olver, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Lozier, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Boisvert, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Clark, editors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' NIST handbook of mathematical functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Department of Commerce, National Institute of Standards and Technology, Washington, DC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Cambridge University Press, Cambridge, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' [46] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Onatski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Testing hypotheses about the number of factors in large factor models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Econometrica, 77(5):1447–1479, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' [47] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Onatski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Moreira, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Hallin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Asymptotic power of sphericity tests for high-dimensional data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Stat, 41(3):1204–1231, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' [48] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Onatski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Moreira, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Hallin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Signal detection in high dimension: The multispiked case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Stat, 42(1):225–254, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' [49] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Passemier and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Yao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' On determining the number of spikes in a high-dimensional spiked popu- lation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Random Matrices: Theory and Applications, 1(01):1150002, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' [50] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Perry, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Wein, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Bandeira, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Moitra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Optimality and sub-optimality of PCA I: Spiked random matrix models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Stat, 46(5):2416–2451, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' A Examples and Simulations In Appendix A, we consider specific examples of spiked random matrices under various settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We first demonstrate with an example the change of the threshold by the improved PCA in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We then provide the details of the proposed tests in Algorithms 2 and 3 with different examples, and the test for rank estimation in Algorithm 4 for these and compute the theoretical errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We also perform the numerical simulation for the proposed tests and compare the numerical errors with the theoretical errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1 Spiked Wigner matrix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1 Improved PCA with Entrywise Transformation Our first example is a spiked Wigner matrix with non-Gaussian noise to which we apply the entrywise transformation for the improved PCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We let the density function of the noise be a bimodal distribution with unit variance, defined as g(x) = gd(x) = 1 √ 2π � e−2(x− √ 3/2)2 + e−2(x+ √ 3/2)2� , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1) 25 which is the density function of a random variable 1 2N + √ 3 2 R, where N is a standard Gaussian random variable and R is a Rademacher random variable, independent to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We sample Zij = Zji independently from the density g and let Wij = Zij/ √ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We let u(ℓ) = (u1(ℓ), u2(ℓ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' , uN(ℓ))T , where √ Nui(ℓ)’s are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Rademacher random variables for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' , N and ℓ = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The data matrix M = UΛ1/2U T , where U = [u(1), u(2), u(3)] and Λ = diag(λ, λ, λ, 0, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' , 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The size of the data matrix is set to be N = 4000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The BBP-transition predicts that the largest eigenvalue of M pops up from the bulk of the spectrum if λ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' With the entrywise transformation defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2), we obtain a transformed matrix � Mij = 1 � FgN h( √ NMij) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2) where h(x) = −g′(x) g(x) = 2 �√ 3 − e4 √ 3x( √ 3−2x) + 2x � 1 + e4 √ 3x (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3) and Fg = � ∞ −∞ (g′(x))2 g(x) dx ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='50810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' From Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3, it is expected that the largest eigenvalue of � M separates from other eigenvalues if λ > 1 Fg ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' In the numerical experiment, we set λℓ = ℓ + 1 Fg ℓ + 1 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4) for ℓ = 1, 2, 3, and we compare the spectrum of the matrices M and � M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' In Figure 1, we find three isolated eigenvalues in the spectrum of � M (right), which are absent in that of M (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5 2 0 5 10 15 20 25 30 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5 2 0 5 10 15 20 25 30 Figure 1: The spectrum of the data matrix (N = 4000) with bimodal noise, before (left) and after (right) the entrywise transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Three eigenvalues pop up from the bulk of the spectrum after the entrywise transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' 26 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2 Spiked Gaussian Wigner matrix We consider the weak detection problem with the simplest case of the spiked Gaussian Wigner matrix where w2 = 2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=', W is a GOE matrix) and the signal u(m) = (u1(m), u2(m), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' , uN(m)) where √ Nui(m)’s are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Rademacher random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Note that the parameters w2 = 2 and w4 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' In the numerical simulation done in Matlab, we generated 10,000 independent samples of the 256 × 256 data matrix M, where we fix k1 = 1 (under H1) and vary k2 from 2 to 5 (under Hk2), with the SNR λ varying from 0 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' To apply Algorithm 2, we compute Lλ = − log det � (1 + λ)I − √ λM � + λN 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5) We accept H1 if Lλ ≤ mk1 + mk2 2 = −k2 + 2 2 log(1 − λ) and reject H1 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The (theoretical) limiting error of the test is erfc �k2 − 1 4 � − log(1 − λ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='6) In Figure 2, we compare the error from the numerical simulation and the theoretical error of the proposed algorithm, which show that the numerical errors of the test closely match the theoretical errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Figure 2: The errors from the simulation with Algorithm 2 (solid) versus the limiting errors (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='6) (dashed) for the setting in Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2 with k2 = 2, 3, 4, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3 Spiked Wigner matrix We next consider a spiked Wigner matrix with non-Gaussian noise, where the density function of the noise matrix is given by g(x) = gd(x) = 1 2 cosh(πx/2) = 1 eπx/2 + e−πx/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='7) 27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='8 + Type II error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5 k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='=2 (,=2 (limiting) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4 k,=3 k,=3 (limiting) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3 =4 (limiting) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2 k,=5 (limiting) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='7 rWe sample Zij = Zji from the density g and let Wij = Zij/ √ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We again let the signal u(m) = (u1(m), u2(m), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' , uN(m)) where √ Nui(m)’s are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Rademacher random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Note that the pa- rameters w2 = 1 and w4 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We again perform the numerical simulation 10,000 samples of the 256 × 256 data matrix M with the SNR λ varying from 0 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='6, where we fix k1 = 1 (under H1) and k2 = 3 (under H2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' In Algorithm 2, we compute Lλ = − log det � (1 + λ)I − √ λM � + λN 2 + √ λ Tr M − λ 4 (Tr M 2 − N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='8) We accept H1 if Lλ ≤ mk1 + mk2 2 = −k2 + 2 2 log(1 − λ) + k2λ 2 − (k2 − 3)λ2 8 and accept H2 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The (theoretical) limiting error of the test is erfc � k2 − 1 4 � − log(1 − λ) + λ − λ2 4 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='9) We can further improve the test by introducing the entrywise transformation given by h(x) = −g′(x) g(x) = π 2 tanh πx 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The Fisher information Fg = π2 8 , which is larger than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We thus construct a transformed matrix � M by � Mij = 2 √ 2 π √ N h( √ NMij) = � 2 N tanh � π √ N 2 Mij � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' If λ > 1 Fg = 8 π2 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='8106, we can apply PCA for strong detection of the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' If λ < 8 π2 , applying Algorithm 3, we compute �Lλ = − log det �� 1 + π2λ 8 � I − � π2λ 8 � M � + π2λN 16 + π √ λ 2 √ 2 Tr � M + π2λ 16 (Tr � M 2 − N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (Here, Fg = Fg,d = π2 8 , GH = π2 16 , and �w4 = 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=') We accept H1 if �Lλ ≤ −k2 + 2 2 log � 1 − π2λ 8 � + k2π2λ 16 − 3π4λ2 512 and accept H2 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The limiting error with entrywise transformation is erfc � k2 − 1 4 � − log � 1 − π2λ 8 � + π2λ 8 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='10) Since erfc(·) is a decreasing function and π2 8 > 1, it is immediate to see that the limiting error in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='10) is strictly smaller than the limiting error in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' In Figure 3, we plot the result of the simulation with k2 = 3, which shows that the numerical error from Algorithm 3 is smaller than that of Algorithm 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' both errors closely match theoretical errors in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='10) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' 28 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='9 1 Type I + Type II error Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2 k2=3 Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2 k2=3(limiting) Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3 k2=3 Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3 k2=3(limiting) Figure 3: The errors from the simulation with Algorithm 2 (blue) and with Algorithm 3 (yellow), respectively, versus the limiting errors (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='9) of Algorithm 2 (red) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='10) of Algorithm 3 (purple), respectively, for the setting in Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4 Rank Estimation We again consider the example in Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2 and apply Algorithm 4 to estimate the rank of the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We again perform the numerical simulation 20,000 samples of the 256 × 256 data matrix M with the SNR λ varying 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='025 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='6 and choose the rank of the signal k uniformly from 0 to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Since we know that the range of the rank k is [0, 4], the (theoretical) limiting error in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='32) changes to P(k = 0) · P � Z > √V0 4 � + 3 � i=1 P(k = i) · P � |Z| > √V0 4 � + P(k = 4) · P � Z > √V0 4 � = � 1 − P(k = 0) + P(k = 4) 2 � × erfc � 1 4 � − log(1 − λ) + � 2 w2 − 1 � λ + � 1 w4 − 1 − 1 2 � λ2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We compute the same test statistic Lλ = − log det � (1 + λ)I − √ λM � + λN 2 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='11) and find the nearest nonnegative integer of the value − Lλ log(1 − λ) − 1 2, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='12) rounding half down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Since P(k = 0) = P(k = 4) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2, the limiting error of the estimation is � 1 − P(k = 0) + P(k = 4) 2 � erfc �1 4 � − log(1 − λ) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='8 · erfc �1 4 � − log(1 − λ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='13) 29 The result of the simulation can be found in Figure 4, where we compare the error from the estimation (Algorithm 4) and the theoretical error in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We can see that the error from the numerical simulation matches closely the theoretical error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='76 Probability of error Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4 limiting error Figure 4: The errors from the simulation with Algorithm 4 (solid) versus the limiting error (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='13) (dashed) for the setting in Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2 Spiked rectangular matrices In this section, we check the performance of the improved PCA and the pre-transformed LSS-based tests for spiked rectangular matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1 Improved PCA with Entrywise Transformation Additive model We consider the data with the non-Gaussian noise whose density function is given by the bimodal dis- tribution in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We sample Zij independently from the density g and let Xij = Zij/ √ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We let u(ℓ) = (u1(ℓ), u2(ℓ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' , uM(ℓ))T and v(ℓ) = (v1(ℓ), v2(ℓ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' , vN(ℓ))T , where √ Mui(ℓ)’s and √ Nvj(ℓ)’s are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Rademacher random variables for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' , M, j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' , N and ℓ = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' When we apply the entrywise transformation, defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3), with α = 0 to the rank-3 spiked mean data matrix, we get �Yij = 1 � FgN h( √ NYij) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='14) where h(x) = −g′(x) g(x) = 2 �√ 3 − e4 √ 3x( √ 3−2x) + 2x � 1 + e4 √ 3x (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='15) and Fg = � ∞ −∞ (g′(x))2 g(x) dx ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='50810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The size of the data matrix is set to be M = 2000, N = 4000, and the ratio d0 = M/N = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' 30 Theoretically, the threshold for the BBP-transition of the largest eigenvalue is √d0 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='7071 with the vanilla PCA, whereas the threshold is lowered to √d0 Fg ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2819 with the improved PCA as predicted by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' For ℓ = 1, 2, 3, we set the SNRs λℓ = ℓ√d0 + √d0 Fg ℓ + 1 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='16) to observe the transitions of the largest eigenvalue after the transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' In Figure 5, we compare the spectrum of the sample covariance matrices, Y Y T (left) and �Y �Y T (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' As in the spiked Wigner case in Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1, we again find three outlier eigenvalues only in the spectrum of �Y �Y T (right), which are absent in that of Y Y T (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' 0 1 2 3 0 5 10 15 20 25 30 35 40 0 1 2 3 0 5 10 15 20 25 30 35 40 Figure 5: The spectrum of the sample covariance matrix (M = 2000, N = 4000) with bimodal noise, before (left) and after (right) the entrywise transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Three eigenvalues pop up from the bulk of the spectrum after the entrywise transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Multiplicative model In the spiked covariance model, to clearly observe the outlier in our simulation setting, a distribution with a larger Fisher information value should be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Thus, we let the density function ga of the noise be the generalized version of the bimodal distribution with unit variance in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1), defined as ga(x) = 1 2 � 2(1 − a2)π � e − (x−a)2 2(1−a2) + e − (x+a)2 2(1−a2) � , which ga is the density function of a random variable � 1 − a2N + aR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' 31 We sample Zij independently from the density ga and let Xij = Zij/ √ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We let u(ℓ) = (u1(ℓ), u2(ℓ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' , uM(ℓ))T and v(ℓ) = (v1(ℓ), v2(ℓ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' , vN(ℓ))T , where √ Mui(ℓ)’s and √ Nvj(ℓ)’s are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Rademacher random vari- ables for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' , M, j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' , N and ℓ = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' When we apply the entrywise transformation, defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3) to the rank-3 spiked covariance data matrix, we get �Yij = 1 � (α2 + 2α + Fg)N ha,α( √ NYij) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='17) where ha,α(x) = −g′ a(x) ga(x) + αx = � (x − a)e 2ax 1−a2 + (x + a) � (1 − a2)(1 + e 2ax 1−a2 ) + αx (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='18) and Fg = � ∞ −∞ (g′ a(x))2 ga(x) dx ≈ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='15583, when a = √ 21/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The size of the data matrix is set to be M = 4000, N = 8000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We also use α = � Fg, and the ratio d0 = M/N = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The threshold for the BBP-transition of the largest eigenvalue is √d0 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='7071 for the vanilla PCA, whereas the threshold changes to λg,ℓ = (1+√ Fg) 2 (2γℓ + � Fgγ2 ℓ ) = √d0 for the transformed PCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (See Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=') For ℓ = 1, 2, 3, we set the SNRs λℓ = ℓ√d0 + 2√d0 1+√ Fg ℓ + 1 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='19) to observe the transitions of the largest eigenvalue after the transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We obtain a result analogous to the additive model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' See Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' 0 1 2 3 0 10 20 30 40 50 60 70 80 0 1 2 3 0 10 20 30 40 50 60 70 80 Figure 6: The spectrum of the sample covariance matrix (M = 4000, N = 8000) with bimodal noise, before (left) and after (right) the entrywise transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Three eigenvalues pop up from the bulk of the spectrum after the entrywise transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' 32 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2 Hypothesis Testing with pre-transformed LSS estimator We now consider an (additive) spiked rectangular matrix with the non-Gaussian noise whose density function is given by (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We let the signal u = (u1, u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' , uM)T and v = (v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' , vN)T , where √ Mui’s and √ Nvj’s are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Rademacher random variables for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' , M and j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Let the data matrix Y = √ λuvT + X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Recall that w4 = 5, Fg = π2 8 , GH = π2 16 , and �w4 = 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The LSS estimators are given by Lω = − log det �� 1 + d0 ω � (1 + ω)I − Y Y T � − ω 2d0 (Tr Y Y T − M) + M � ω d0 − log � ω d0 � − 1 − d0 d0 log(1 + ω) � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='20) and �Lω = − log det �� 1 + 8d0 ωπ2 � (1 + ω π2 8 )I − �Y �Y T � + π2ω 8d0 (Tr �Y �Y T − M) + M �ωπ2 8d0 − log �ωπ2 8d0 � − 1 − d0 d0 log � 1 + ω π2 8 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='21) With critical values mω = − log � 1 − ω2 d0 � + 3ω2 4d0 and �mω = − log � 1 − ω2π4 64d0 � − 3π4ω2 256d0, the errors are erfc � 1 4 � − log � 1 − ω2 d0 � − ω2 2d0 � and erfc � 1 4 � − log � 1 − π4ω2 64d0 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' In Figure 7, we plot empirical average (after 1,000 Monte Carlo simulations) of the error of the proposed test and the theoretical (limiting) error, varying the SNR ω from 0 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5, with M = 256 and N = 512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' It can be checked that the error of the proposed test closely matches the theoretical error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='95 1 simulation limiting error simulation-transformed limiting error-tansformed Figure 7: The error from the simulation (solid) and the theoretical limiting error (dashed), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' 33 B Proof of Theorems for improved PCA In this section, we rigorously prove Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5 in Section 3, which are about the detection threshold of the improved PCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1 Preliminaries We first introduce the following notions, which provide a simple way of making precise statements regarding the bound up to small powers of N that holds with probability higher than 1 − N −D for all D > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Definition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1 (Overwhelming probability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We say that an event (or family of events) Ω holds with overwhelming probability if for all (large) D > 0 we have P(Ω) ≤ N −D for any sufficiently large N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Definition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2 (Stochastic domination).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Let ξ = � ξ(N)(u) : N ∈ N, u ∈ U (N)� , ζ = � ζ(N)(u) : N ∈ N, u ∈ U (N)� be two families of random variables, where U (N) is a possibly N-dependent parameter set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We say that ξ is stochastically dominated by ζ, uniformly in u, if for all (small) ϵ > 0 and (large) D > 0 sup u∈U (N) P � |ξ(N)(u)| > N ϵζ(N)(u) � ≤ N −D for any sufficiently large N ≥ N0(ε, D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Throughout this appendix, the stochastic domination will always be uniform in all parameters, including matrix indices and the spectral parameter z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We write ξ ≺ ζ or ξ = O≺(ζ), if ξ is stochastically dominated by ζ, uniformly in u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' For a Wigner matrix W, we will use the following result for the resolvents, which is called an isotropic local semicircle law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3 (Isotropic local semicircle law).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Suppose that z ∈ R outside an open interval containing [−2, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Let ssc(z) be the Stieltjes transform of the Marchenko–Pastur law, which is also given by ssc(z) = −z + √ z2 − 4 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1) Then, ⟨u(ℓ1), (W − zI)−1u(ℓ2)⟩ = ssc(z)⟨u(ℓ1), u(ℓ2)⟩ + O≺(N − 1 2 ) See Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3 of [36] (also Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='7 of [22]) for the proof of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Further, for a rectangular matrix X, we will use the following analogous result for the resolvents, which is called an isotropic Marchenko–Pastur law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4 (Isotropic local Marchenko–Pastur law).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Suppose that z ∈ R outside an open interval containing [d−, d+].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Let s(z) be the Stieltjes transform of the Marchenko–Pastur law, which is also given by s(z) = (1 − d0 − z) + � (1 − d0 − z)2 − 4d0z 2d0z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2) Then, ⟨v(ℓ1), (XT X − zI)−1v(ℓ2)⟩ = − � 1 zs(z) + 1 � ⟨v(ℓ1), v(ℓ2)⟩ + O≺(N − 1 2 ) 34 and ⟨XT u(ℓ1), (XT X − zI)−1XT u(ℓ2)⟩ = (zs(z) + 1)⟨u(ℓ1), u(ℓ2)⟩ + O≺(N − 1 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' See Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5 of [17] (also Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='7 of [18]) for the proof of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The following concentration inequality will be frequently used in the proof, which is sometimes called the large deviation estimate in random matrix theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5 (Large deviation estimate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Let � ξ(N) i � and � ζ(N) i � be independent families of random variables and � a(N) ij � and � b(N) i � be deterministic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' here N ∈ N and i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Suppose that complex-valued random variables ξ(N) i and ζ(N) i are independent and satisfy for all p ≥ 2 that Eξ = 0 , E|ξ|p ≤ Cp NBp−2 (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3) for some B ≤ N 1/2 and some (N-independent) constant Cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Then we have the bounds � i biξi ≺ � 1 N � i |bi|2 �1/2 + maxi |bi| B , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4) � i,j aijξiζj ≺ � 1 N 2 � i̸=j |aij|2 �1/2 + maxi̸=j |aij| B + maxi |aii| B2 , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5) � i̸=j aijξiξj ≺ � 1 N 2 � i̸=j |aij|2 �1/2 + maxi̸=j |aij| B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='6) If the coefficients a(N) ij and b(N) i depend on an additional parameter u, then all of these estimates are uniform in u, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' N0 = N0(ε, D) in the definition of ≺ depends not on u but only on the constant C from (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' If B = N 1/2, the bounds can further be simplified to � i biξi ≺ � 1 N � i |bi|2 �1/2 , � i,j aijξiζj ≺ � 1 N 2 � i,j |aij|2 �1/2 , � i̸=j aijξiξj ≺ � 1 N 2 � i̸=j |aij|2 �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='7) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' These estimates are an immediate consequence of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='8 in [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Finally, we recall that, for our prior, |⟨u(ℓ1), u(ℓ2)⟩ − δℓ1ℓ2|, |⟨v(ℓ1), v(ℓ2)⟩ − δℓ1ℓ2| ≺ N −φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2 Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3 We first prove the behavior of the k largest eigenvalues described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2, which we will call the BBP result, in our setting, following the strategy of [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' M − zI = W + UΛ1/2U T − zI = (W − zI)(I + (W − zI)−1(UΛ1/2U T )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='8) Thus, if z is an eigenvalue of M but not of W, then it satisfies det(I + (W − zI)−1UΛ1/2U T ) = 0, which also implies that −1 is an eigenvalue of T ≡ T(z) := (W − zI)−1UΛ1/2U T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' 35 We then see that T(W − zI)−1u(ℓ) = (W − zI)−1UΛ1/2U T (W − zI)−1u(ℓ) = � λℓ⟨u(ℓ), (W − zI)−1u(ℓ)⟩(W − zI)−1u(ℓ) + O≺(N −φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=', (W − zI)−1u(ℓ) must be a eigenvector for T with the corresponding eigenvalue −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Thus, by Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3, � λℓssc(z) = −1 + O≺(N −φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' It is elementary to check that the solution of the above equation is z = √λℓ + 1 √λℓ + O≺(N −φ) if and only if λℓ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We now turn to the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' For the spike ∥U∥∞ ≺ N −φ, suppose that a function q and its all derivatives are polynomially bounded in the sense of Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Following the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='8 in [50], we have the following local linear estimation of q( √ NMij) by q( √ NMij) = q( √ NWij) + √ λNuiuT j E[q′( √ NWij)] + Rij, where the error Rij is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Set Mq := E[q′( √ NWij)], Vq := E[q( √ NWij)2], �λ := λM 2 q /Vq, and Qij := 1 � NVq q( √ NWij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Then the spectrum of the transformed matrix is determined by the matrix Q + U ˆΛ1/2U T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Since Q is also Wigner matrix with NE[Q2 ij] = 1, by repeating the same process above, we get the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3 Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4 We first prove the behavior of the k largest eigenvalues described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2, which we will again call the BBP result, in our setting, following the strategy of [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Note that the k largest eigenvalue of Y Y T is equal to the k largest eigenvalues of Y T Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Consider the identity Y T Y − zI = (X + UΛ1/2V T )T (X + UΛ1/2V T ) − zI = (XT X − zI)T(z) (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='9) where T ≡ T(z) := (XT X − zI)−1(XT UΛ1/2V T + V Λ1/2U T X + V Λ1/2U T UΛ1/2V T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Thus, if z is an eigenvalue of Y Y T but not of XXT , then it satisfies det(T(z)) = 0, which also implies that −1 is an eigenvalue of T(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' 36 Note that since ∥X∥, ∥(XT X − zI)−1∥ ≺ 1, from Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5, ⟨b, (XT X − zI)−1XT a⟩ = � i,j � (XT X − zI)−1XT � ij biaj ≺ � � 1 N 2 � i̸=j ��� � (XT X − zI)−1XT � ij ��� 2 � � 1/2 + N −φ max i,j ��� � (XT X − zI)−1XT � ij ��� ≺ � 1 N ∥(XT X − zI)−1XT ∥2 �1/2 + N −φ∥(XT X − zI)−1XT ∥ ≺ N −φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Then the matrix T satisfies T · (XT X − zI)−1XT u(ℓ) = (XT X − zI)−1XT UΛ1/2(V T (XT X − zI)−1XT u(ℓ)) + (XT X − zI)−1V Λ1/2(U T X(XT X − zI)−1XT u(ℓ)) + (XT X − zI)−1V Λ1/2(U T U)Λ1/2(V T (XT X − zI)−1XT u(ℓ)) = � λℓ⟨u(ℓ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' X(XT X − zI)−1XT u(ℓ)⟩(XT X − zI)−1v(ℓ) + θ1(ℓ) and T · (XT X − zI)−1v(ℓ) = (XT X − zI)−1XT UΛ1/2(V T (XT X − zI)−1v(ℓ)) + (XT X − zI)−1V Λ1/2(U T X(XT X − zI)−1v(ℓ)) + (XT X − zI)−1V Λ1/2(U T U)Λ1/2(V T (XT X − zI)−1v(ℓ)) = � λℓ⟨v(ℓ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (XT X − zI)−1v(ℓ)⟩(XT X − zI)−1XT u(ℓ) + λℓ⟨v(ℓ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (XT X − zI)−1v(ℓ)⟩(XT X − zI)−1v(ℓ) + θ2(ℓ) where ∥θ1(ℓ)∥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' ∥θ2(ℓ)∥ = O≺(N −φ) since ∥U T U − I∥F ≺ N −φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' In particular, k extremal eigenvectors of T are a linear combination of (XT X−zI)−1XT u(ℓ) and (XT X− zI)−1v(ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Suppose that aℓ(XT X−zI)−1XT u(ℓ)+bℓ(XT X−zI)−1v(ℓ) is an eigenvector of T with the corresponding eigenvalue −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Thus, from Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4, − � aℓ(XT X − zI)−1XT u(ℓ) + bℓ(XT X − zI)−1v(ℓ) � = T � aℓ(XT X − zI)−1XT u(ℓ) + bℓ(XT X − zI)−1v(ℓ) � = −bℓ � λℓ � 1 zs(z) + 1 � (XT X − zI)−1XT u(ℓ) + aℓ � λℓ(zs(z) + 1)(XT X − zI)−1v(ℓ) − bℓλℓ � 1 zs(z) + 1 � (XT X − zI)−1v(ℓ) + �θ(ℓ) (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='10) for some �θ(ℓ), which is a linear combination of (XT X −zI)−1XT u(ℓ) and (XT X −zI)−1v(ℓ) with ∥�θ(ℓ)∥ = O≺(N −φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Since U, V , and X are independent, (XT X − zI)−1XT u(ℓ) and (XT X − zI)−1v(ℓ) are linearly inde- 37 pendent with overwhelming probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Thus, from (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='10), −aℓ = −bℓ � λℓ � 1 zs(z) + 1 � + O≺(N −φ), −bℓ = aℓ � λℓ(zs(z) + 1) − bℓλℓ � 1 zs(z) + 1 � + O≺(N −φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' It is then elementary to check that λℓ(zs(z) + 1) + 1 = O≺(N −φ), which has the solution z = (1 + λℓ) � 1 + d0 λℓ � + O≺(N −φ) if and only if λℓ > √d0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' This proves the BBP result in our setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We now turn to the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' To simplify the exposition, we focus on the case that SNRs are the same i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=', Λ = λI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' For the spike prior in Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1, suppose that a function q and its all derivatives are polynomially bounded in the sense of Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Following the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='8 in [50], we define the error term from the local linear estimation of q( √ NYij) by q( √ NYij) = q( √ NXij) + √ λNuivT j q′( √ NXij) + Rij where Rij = 1 2q′′( √ NXij + eij)λN(uivT j )2 for some |eij| ≤ | √ λNuivT j |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The Frobenius norm of R is bounded as ∥R∥2 F = Tr RT R = λ2N 2 4 M � i=1 N � j=1 (uivT j )4q′′( √ NXij + eij)2 ≤ λ2N 2−4φ 4 M � i=1 N � j=1 (uivT j )2q′′( √ NXij + eij)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Since q′′ is polynomially bounded, q′′( √ NXij + eij) is uniformly bounded by an N-independent constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Thus, with overwhelming probability, ∥R∥2 ≤ ∥R∥2 F ≤ Cλ2N 2−4φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Next, we approximate q( √ NXij) by its mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Let Eij = q′( √ NXij) − E[q′( √ NXij)], ∆ij = √ λNuivT j Eij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Then, ∥∆∥ ≺ N 1 2 −2φ∥E∥ and, since the entries of matrix E are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=', centered and with finite moments, its norm ∥E∥ = O( √ N) with overwhelming probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (See, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=', [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=') Thus, ∥∆∥ = O≺(N 1−2φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Set Mq := E[q′( √ NXij)], Vq := E[q( √ NXij)2], �λ := λM 2 q /Vq, 38 and Qij := 1 � NVq q( √ NXij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We have proved so far that the difference of the largest eigenvalue of Q + �λ 1 2 UV T and that of the matrix � 1 � NVq q( √ NYij) � is O≺(N 1 2 −2φ), which is o(1) with overwhelming probability for φ > 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' It is directly applicable to the case that Λ in our model with �Λℓℓ := λℓM 2 q /Vq since the above process does not require any information of the SNRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The BBP result holds the matrix Q+U �Λ 1 2 V T , which is another (additive) spiked rectangular matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' This shows that the BBP result also holds for �Y with SNR matrix �Λ := M 2 q Vq Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' This proves Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4 Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5 Recall that the spike prior satisfies the technical conditions in Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1 with φ > 1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' For the sake of brevity, we assume that Λ = λI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' As in the additive case, we further assume that a function q and its all derivatives are polynomially bounded and consider the local linear approximation of q( √ NYij), q( √ NYij) = q( √ NXij) + γ √ NE[q′( √ NXij)] � ℓ uiuT ℓ Xℓj + Rij + γ∆ij, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='11) where Rij = 1 2q′′�√ NXij + θγ � ℓ uiuT ℓ √ NXℓj � � γ � ℓ uiuT ℓ √ NXℓj �2 for some θ ∈ [−1, 1] and ∆ij = √ NEij � ℓ uiuT ℓ Xℓj, Eij = q′( √ NXij) − E[q′( √ NXij)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' For any unit vectors a = (a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' , aM) and b = (b1, b2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' , bN), aT ∆b = � s � i,j aiui(s)Eijbj �� ℓ uℓ(s) √ NXℓj � = � s � i,j aiui(s)2bjEij √ NXij + � s � i,j aiui(s)Eijbj � �� ℓ̸=i uℓ(s) √ NXℓj � � From the concentration inequalities such as Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5, � ℓ uiuT ℓ √ NXℓj = � s � ℓ ui(s)uℓ(s)T √ NXℓj ≺ � s |ui(s)| �� ℓ uℓ(s)2 �1/2 ≺ N −φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='12) Recall that ∥E∥ = O( √ N) with overwhelming probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Note that, by Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1, the density function q have to be an odd function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Further, since q is an odd function (hence xq′(x) is an odd function 39 of x), the norm of the matrix whose (i, j)-entry is Eij √ NXij is also O( √ N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Thus, aT ∆b ≺ N −2φ + N 1 2 −φ, which shows that ∥∆∥ ≺ N 1 2 −φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Moreover, since q′′ is polynomially bounded, following the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4 with (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='12), ∥R∥2 ≤ ∥R∥2 F ≤ CN 2−4φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Thus, as in the additive case, the error terms Rij and ∆ij in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='11) are negligible when finding the limit of the extreme eigenvalues of the transformed matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Set Mq := E[q′( √ NXij)], Vq := E[q( √ NXij)2], Eq = E[ √ NXijq( √ NXij)], �γ := γMq/ � Vq, and Qij := 1 � NVq q( √ NXij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' With the approximation (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='11), we now focus on the largest eigenvalue of (Q + �γUU T X)T (Q + �γUU T X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Note that the assumption on the polynomial boundedness of q implies that the matrix Q is also a rectangular matrix satisfying the assumptions in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Let G(z) and G(z) be the resolvents G ≡ G(z) := (QQT − zI)−1, G ≡ G(z) := (QT Q − zI)−1 for z ∈ R outside an open interval containing [d−, d+].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We note that the following identities hold for G(z) and G(z): G(z)Q = QG(z), QT G(z)Q = I + zG(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='13) As in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4, we consider (Q + �γUU T X)T (Q + �γUU T X) − zI = (QT Q − zI)(I + (QT Q − zI)−1(�γXT UU T Q + �γQT UU T X + �γ2XT UU T UU T X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='14) Let L ≡ L(z) = G(z)(�γXT UU T Q + �γQT UU T X + �γ2XT UU T UU T X), Then, as in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4, if z is an eigenvalue of (Q + �γUU T X)T (Q + �γUU T X) (but not of QT Q), −1 is an eigenvalue of L(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Again, the rank of L is at most 2k, with L · GQT U = �γGXT UU T QGQT U + �γGQT UU T XGQT U + �γ2GXT UU T UU T XGQT U, L · GXT U = �γGXT UU T QGXT U + �γGQT UU T XGXT U + �γ2GXT UU T UU T XGXT U, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='15) and an eigenvector of L is a linear combination of GQT u(ℓ) and GXT u(ℓ) for 1 ≤ ℓ ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' In the simplest case where Q is the identity mapping, Q = X, hence the rank of L is k, and the eigenvalue equation (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='15) is simplified to L · GQT U = �γGQT U(U T QGQT U) + �γGQT U(U T QGQT U) + �γ2GQT U(U T UU T QGQT U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='16) 40 In this case, GQT u(ℓ) are eigenvectors of L corresponding to the eigenvalue −1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=', L · GQT u(ℓ) = −GQT u(ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The right side of (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='16) can be approximated as follows, which is a direct consequence of the isotropic local Marchenko–Pastur law (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=', Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5 of [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' With the isotropic local Marchenko–Pastur law, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='16) can be approximated by a deterministic vector equation on z (and s(z)), and the location of the k largest eigenvalues can be proved by solving the equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' In a general case where Q is not a multiple of X and the vectors GQT u(ℓ) and GXT u(ℓ) are linearly independent, however, the eigenvalue equation (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='15) contains other matrices U T QGQT U, U T QGXT U, and U T XGQT U, which cannot be estimated by Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' For these matrices, we use the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Suppose that the assumptions in Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Then, ⟨u(ℓ1), XGQT u(ℓ2)⟩ = ⟨u(ℓ1), QGXT u(ℓ2)⟩ = � Eq � Vq (zs(z) + 1) � δℓ1ℓ2 + O≺(N −φ) and ⟨u(ℓ1), XGXT u(ℓ2)⟩ = � E2 q Vq zs(z) � d0s(z) + d0 − 1 z �2 + d0s(z) + d0 − 1 z � δℓ1ℓ2 + O≺(N −φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We defer the proof to Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' With Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='6, we are ready to finish the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' From the definition of s(z) in Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4, we notice that s(z) = 1 1 − d0 − d0zs(z) − z , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='17) or z � d0s(z) + d0 − 1 z � = − 1 s(z) − z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='18) Set σ(z) := zs(z) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' By applying Lemmas B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4 and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='6 to (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='16), for 1 ≤ ℓ ≤ k L · GQT u(ℓ) = �γ⟨u(ℓ), QGQT u(ℓ)⟩ · GXT u(ℓ) + �γ⟨u(ℓ), XGQT u(ℓ)⟩ · GQT u(ℓ) + ∥u(ℓ)∥2�γ2⟨u(ℓ), XGQT u(ℓ)⟩ · GXT u(ℓ) = �γσ(z)GXT u(ℓ) + �γσ(z) Eq � Vq GQT u(ℓ) + �γ2 Eq � Vq σ(z)GXT u(ℓ) + θ1(ℓ) , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='19) and L · GXT u(ℓ) = �γ⟨u(ℓ), QGXT u(ℓ)⟩ · GXT u(ℓ) + �γ⟨u(ℓ), XGXT u(ℓ)⟩ · GQT u(ℓ) + ∥u(ℓ)∥2�γ2⟨u(ℓ), XGXT u(ℓ)⟩ · GXT u(ℓ) = �γσ(z) Eq � Vq GXT u(ℓ) + �γ �� σ(z) + σ(z) σ(z) − 1 � E2 q Vq − σ(z) σ(z) − 1 � GQT u(ℓ) + �γ2 �� σ(z) + σ(z) σ(z) − 1 � E2 q Vq − σ(z) σ(z) − 1 � GXT u(ℓ) + θ2(ℓ) , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='20) for some θ1(ℓ), θ2(ℓ), which are linear combinations of GQT u(ℓ) and GXT u(ℓ), with ∥θ1(ℓ)∥, ∥θ2(ℓ)∥ = O≺(N −φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Suppose that aℓGQT u(ℓ)+bℓGXT u(ℓ) is an eigenvector of L with the corresponding eigenvalue −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' From 41 (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='19), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='20), and the linear independence between GQT u(ℓ) and GXT u(ℓ), we find the relation −aℓ = aℓ�γσ(z) Eq � Vq + bℓ�γσ(z)2 σ(z) − 1 · E2 q Vq − bℓ�γσ(z) σ(z) − 1 + O(N −φ), −bℓ = aℓ�γσ(z) + aℓ�γ2σ(z) Eq � Vq + bℓ�γσ(z) Eq � Vq + bℓ�γ2σ(z)2 σ(z) − 1 · E2 q Vq − bℓ�γ2σ(z) σ(z) − 1 + O(N −φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We then find that bℓ aℓ � 1 + �γσ(z) Eq � Vq � + �γσ(z) − �γ = O(N −φ) and aℓ � 1 + �γσ(z) Eq � Vq � = bℓ � �γσ(z) σ(z) − 1 � 1 − σ(z) · E2 q Vq �� + O(N −φ), which implies that 1 + 2�γσ(z) Eq � Vq + �γ2σ(z) = 1 + � 2γMqEq + γ2M 2 q Vq � σ(z) = O(N −φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='21) From the explicit formula for s, it is not hard to check that (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='21) holds if and only if λq := 2γMqEq + γ2M 2 q Vq > � d0 and z = (1 + λq) � 1 + d0 λq � + O(N −φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='22) We see that it is valid for general Λ in our model, since the above process also does not require any information of the SNRs as in the additive case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Now, the desired theorem follows from the direct computation for the case q = hαg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' see also Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5 Optimal entrywise transformation B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1 Additive model Recall that E[q′( √ NWij)] = E[q′( √ NXij)] = Mq, E[q( √ NWij)2] = E[q( √ NXij)2] = Vq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Following the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4 in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3, it is not hard to see that the effective SNR is maximized by optimizing M 2 q /Vq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Such an optimization problem was already considered in [50] for the spiked Wigner matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' For the sake of completeness, we solve this problem by using the calculus of variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Recall the density of random variables √ NWij and √ NXij is g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' To optimize q, we need to maximize �� ∞ −∞ q′(x)g(x)dx �2 / �� ∞ −∞ q(x)2g(x)dx � = �� ∞ −∞ q(x)g′(x)dx �2 / �� ∞ −∞ q(x)2g(x)dx � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='23) Putting (q + εη) in place of q in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='23) and differentiating with respect to ε, we find that the optimal q 42 satisfies �� ∞ −∞ η(x)g′(x)dx � �� ∞ −∞ q(x)2g(x)dx � = �� ∞ −∞ q(x)η(x)g(x)dx � �� ∞ −∞ q(x)g′(x)dx � (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='24) for any η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' It is then easy to check that q = −Cg′/g is the only solution of (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Since the value in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='23) does not change if we replace q by Cq, and the effective SNR is increased with the entrywise transform −g′/g is the optimal entrywise transformation for PCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2 Multiplicative model As we can see from the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5 in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4, we need to maximize 2 �� ∞ −∞ xq(x)g(x)dx � �� ∞ −∞ q′(x)g(x)dx � + γ �� ∞ −∞ q′(x)g(x)dx �2 �� ∞ −∞ q(x)2g(x)dx � = −2 �� ∞ −∞ xq(x)g(x)dx � �� ∞ −∞ q(x)g′(x)dx � + γ �� ∞ −∞ q(x)g′(x)dx �2 �� ∞ −∞ q(x)2g(x)dx � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='25) Putting (q + εη) in place of q in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='23) and differentiating with respect to ε, we find that the optimal q satisfies − 2 �� xηg � �� qg′ � �� q2g � − 2 �� xqg � �� ηg′ � �� q2g � + 2γ �� ηg′ � �� qg′ � �� q2g � + 4 �� qηg � �� xqg � �� qg′ � − 2γ �� qg′ �2 �� qηg � = 0 (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='26) which is written with slight abuse of notation such as � xηg = � ∞ −∞ xη(x)g(x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Since the equation contains the terms � xηg, � ηg′, � qηg, it is natural to consider an ansatz q(x) = −g′(x) g(x) + αx (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='27) for a constant α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Collecting the terms involving � xηg and the terms involving � ηg′, we get 2(Fg + α)(Fg + 2α + α2) − 4α(1 + α)(Fg + α) − 2αγ(Fg + α)2 = 0 and −2(1 + α)(Fg + 2α + α2) − 2γ(Fg + α)(Fg + 2α + α2) + 4(1 + α)(Fg + α) + 2γ(Fg + α)2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We can then check that α = αg = −γFg + � 4Fg + 4γFg + γ2F 2g 2(1 + γ) , 43 and hence (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='26) is satisfied with q(x) = −g′(x) g(x) + −γFg + � 4Fg + 4γFg + γ2F 2g 2(1 + γ) x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The corresponding effective SNR λhαg ≡ λg = γ + γ2Fg 2 + γ � 4Fg + 4γFg + γ2F 2g 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' For a general α, when the entrywise transform hα is applied, the effective SNR λhα = 2γ(1 + α)(Fg + α) + γ2(α + Fg)2 α2 + 2α + Fg , In particular, if α = � Fg, λh√ Fg = γ(1 + � Fg) + γ2 2 (Fg + � Fg) ≥ 2γ + γ2 = λ where the inequality is strict if Fg > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='6 Proof of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='6 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1 Key ingredient: Entrywise local estimates Recall the definition of the random matrices X and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The couple of random matrices (X, Q) is one example of the following concept for a coupled random matrices: Definition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='7 (Entrywise correlated random matrices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Suppose that A and B are M × N random rectan- gular matrices in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2 satisfying the following conditions: For all 1 ≤ a, b ≤ M and 1 ≤ α, β ≤ N, Aaα and Bbβ are dependent only when a = b and α = β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' For all a, α, E[Aaα] = E[Baα] = 0, NE[A2 aα] = wA, NE[B2 aα] = wB, and NE[AaαBaα] = wAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' For any positive integer p, there exists Cp, independent of N, such that N p 2 E[Ap aα], N p 2 E[Bp aα] ≤ Cp for all a, α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' A couple of random matrices (A, B) is called the entrywise correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The key estimates in the proof of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='6 are the exact bounds on the entries of K := Q(QQT − zI)−1XT and K := X(QQT − zI)−1XT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We prove the following lemma for the entrywise correlated random matrices (A, B), which exactly contains the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Let (A, B) be the entrywise correlated random matrices with wB = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' For z ∈ R outside an open interval containing [d−, d+], |(A(BT B − zI)−1AT )ij − (wAs(z) + w2 ABzs(z)s(z)2)δij| = O≺(N −1/2), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='28) 44 |(A(BT B − zI)−1BT )ij − (wABs(z) + wABzs(z)s(z)2)δij| = O≺(N −1/2) (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='29) and |(B(BT B − zI)−1AT )ij − (wABs(z) + wABzs(z)s(z)2)δij| = O≺(N −1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='30) Remark B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Recall that σ(z) = zs(z) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' For (X, Q), since wX = wQ = 1 and wXQ = Eq/ � Vq, we have the following: For z ∈ R outside an open interval containing [d−, d+], |Kij − �s(z)δij| = O≺(N −1/2), |Kij − ˇs(z)δij| = O≺(N −1/2), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='31) where �s(z) := σ(z) Eq � Vq , ˇs(z) := zs(z) � d0s(z) + d0 − 1 z �2 E2 q Vq + � d0s(z) + d0 − 1 z � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='32) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2 Linearization We consider G ≡ GB(z) = (BBT − zI)−1, G ≡ GB(z) = (BT B − zI)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' In the proof of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='8, we use the formalism known as the linearization to simplify the computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We define an (M + N) × (M + N) symmetric matrix HB by HB ≡ HB(z) = �−zIM B BT −IN � , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='33) where IM and IN are the identity matrices with size M and N, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Let RB(z) = HB(z)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (For the invertibility of HB(z), we refer to Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1 in [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=') By Schur’s complement formula, RQ(z) = � GB(z) GB(z)B BT GB(z) zGB(z) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='34) Therefore, Rab(z) = (BBT − zI)−1 ab = Gab(z), Rαβ(z) = z(BT B − zI)−1 α−M,β−M = zGα−M,β−M(z), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='35) and Rαa(z) = Raα(z) = (GB)a,α−M(z), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='36) where we use lowercase Latin letters a, b, c, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' for indices from 1 to M and Greek letters α, β, γ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' for indices from (M + 1) to (M + N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We also use uppercase Latin letters A, B, C, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' for indices from 1 to (M + N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' In the rest of Appendix B, we omit the subscript Q for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' For T ⊂ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' , M + N}, we define the matrix minor H(T) by (H(T))AB := 1{A,B /∈T}HAB .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='37) Moreover, for A, B /∈ T we define R(T) AB(z) := (H(T))−1 AB, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='38) In the definitions above, we abbreviate ({A}) by (A);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' similarly, we write (AB) instead of ({A, B}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We have the following resolvent (decoupling) identities for the matrix entries of R and R(T), which are 45 elementary consequences of Schur’s complement formula;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1 of [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='10 (Resolvent identities for R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Suppose that z ∈ R is outside an open interval containing [d−, d+].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' For a ̸= b, Rab = −Raa � α HaαR(a) αb = −Rbb � β R(b) aβHβb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' For α ̸= β, Rαβ = −Rαα � a HαaR(α) aβ = −Rββ � b R(β) αb Hbβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' For any a and α, Raα = −Raa � β HaβR(a) βα = −Rαα � b R(α) ab Hbα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' For A, B ̸= C, RAB = R(C) AB + RACRCB RCC .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Throughout this section, we will frequently use the estimate that all entries of X and Q (and hence all off-diagonal entries of W) are O≺(N −1/2), which holds since all moments of the entries of √ NQ and √ NX are bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' For the entries of R, we have the following estimates: Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Let s(z) = � d0s(z) + d0 − 1 z � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='39) For z ∈ R outside an open interval containing [d−, d+], |Rij(z) − s(z)δij| , |Rµν(z) − zs(z)δµν| , |Riµ(z)| ≺ N −1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='40) Proof of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The first two estimates can be checked from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5 (and Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='7) in [17] with the deterministic unit vectors v = ei and w = ej where ei ∈ RN or RM is a standard basis vector whose i-th coordinate is 1 and all other coordinates are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' For the last estimate, we apply Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='10 to find that Riµ(z) = −Rii � α HiαR(i) αµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Since Hiα and R(i) αµ are independent, R(i) αµ ≺ N −1/2 for α ̸= µ, and R(i) µµ = Θ(1) with overwhelming probability, we find from Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5 that � α HiαR(i) αµ ≺ � 1 N � α |R(i) αµ|2 �1/2 ≺ N −1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Proof of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Throughout this section, for the sake of brevity, we will use the notation Baα := Ba,(α−M) = Haα, Aaα := Aa,(α−M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We begin by estimating the diagonal entry (BGAT )ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' From Schur’s complement formula, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='35), we can decompose it into (BGAT )ii = 1 z � α HiαRααAiα + 1 z � α̸=β HiαRαβAiβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='41) 46 From concentration inequalities it is not hard to see that � α BiαAiα = E[BiαAiα] + O≺(N −1/2) = wAB + O≺(N −1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Applying Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='11, we find for the first term in the right side of (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='41) that 1 z � α HiαRααAiα = wABs(z) + O≺(N −1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='42) We next estimate the second term in the right side of (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We expand it with the resolvent identities in Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='10 as follows: � α̸=β HiαRαβAiβ = � α̸=β HiαR(i) αβAiβ + � α̸=β Hiα RαiRiβ Rii Aiβ = � α̸=β HiαR(i) αβAiβ + � α̸=β Hiα RαiRiβ s(z) Aiβ + O≺(N −1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='43) Here, in the estimate for the second term, we simply counted the power (of N) as it involves two indices for the sum (hence O(N 2) terms) of Hiα, Rαi, Riβ, Aiβ ≺ N −1/2, hence � α̸=β HiαRαiRiβAiβ = O≺(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Applying Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5 to the first term in the right side of (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='43), � α̸=β HiαR(i) αβAiβ ≺ � � 1 N 2 � α,β |R(i) αβ|2 � � 1/2 ≺ N −1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' For the second term in the right side of (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='43), we further expand it to find � α̸=β HiαRαiRiβAiβ = � α̸=β HiαRαiRiβAiβ = − � α̸=β Hiα � Rii � µ R(i) αµHµiRiβAiβ � Note that � µ R(i) αµHµi ≺ N −1/2, as in the proof of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Since |Rij − s(z)| ≺ N −1/2, Riβ = R(α) iβ + RiαRαβ Rαα = R(α) iβ + N −1, we have − � α̸=β Hiα � Rii � µ R(i) αµHµiRiβAiβ � = −s(z) � α̸=β Hiα �� µ R(i) αµHµiR(α) iβ Aiβ � + O≺(N −1/2) = −s(z) � α̸=β Hiα � � � µ:µ̸=α R(i) αµHµiR(α) iβ Aiβ � � − s(z) � α̸=β (Hiα)2R(i) ααR(α) iβ Aiβ + O≺(N −1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='44) 47 Applying Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5 again to the first term in the right side of (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='44), � α̸=β Hiα � � � µ:µ̸=α R(i) αµHµiR(α) iβ Aiβ � � ≺ � � � 1 N � α ������ � β:β̸=α � � � µ:µ̸=α R(i) αµHµi � � R(α) iβ Aiβ ������ 2� � � 1/2 ≺ � � � 1 N � α � � � β:β̸=α N −1/2 ���R(α) iβ Aiβ ��� � � 2� � � 1/2 ≺ N −1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Similarly, by expanding R(α) iβ , we find for the second term in the right side of (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='44) that −s(z) � α̸=β (Hiα)2R(i) ααR(α) iβ Aiβ = zs(z)s(z) � α̸=β (Hiα)2R(α) ii � ν:ν̸=α H(α) iν R(iα) νβ Aiβ + O≺(N −1/2) = zs(z)2s(z) � α̸=β (Hiα)2 � ν:ν̸=α,β HiνR(iα) νβ Aiβ + zs(z)2s(z) � α̸=β (Hiα)2HiβR(iα) ββ Aiβ + O≺(N −1/2) = z2s(z)2s(z)2 � α̸=β (Hiα)2HiβAiβ + O≺(N −1/2), where we used Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5 to find � ν̸=β:ν,β̸=α HiνR(iα) νβ Aiβ ≺ � � 1 N 2 � ν̸=β:ν,β̸=α ���R(iα) νβ ��� 2 � � 1/2 ≺ N −1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Thus, since wB = 1, � α̸=β HiαRαiRiβAiβ = z2s(z)2s(z)2 � α̸=β (Hiα)2HiβAiβ + O≺(N −1/2) = z2s(z)2s(z)2wAB + O≺(N −1/2), and putting it back to (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='43) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='41), together with (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='42), we conclude that (AGB)ii = wABs(z) + wABzs(z)s(z)2 + O≺(N −1/2) = wABσ(z) Eq � Vq + O≺(N −1/2), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='45) where we used the identity zs(z)s(z) = −σ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' In the same manner, we also find that (AGA)ii = 1 z � α AiαRααAiα + zs(z)s(z)2 � α̸=β AiαHiαHiβAiβ + O≺(N −1/2) = wAs(z) + w2 ABzs(z)s(z)2 + O≺(N −1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='46) We next estimate the off-diagonal entry (AGB)ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We expand it as (AGB)ij = 1 z � α,β HiαRαβAjβ = 1 z � α,β HiαR(i) αβAjβ + 1 z � α,β Hiα RαiRiβ Rii Ajβ = 1 z � α,β HiαR(ij) αβ Ajβ + 1 z � α,β Hiα R(i) αjR(i) jβ R(i) jj Ajβ + 1 z � α,β Hiα R(j) αi R(j) iβ R(i) jj Ajβ + O≺(N −1/2) (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='47) 48 From Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5, � α,β HiαR(ij) αβ Ajβ ≺ N −1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We also have � α,β Hiα R(i) αjR(i) jβ R(i) jj Ajβ ≺ � � � 1 N � α ������ � β R(i) αjR(i) jβ R(i) jj Ajβ ������ 2� � � 1/2 ≺ � � � 1 N � α ������ � β N −3/2 ������ 2� � � 1/2 ≺ N −1/2 and a similar estimate holds for the third term in the right side of (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Thus, (AGB)ij ≺ N −1/2 In the same manner, we also find that (AGA)ij ≺ N −1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Together with (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='45) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='46), this proves Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3 Isotropic local law We also assume that wB = 1 and use the same notation in previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Then our goal is to prove the following statement: Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Let (A, B) be the entrywise correlated random matrices where wB = 1 and x, y are determin- istic and ℓ2 - normalized vectors in RM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Then, for z ∈ R outside an open interval containing [d−, d+], ⟨x, A(BT B − zI)−1AT y⟩ = (wAs(z) + w2 ABzs(z)s(z)2)⟨x, y⟩ + O≺(N −1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Proof of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Note that, due to polarization identity, we suffice to prove for ⟨x, A(BT B−zI)−1AT x⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Recall that we have (A(BT B − zI)−1AT )ij = (wAs(z) + w2 ABzs(z)s(z)2)δij + O≺(N −1/2) and (A(BT B − zI)−1BT )ij = (B(BT B − zI)−1AT )ij = (wABs(z) + wABzs(z)s(z)2)δij + O≺(N −1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Once the entrywise local law is given, the proof of the isotropic (or anisotropic) type law follows exactly as in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' To be more precisely, we can write ⟨x, A(BT B − zI)−1AT x⟩ = � i xi(AGAT )iixi + � i̸=j xi(AGAT )ijxj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Then the entrywise local law implies � i xi(AGAT )iixi − (wAs(z) + w2 ABzs(z)s(z)2)⟨x, x⟩ = � i x2 i � (AGAT )ii − (wAs(z) + w2 ABzs(z)s(z)2) � ≺ N −1/2, 49 and so the main difficulty is to control the off-diagonal part ZAB := � α,β � i̸=j xiAiαGαβAjβxj = O≺(N −1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' For instance, for the sample covariance matrix case ⟨x, BGBT x⟩ = (zs(z) + 1)⟨x, x⟩ + O≺(N −1/2) = (s(z) + zs(z)s(z)2)⟨x, x⟩ + O≺(N −1/2) was proved in [17] by proving the following bound for higher moments E|ZB|p ≺ N −p/2 (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='48) for any large and even p, where ZB := � i̸=j xi(BGBT )ijxj = z � i̸=j xiGijxj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' In particular, the (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='48) have proved by using the standard maximal expansion method in [17] and [2], which only requires the independence between each element, the boundedness of the moment of each entries, and the entrywise local law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Thus, from the definition of the entrywise correlated random matrices (A, B), it can be expected that E|ZAB|p ≺ N −p/2 also holds for any large and even p, by expanding maximally Gαβ instead of Gab as in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Then, we can conclude the proof by using Markov inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' To prove such an argument , we only need to check what is changing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' First, we express the p-th moment of ZAB by E|ZAB|p = E � b11̸=b12 · · � bp1̸=bp2 � � p/2 � k=1 xbk1(AGAT )bk1bk2xbk2 � � � � p � k=p/2+1 xbk1(AGAT )bk1bk2xbk2 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='49) Let T = {bk1} ∪ {bk2} be the set of indices of x appearing in the fixed summand of the representation of the p-th moment of ZAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Then our goal is to decompose the off-diagonal entry of the matrix (AGAT ) into the two parts by using Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='10, where one consists of the finite number of the maximally expanded term and the other consists of the terms containing a sufficiently large number of off-diagonal entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We note that the latter case is small enough due to the entrywise local laws of off-diagonal entries, and so the leading order term contained in the formal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Step 1 : The maximal expansion for the off-diagonal entries of (AGBA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' In our case, the maximally expanded terms (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4 of [17]) refer to terms that have one of the following forms: (AG(T\\a,b)AT )ab, (AG(T\\a,b)BT )ab, (BG(T\\a,b)AT )ab or (BG(T\\a,b)BT )ab = z(G(T\\a,b))ab, for some a ̸= b ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' To proceed, we use the following operation successively : Operation (a) Let T ⊂ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' M} be a set of indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' 50 For a ̸= b and c /∈ T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (AG(T )AT )ab = (AG(T c)AT )ab + (AG(T )BT )ac(BG(T )AT )cb z(G(T ))cc For a ̸= b and c /∈ T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (AG(T )BT )ab = (AG(T c)BT )ab + (AG(T )BT )ac(BG(T )BT )cb z(G(T ))cc = (AG(T c)BT )ab + (AG(T )BT )ac(G(T ))cb (G(T ))cc For a ̸= b /∈ T and a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' b ̸= c 1 z (BG(T )BT )ab = (G(T ))ab = (G(T c))ab + (G(T ))ac(G(T ))cb (G(T ))cc For a ̸= b /∈ T 1 (G(T ))aa = 1 (G(T b))aa − (G(T ))ab(G(T ))ba (G(T ))aa(G(T b))aa(G(T ))bb We then observe that the expanded terms contains at most two crossed terms (AG(T\\a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='b)BT )ab,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (BG(T\\a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='b)AT )ab and each expansions produce two types of terms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' the first one has one more additional upper index,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' and the second one at least one more additional off-diagonal entry of BGAT ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' AGBT or BGBT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Moreover, we also remark that the denominators are always the diagonal entries of the resolvent G(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' It can be seen that the above expansion formulas eventually play the same role as operation (a) in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Therefore, to obtain the desired decomposition, we only need to iterate the operation (a) until it can no longer be expanded or contains sufficiently many off-diagonal entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Step 2 : The further expansions for the maximally expanded off-diagonal entries We further expand the maximally expanded term by using the following operations : Operations (b) (and (c)) For a ̸= b ∈ T (G(T\\a,b))ab = z(G(T\\a,b))aa(G(T\\b))bb(BG(T)BT )ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Furthermore, we use the following type expansion, which is from the above formula, to the terms (G(T\\a,b))aa and (G(T\\a,b))bb (G(T\\a,b))aa = (G(T\\a))aa + (G(T\\a,b))ab(G(T\\a,b))ba (G(T\\a,b))bb = (G(T\\a))aa + z2(G(T\\a,b))aa(G(T\\a))aa(G(T\\b))bb(BG(T)BT )2 ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Then, this expansion splits such not-maximally expanded term into two parts, one is maximally ex- panded and the other is a monomial expressed as the product of itself, the diagonal entry, and the maximally expanded terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' In particular, it can be seen that the number of the off-diagonal entries included in the latter monomial increases by exactly two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' 51 For a ̸= b ∈ T (AG(T\\a,b)AT )ab = (AG(T)AT )ab + z(G(T\\b))bb(AG(T)BT )ab(BG(T)AT )bb + (AG(T\\a,b)BT )aa(BG(T\\a,b)AT )ab z(G(T\\a,b))aa = (AG(T)AT )ab + z(G(T\\b))bb(AG(T)BT )ab(BG(T)AT )bb + z(G(T\\a,b))aa(AG(T)BT )aa � (BG(T)AT )ab + z(G(T\\b))bb(BG(T)BT )ab � + z2(G(T\\a,b))aa(G(T\\b))bb(AG(T)BT )ab(BG(T)BT )ab× � (BG(T)AT )ab + z(G(T\\b))bb(BG(T)BT )ab � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' since (AG(T\\a,b)BT )aa = −z(G(T\\a,b))aa(AG(T\\b)BT )aa = −z(G(T\\a,b))aa � (AG(T)BT )aa + z(G(T\\b))bb(AG(T)BT )ab(BG(T)BT )ab � and (BG(T\\a,b)AT )ab = −z(G(T\\a,b))aa � (BG(T)AT )ab + z(G(T\\b))bb(BG(T)BT )ab � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The expansion of the first two monomials terminated since every term were maximally expanded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' After this, for any fixed positive integer ℓ, we expand the term which contains the term (G(T\\a,b))aa until the last term is a monomial containing ℓ or more off-diagonal entries by applying the first formula recursively to the not-maximally expanded diagonal entry (G(T\\a,b))aa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' For a ̸= b ∈ T (AG(T\\a,b)BT )ab = −z(G(T\\b))bb(AG(T)BT )ab + (AG(T\\a,b)BT )aa(G(T\\a,b))ab (G(T\\a,b))aa = −z(G(T\\b))bb(AG(T)BT )ab − z2(G(T\\a,b))aa(AG(T)BT )aa(G(T\\b))bb(BG(T)BT )ab − z3(G(T\\a,b))aa(G(T\\b))2 bb(AG(T)BT )ab(BG(T)BT )ab(BG(T)BT )ab Even in this case, we also expand the second and third monomials recursively by applying the first formula to not-maximally expanded diagonal entry (G(T\\a,b))aa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' In particular, we have two observations from the above operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The expansions of the maximally expanded off-diagonal entry consist of the monomials containing only an odd number of off-diagonal entries: (BG(T)AT )ab, (AG(T)BT )ab and (BG(T)BT )ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The diagonal entries (AG(T)BT )aa = (BG(T)AT )aa for a ∈ T, appear in the expanded term by implement- ing operation (b) and (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' These terms can be interpreted as a loop of the vertex a in the structure of the graph considered in [17], since like the maximally expended diagonal entry, these terms are com- parable to wABs(z) by the entrywise local law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Therefore, similar to the maximally expanded diagonal entry, terms of such types have no effect on the partial expectation techniques in subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='13 of [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' This part will be explained in more detail in the next step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' As with the previous step, from the explanations depicted in each expansion formula, we can see that the above expansions eventually play the same role as operations (b) and (c) in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' 52 Step 3 : The further expansions for the maximally expanded diagonal entries Finally, unless we end up with an expression that includes a sufficiently large numbers of off-diagonal resolvent entries (such trivial leaves are dealt with separately in Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='11 of [17]), we need to expand the maximally expanded diagonal elements (AG(T)BT )aa = (BG(T)AT )aa and (G(T\\a))aa for a ∈ T appearing in the non-trivial leaves (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='12 ∼ 14 of [17]), where we need to slightly adjust the proof to the setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' These terms corresponds to the maximally expanded diagonal G-edge in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' First, for c ∈ T, 1 (G(T\\c) B )cc = −z − z(BG(T) B BT )cc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='50) We note that |(G(T))µµ − s(z)| ≺ N −1/2 by following the proof of the entrywise local law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Using (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='50) and the facts zs(z)s(z) = −(zs(z) + 1) and |s(z)| ≍ 1, we see that 1 (G(T\\c))cc = 1 s(z) − z � (BG(T)BT )cc − s(z) � and this implies that (G(T\\c))cc = ℓ−1 � k=0 (s(z))k+1zk � (BG(T)BT )cc − s(z) �k + O≺(N −ℓ/2) for any integer ℓ ≥ 1 since (BG(T)BT )cc − s(z) is O≺(N −1/2), by using Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Similarly, for a ∈ T, we see that 1 (AG(T)BT )aa = 1 wABs(z) − (AG(T) B BT )aa − wABs(z) wABs(z)(AG(T) B BT )aa (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='51) and so (AG(T)BT )aa = wABs(z) − (AG(T)BT )aa wABs(z)−(AG(T)BT )aa wABs(z) 1 − wABs(z)−(AG(T)BT )aa wABs(z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' By using the estimate (AG(T)BT )aa − wABs(z) = � µ̸=ν Aaµ(G(T))µνBaν + � µ AaµBaµ � (G(T))µµ − s(z) � + s(z) � 1 N � µ (NAaµBaµ − wAB) � ≺ N −1/2, we have the following series expansion for any integers ℓ ≥ 1, (AG(T)BT )aa = wABs(z) − (AG(T)BT )aa1(ℓ ≥ 2) ℓ−1 � k=1 (wABs(z))−k � wABs(z) − (AG(T)BT )aa �k + O≺(N −ℓ/2) which corresponds to the term (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='42) in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' This way we end up with an expression where only contains the resolvent terms of the type (AG(T)AT )ab, (AG(T)BT )ab, (BG(T)AT )ab or (BG(T)BT )ab = (G(T))ab, for some a ̸= b ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' In other words, the x indices and the indices of the resolvent entries are completely decoupled;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' only explicit products of entries of (A, B) 53 represent the connections between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Step 4 : Sketch of the rest of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Through previous steps, for our case (AGAT ), we observed the modified version of the operations, which are done for the resolvents G and G in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' After with these modifications, it can be seen that the rest procedures (Step 6 ∼ 8 in [17]) of the proof for the non-trivial leaves with the stopping rule, which relies on the number of off-diagonal terms (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='7 of [17]), are also valid for the ZAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' More precisely, by using the entrywise laws and H¨older’s inequality, the same estimation also holds for the trivial leave as in Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Furthermore, the most of the finitely generated non-trivial leaves have a decay N −p/2 also by applying the same argument in the case of the trivial leaves (Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='12 in [17]), and the remaining leading order non-trivial leaves have the same decay by applying the partial expectation method (Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='13 in [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We conclude the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Proof of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' From the above version of an isotropic law, we also arrive at the isotropic version of the entrywise law in Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='8 by taking A = Q + X and B = Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Then, it is easy to check that wA = 2 � 1 + Eq � Vq � , wAB = 1 + Eq � Vq , wB = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Precisely, applying Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='12 directly, we see that 2⟨u, X(QT Q − zI)−1QT u⟩ = ⟨u, X(QT Q − zI)−1QT u⟩ + ⟨u, Q(QT Q − zI)−1XT u⟩ = ⟨u, A(BT B − zI)−1AT u⟩ − ⟨u, B(BT B − zI)−1BT u⟩ − ⟨u, X(BT B − zI)−1XT u⟩ = 2s(z) � 1 + Eq � Vq � + zs(z)s(z)2 � 1 + Eq � Vq �2 − s(z) − zs(z)s(z)2 − s(z) − zs(z)s(z)2 E2 q Vq = 2 Eq � Vq (s(z) + zs(z)s(z)2) = 2 Eq � Vq (zs(z) + 1) with O≺(N −φ) error terms, and it exactly matches the entrywise law since correlation wXQ = Eq √ Vq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Thus, we conclude that the improved PCA via the entrywise transform holds for the spike U s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' ∥U T U − Ik∥F , ∥U∥∞ ≺ N −φ, where φ > 1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' C Proof of CLTs In Appendix C, we prove the CLT for the LSS of spiked random matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The proof of the CLT for the LSS is based on the strategy of [6] in which the LSS is first written as a contour integral of the resolvent of a spiked Wigner matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Then, the averaged trace of the resolvent converges to a Gaussian process, which also implies that the limiting distribution of the LSS is Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' It is the biggest obstacle in adapting the proof in [6] for spiked matrices that the martingale CLT and covariance computation are hard to be reproduced with spikes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' even with the special choice of rank-1 spike the proof for the CLT is very tedious as in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' In [22], the interpolation between a general rank-1 spike 54 and the special rank-1 spiked was introduced to compare the LSS, based on an ansatz that the mean and the variance of the LSS do not depend on the choice of the spike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' In this paper, since we do not have a reference matrix to be compared with as in the rank-1 case, we introduce a direct interpolation between a spiked random matrices of general rank and a matrix without any spikes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' With the interpolation, we find the change of the mean in the limiting Gaussian distribution and also prove that its variance is invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1 Proof of CLTs for spiked random matrices Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We adapt the proof of Theorem 5 in [22] with the following change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Instead of inter- polating the spiked Wigner matrices M with the original signal and with the signal with all 1’s considered in [9], we directly interpolate M and W and track the change of the mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Consider the following interpolating matrix M(θ) = θ √ λUU T + W and the corresponding eigenvalues {µi(θ)}N i=1 of M(θ) for θ ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Let Γ be a rectangular contour in the proof of Theorem 5 in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Applying Cauchy’s integral formula, we have N � i=1 f(µi(1)) − N � 2 −2 √ 4 − x2 2π f(x) dx = − N 2πi � Γ f(z) � sN(1, z) − ssc(z) � dz (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1) where ssc(z) = −z+ √ z2−4 2 is the Stieltjes transform of the Wigner semicircle law and sN(θ, z) is the Stieltjes transform of the empirical spectral distribution (ESD) of M(θ) for θ ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Note that the normalized trace of the resolvent satisfies 1 N Tr R(θ, z) = 1 N N � i=1 1 µi(θ) − z = sN(θ, z) (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2) where R(θ, z) is the resolvent corresponding to M(θ), defined as R(θ, z) := (M(θ) − zI)−1 (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3) for z ∈ C+ and θ ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The change of the mean in the CLT for W and the CLT for M can be computed by tracking the change of the corresponding resolvent in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3), since (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1) can be decomposed by N � i=1 f(µi(1)) − N � 2 −2 √ 4 − x2 2π f(x) dx = − 1 2πi � Γ f(z) � Tr R(1, z) − Tr R(0, z) � dz (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4) − 1 2πi � Γ f(z) � Tr R(0, z) − Nssc(z) � dz (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5) and the fluctuation result of (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5) is already given in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Set Γε = {z ∈ C : minw∈Γ |z − w| ≤ ε}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Choose ε so that min w∈Γε,x∈[−2,2] |x − w| > 2ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' 55 Following the proof of Theorem 5 in [22], on z ∈ Γε 1/2 := Γε ∩ {z ∈ C : |Imz| > N −1/2}, we first find that ∂ ∂θ Tr R(θ, z) = − k � m=1 √ λ ∂ ∂z � x(m)T R(θ, z)u(m) � = −k ∂ ∂z � √ λssc(z) 1 + θ √ λssc(z) � + O(N − 1 2 ) = − k √ λs′ sc(z) (1 + θ √ λssc(z))2 + O(N − 1 2 ) (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='6) with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' More precisely, since the elementary resolvent expansion implies R(0, z) − R(θ, z) = θ √ λR(θ, z) � k � ℓ=1 u(ℓ)u(ℓ)T � R(0, z), (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='7) we then find that � u(m)T R(0, z)u(m) � = � u(m)T R(θ, z)u(m) � + θ √ λ k � ℓ=1 � u(m)T R(θ, z)u(ℓ) � � u(ℓ)T R(0, z)u(m) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' From the rigidity of the eigenvalues, we have a deterministic bound for resolvent | � u(m)T R(θ, z)u(ℓ) � | ≤ ∥R(θ, z)∥ ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='8) Since columns of spike {u(ℓ)}k ℓ=1 are orthonormal, the isotropic local law for R(0, z) implies that � u(m)T R(0, z)u(ℓ) � = s(z)δmℓ + O(N −1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='9) uniformly on z ∈ Γε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We then obtain that � u(m)T R(0, z)u(m) � = � u(m)T R(θ, z)u(m) � � 1 + θ √ λ � u(m)T R(0, z)u(m) �� + O(N − 1 2 ) and so � u(m)T R(θ, z)u(m) � = ssc(z) 1 + θ √ λssc(z) + O(N − 1 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' This proves (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Moreover, on Γε, we easily check that the exactly same argument holds for a finite rank perturbation of Wigner matrix (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' interlacing and rigidity properties).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Thus, we conclude that (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4) is k 2πi � Γ √ λs′ sc(z) 1 + √ λssc(z) f(z)dz + o(1) with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Finally, following the computation in the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4 in [9], we then find that the difference between the LSS of M and the LSS of W is k ∞ � ℓ=1 √ λℓτℓ(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='10) This proves the desired theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' 56 Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The proof of the CLT for the spiked rectangular matrices is quite similar to the case of spiked Wigner matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We first consider the interpolating matrix for the additive model, defined as Y (θ) = θ √ λUV T + X (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='11) for θ ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Note that Y (0) = X and Y (1) = Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Denote by µ1(θ) ≥ µ2(θ) ≥ · · · ≥ µM(θ) the eigenvalues of Y (θ)Y (θ)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We also define the resolvent G(θ, z) = (Y (θ)Y (θ)T − zI)−1, G(θ, z) = (Y (θ)T Y (θ) − zI)−1 (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='12) for z ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We choose (N-independent) constants a− < d−, a+ > d+, and v0 ∈ (0, 1) so that the function f is analytic on the rectangular contour Γ whose vertices are (a− ± iv0) and (a+ ± iv0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' With overwhelming probability, all eigenvalues of Y (θ)Y (θ)T are contained in Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Applying Cauchy’s integral formula, we find that M � i=1 f(µi(1)) − M � i=1 f(µi(0)) = − � 1 2πi � Γ f(z) (Tr G(1, z) − Tr G(0, z)) dz � (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='13) To estimate the difference Tr G(1, z) − Tr G(0, z), we consider its derivative ∂ ∂θ Tr G(θ, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Note that ∂Gab(θ) ∂Yij(θ) = −Gai(θ)(Y (θ)T G(θ))jb − (G(θ)Y (θ))ajGib(θ), dYij(θ) dθ = √ λuivT j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='14) Thus, by chain rule ∂ ∂θ Tr G(θ, z) = M � a=1 M � i=1 N � j=1 ∂Yij(θ) ∂θ ∂Gaa(θ) ∂Yij(θ) = − M � a=1 M � i=1 N � j=1 √ λuivT k [Gai(θ)(Y (θ)T G(θ))ja + (G(θ)Y (θ))ajGia(θ)] = −2 M � a=1 M � i=1 N � j=1 M � b=1 √ λuivT j [Ybj(θ)Gba(θ)Gai(θ)] (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='15) From the fact � ∂ ∂z G(θ) � bi = (G(θ)2)bi = � a Gba(θ)Gai(θ), we then find that ∂ ∂θ Tr G(θ, z) = −2 √ λ ∂ ∂z M � i=1 N � j=1 uivT j (G(θ)Y (θ))ij = −2 √ λ ∂ ∂z k � ℓ=1 ⟨u(ℓ), G(θ)Y (θ)v(ℓ)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='16) It remains to estimate ∂ ∂z⟨u(ℓ), G(θ)Y (θ)v(ℓ)⟩ for 1 ≤ ℓ ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We suffices to estimate the desired term for fixed ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' From now, we omit ℓ-dependency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Note that ⟨u, G(θ)Y (θ)v⟩ = θ √ λ⟨u, G(θ)u⟩ + ⟨u, G(θ)Xv⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' 57 We consider the resolvent expansion G(0, z) − G(θ, z) = G(θ, z) (H(θ) − H(0)) G(0, z) = G(θ, z) (θ2λuuT + θ √ λXvuT + θ √ λuvT XT ) G(0, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='17) Taking inner products with u and v, we obtain ⟨u, G(0)u⟩ = ⟨u, G(θ)u⟩ + θ2λ⟨u, G(θ)u⟩⟨u, G(0)u⟩ + θ √ λ⟨u, G(θ)Xv⟩⟨u, G(0)u⟩ + θ √ λ⟨u, G(0)Xv⟩⟨u, G(θ)u⟩ (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='18) and ⟨u, G(0)Xv⟩ = ⟨u, G(θ)Xv⟩ + θ2λ⟨u, G(θ)Xv⟩⟨u, G(0)Xv⟩ + θ √ λ⟨u, G(θ)Xv⟩⟨u, G(0)Xv⟩ + θ √ λ⟨v, XT G(0)Xv⟩⟨u, G(θ)u⟩, (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='19) where we omitted z-dependence for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We then use the following result to control the terms in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='18) and (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Recall the definition of s(z) and s(z) in Lemmas B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4 and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Moreover, we consider the same linearization HX(z) of the matrix X and its inverse RX(z) = HX(z)−1 as in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='33) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1 (Isotropic local law).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' For an N-independent constant ε > 0, let Γε be the ε-neighborhood of Γ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=', Γε = {z ∈ C : min w∈Γ |z − w| ≤ ε}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Choose ε small so that the distance between Γε and [d−, d+] is larger than 2ε, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=', min w∈Γε,x∈[d−,d+] |x − w| > 2ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='20) Then, for any unit vectors x, y ∈ CM+N independent of X, |⟨x, (RX(z) − Π(z))y⟩| ≺ N −1/2, (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='21) uniformly on z ∈ Γε, where Π(z) = �s(z) · IM 0 0 zs(z) · IN � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='22) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' See Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='6, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='7, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='9, and Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='10 in [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Note that Im s(z), Im s(z) = Θ(η) on the vertical part of Γε, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=', the neighborhood of the line segment joining (a++iv0) and (a+−iv0) (respectively (a− + iv0) and (a− − iv0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Set A := ⟨u, G(0, z)u⟩, B := ⟨u, G(0, z)Xv⟩, C := ⟨v, XT G(0, z)Xv⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Recall that RX(z) = � G(0, z) G(0, z)X XT G(0, z) zG(0, z) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='23) Then, as consequences of Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1 with appropriate choices of the deterministic vectors, A = s(z) + O≺(N −1/2), C = ⟨v, zG(0, z)v⟩ + 1 + O(N −1/2) = d0(zs(z) + 1) + O≺(N −1/2), (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='24) 58 and B = O≺(N −1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We thus have from (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='18) and (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='19) that ⟨u, G(θ)Xv⟩ = −θd0 √ λs(z)(zs(z) + 1) θ2λzs(z) + θ2λ + 1 + O≺(N −1/2) ⟨u, G(θ)u⟩ = s(z) θ2λzs(z) + θ2λ + 1 + O≺(N −1/2) (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='25) and hence ⟨u, G(θ)Y (θ)v⟩ = θ √ λ⟨u, G(θ)u⟩ + ⟨u, G(θ)Xv⟩ = θ √ λzs(z) + θ √ λ θ2λzs(z) + θ2λ + 1 + O≺(N −1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='26) Note that this estimate is uniform on θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Differentiating it with respect to z and plugging it back to (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='16), we get ∂ ∂θ Tr G(θ, z) = −k 2θλ d dz(zs(z) + 1) (θ2λzs(z) + θ2λ + 1)2 + O≺(N −1/2) and, integrating over θ, we obtain Tr G(1, z) − Tr G(0, z) = � 1 0 ∂ ∂θ Tr G(θ, z)dθ = −k d dzλ(zs(z) + 1) λzs(z) + λ + 1 + O≺(N −1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='27) We now invoke the following relation between the Stieltjes transforms for Marchenko–Pastur law and the Wigner semicircle law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Let ssc(z) = −z + √ z2 − 4 2 be the Stieltjes transform of the Wigner semicircle law and ϕ(z) = 1 √d0 (z − (1 + d0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Then � d0(zs(z) + 1) = ssc(ϕ(z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='28) We thus have 1 2πi � Γ f(z)λ d dz(zs(z) + 1) λzs(z) + λ + 1 dz = 1 2πi � Γ �f(ϕ(z)) λs′ sc(ϕ(z))ϕ′(z) λssc(ϕ(z)) + √d0 dz = 1 2πi � �Γ �f(ϕ) λs′ sc(ϕ) λssc(ϕ) + √d0 dϕ (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='29) where we let f(√d0z + 1 + d0) = �f(z) and �Γ = ϕ(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (Note that �Γ contains the interval [−2, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=') So far, we have proved that M � i=1 f(µi(1)) − M � i=1 f(µi(0)) = k 2πi � �Γ �f(ϕ) λs′ sc(ϕ) λssc(ϕ) + √d0 dϕ + O≺(N −1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='30) Since the difference in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='30) is the sum of a deterministic term and a random term stochastically dominated 59 by N −1/2, we can see that the CLT holds for the LSS with the non-null model Y (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Moreover, the variance is the same as that of the null model, which is VY (f) = 2 ∞ � ℓ=1 ℓτℓ( �f)2 + (w4 − 3)τ1( �f)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='31) (See, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=', [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=') The change of the mean is the first term in the right side of (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='30), which can be computed by following the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4 in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We obtain mY (f) = �f(2) + �f(−2) 4 − 1 2τ0( �f) + (w4 − 3)τ2( �f) + k ∞ � ℓ=1 � λ √d0 �ℓ τℓ( �f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='32) This proves the first part of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2 for the additive model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' For the multiplicative model, we will follow the same strategy as in the additive model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Let Y (θ) = X + θγUU T X (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='33) for θ ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Note that Y (0) = X and Y (1) = Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We denote by µ1(θ) ≥ µ2(θ) ≥ · · · ≥ µM(θ) the eigenvalues of Y (θ)Y (θ)T , and also let G(θ, z) = (Y (θ)Y (θ)T − zI)−1, G(θ, z) = (Y (θ)T Y (θ) − zI)−1 (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='34) for z ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We have the relations ∂Gab(θ) ∂Yij(θ) = −Gai(θ)(Y (θ)T G(θ))jb − (G(θ)Y (θ))ajGib(θ), ∂Yij(θ) ∂θ = γ M � c=1 uiuT b Xbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='35) Following (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='15)-(C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='16), we get ∂ ∂θ Tr G(θ, z) = −γ M � a=1 M � i=1 N � j=1 M � b=1 uiuT b Xbj[Gai(θ)(Y (θ)T G(θ))ja + (G(θ)Y (θ))ajGia(θ)] = −2γ M � a=1 M � i=1 N � j=1 M � b=1 uiuT b Xbj[(Y (θ)T G(θ))jaGai(θ)] = −2γ ∂ ∂z M � i=1 N � j=1 M � b=1 uiuT b Xbj(G(θ)Y (θ))ij = −2γ ∂ ∂z k � ℓ=1 ⟨u(ℓ), G(θ)Y (θ)XT u(ℓ)⟩ = −2γ ∂ ∂z k � ℓ=1 ⟨u(ℓ), G(θ)Y (θ)Y (0)T u(ℓ)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='36) Moreover, since Y (0) = X = (I + θγUU T )−1Y (θ) = � I − θγ 1 + θγ UU T � Y (θ), (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='37) 60 we have ⟨u(ℓ), G(θ)Y (θ)Y (0)T u(ℓ)⟩ = ⟨u(ℓ), G(θ)Y (θ)Y (θ)T (I + θγUU T )−1u(ℓ)⟩ = ⟨u(ℓ), (I + zG(θ))(I + θγUU T )−1u(ℓ)⟩ = 1 1 + θγ + z 1 + θγ ⟨u(ℓ), G(θ)u(ℓ)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='38) To estimate the term ⟨u(ℓ), G(θ)u(ℓ)⟩, we use the following Anisotropic local law in [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2 (Anisotropic local law).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Let Γε be the ε-neighborhood of Γ as in Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Then, for any unit vectors x, y ∈ CM independent of X, the following estimate holds uniformly on z ∈ Γε : ���� � x, � G(θ, z) + � zI + zs(z)(I + θγUU T )2�−1� y ����� ≺ N − 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='39) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The proof of Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2 is the same as that of Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Now, as in the additive case, we drop the ℓ-dependency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' From Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2, we find that ⟨u, G(θ)u⟩ = − � u, � zI + zs(z)(I + θγUU T )2�−1 u � + O(N −1/2) = − 1 (1 + θγ)2z(1 + s(z)) + O(N −1/2), (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='40) and plugging it into (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='38), we obtain ⟨u, G(θ)Y (θ)Y (0)T u⟩ = 1 1 + θγ − 1 (1 + θγ)(1 + (1 + θγ)2s(z)) + O(N −1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='41) We thus get ∂ ∂θ Tr G(θ, z) = −2kγ (1 + θγ)s′(z) (1 + (1 + θγ)2s(z))2 + O(N −1/2), (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='42) and integrating it yields Tr G(1, z) − Tr G(0, z) = −k λs′(z) (1 + s(z))(1 + (1 + λ)s(z)) + O(N −1/2) = −λk d dz(zs(z) + 1) λzs(z) + λ + 1 + O(N −1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='43) Since (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='43) coincides with (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='27), the rest of the proof is exactly the same as in the additive case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' This finishes the proof of the first part of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2 Proof of CLTs for entrywise transformed matrices Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We adapt the proof of Theorem 7 in [22] with the following changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Let S be the variance matrix of the transformed matrix � M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We then find that Sij = E[� M 2 ij] − (E[� Mij])2 = 1 N + λ(GH − Fg)(uiuT j )2 + O(N 1−8φ) and Sii = E[� M 2 ii] − (E[� Mii])2 = w2 N + λ(Gg,d − Fg,d)(uiuT i )2 + O(N 1−8φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' 61 Normalizing and centering each entry of the matrix � M, we arrive at another Wigner matrix � W where � Wij = 1 � NSij (� Mij − E� Mij), � Wii = � w2 NSii (� Mii − E� Mii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Interpolating � W and � M − E[� M] by � W(θ) = (1 − θ)� W + θ(� M − E[� M]), � W(θ) is a general Wigner-type matrix with the corresponding quadratic vector equation − 1 mi(θ, z) = z + N � j=1 E[� Wij(θ)2] · mj(θ, z) where mi(θ, z)δij is the limiting distribution of the (i, j)-element of the resolvent R � W (θ, z) = (� W(θ) − zI)−1 for 0 ≤ θ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Recall the ssc(z) is the Stieltjes transform of the Wigner semicircle law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We also directly check that mi(θ, z) = ssc(z) + C1(uiuT i ) + C2N −1 = ssc(z) + O(N −2φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Moreover, the anisotropic local law for the general Wigner-type matrix in [2] implies that uniformly on z ∈ Γε 1/2 (u(m)T R � W (θ, z)u(ℓ)) = ssc(z)δmℓ + O(N −1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Following the proof of Lemmas B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2 and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3 in [22], we check that Uniformly on z ∈ Γε 1/2, Tr R � W (1, z) − Tr R � W (0, z) = kλ(GH − Fg)s′ sc(z)ssc(z) + O(N 1N −4φ) (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='44) Uniformly on z ∈ Γε\\Γε 1/2, | Tr R � W (1, z) − Tr R � W (0, z)| = O(N 1N −2/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='45) Compared with the bound shown in [22], we give the following remark: The error bound in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='44) is better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' This sharper bound can be obtained by using the fact � a ua(ℓ)2 = 1 instead of ��� a ua(ℓ)2�� ≤ N∥u(ℓ)∥2 ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Our next step is to consider � M = � W(1) + E[� M].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Since � M = � W(1) + � λFgUU T + diag(d1, · · · , dN) + E where di = E[� Mii] − � λFg(UU T )ii, we then find that Tr(� M − zI)−1 − Tr R � W (0, z) = kλ(GH − Fg)s′ sc(z)ssc(z) − k � λFgs′ sc(z) 1 + � λFgssc(z) − k √ λ( � Fg,d − � Fg)s′ sc(z) + O(N −1/2) uniformly on z ∈ Γε 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Thus, we obtain the desired CLT by applying Cauchy’s integral formula as in the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' 62 Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Since the proof of the transformed CLT for the spiked Wigner matrix follows the proof in [22], we only describe the process briefly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' On the other hand, there is no technical reference for the spiked rectangular matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' As we mentioned before, our consideration is only the additive case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We consider the optimal entrywise transformation defined by a function h(w) := −g′(w) g(w) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='46) If λ = 0, it is immediate to see that for all i, j E[h( √ NYij)] = � ∞ −∞ h(w)g(w)dw = − � ∞ −∞ g′(w)dw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Further, with λ = 0, as shown in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2 of [50], Fg := E[h( √ NYij)2] = � ∞ −∞ h(w)2g(w)dw = � ∞ −∞ g′(w)2 g(w) dw ≥ 1, (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='47) where the equality holds if and only if √ NXij is a standard Gaussian (hence h(w) = w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We define a transformed matrix �Y as follows: the terms of �Y are defined by �Yij = 1 � FgN h( √ NYij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='48) Note that the entries of �Y are independent up to symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Since g is smooth, h is also smooth and all moments of √ N �Yij are O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Thus, applying a high-order Markov inequality, it is immediate to find that �Yij = O(N − 1 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1 Decomposition of the transformed matrix We first estimate the mean and the variance of entry by using the comparison method with the pre- transformed entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' For all i, j, we find that E[�Yij] = 1 � FgN � ∞ −∞ h(w)g � w − √ NλuivT j � dw = − 1 � FgN � ∞ −∞ g′(w) g(w) � g � w − √ NλuivT j � − g(w) � dw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='49) In the Taylor expansion g � w − √ NλuivT j � − g(w) = 4 � ℓ=1 g(ℓ)(w) ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' � − √ NλuivT j �ℓ + g(5) � w − θ √ NλuivT j � 5!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' � − √ NλuivT j �5 (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='50) 63 for some θ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Note that the second term and the fourth term in the summation are even functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Since g′/g is an odd function, we find that E[�Yij] = 1 � Fg √ λuivT j � ∞ −∞ g′(w)2 g(w) dw + C3N �√ λuivT j �3 + O(N 2(uivT j )5) = � λFguivT j + C3N �√ λuivT j �3 + O(N 2(uivT j )5) (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='51) for some (N-independent) constant C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Similarly, since � g′ g �2 is even, E[�Y 2 ij] = 1 FgN � ∞ −∞ �g′(w) g(w) �2 g � w − √ NλuivT j � dw = 1 N + 1 FgN � ∞ −∞ �g′(w) g(w) �2 � g � w − √ NλuivT j � − g(w) � dw = 1 N + 1 2Fg �√ λuivT j �2 � ∞ −∞ g′(w)2g′′(w) g(w)2 dw + O(N(uivT j )4) = 1 N + λGH(uivT j )2 + O(N(uivT j )4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='52) where GH = 1 2Fg � ∞ −∞ g′(w)2g′′(w) g(w)2 dw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The evaluation of the mean and the variance shows that the transformed matrix �Y is not a spiked rectangular matrix when λ > 0, since the variances of the entries are not identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Our strategy is to approximate �Y as a spiked generalized rectangular Gram matrix for which the variances of the each entries is 1/N in high-dimensional regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Let S be the variance matrix of �Y defined as Sij = E[�Y 2 ij] − (E[�Yij])2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='53) From (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='51) and (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='52), Sij = 1 N + (GH − Fg) �√ λuivT j �2 + O(N∥U∥4 ∞∥V ∥4 ∞), (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='54) which shows that �Y is indeed approximately a spiked generalized Gram matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2 CLT for a random Gram matrix We use the local law for general rectangular Gram matrices in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Consider an another M × N rectangular matrix A = (Aij) defined by Aij = 1 � NSij (�Yij − E[�Yij]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='55) Note that E[Aij] = 0, E[A2 ij] = 1 N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Then the matrix A is a usual rectangular matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We set GA(z) = (AAT − zI)−1 (z ∈ C+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='56) 64 Next, we introduce an interpolation for A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' For 0 ≤ θ ≤ 1, we define a matrix A(θ) by Aij(θ) = (1 − θ)Aij + θ(�Yij − E[�Yij]) = � 1 − θ + θ � NSij � Aij = � 1 + θNλ(GH − Fg)(uivT j )2 2 + O(N 2(uivT j )4) � Aij (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='57) Note that A(0) = A and A(1) = �Y − E[�Y ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' For 0 ≤ θ ≤ 1, A(θ) is a random Gram matrix considered in [4] satisfying the conditions (A)–(D) therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Moreover, if we let GA(θ, z) = (A(θ)A(θ)T − zI)−1 (z ∈ C+) (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='58) and Sij(θ) = E[Aij(θ)2], then Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='7 of [4] asserts that the limiting distribution of GA ij(z) is si(z)δij, where si(θ, z) is the unique solution to the system of quadratic vector equations − 1 si(θ, z) = z + N � j=1 Sij(θ) zsj(θ, z) (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='59) and − 1 sj(θ, z) = z + M � i=1 Sij(θ) zsi(θ, z) (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='60) Remark C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Recall that s(z) is the Stieltjes transform of the Marchenko-Pastur measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We can then find that si(θ, z) = s(z)+C1(uiuT i )+C2N −1 = s(z)+O(N −1/2) and sj(θ, z) = s(z)+C1(vjvT j )+C2N −1 = s(z) + O(N −1/2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' see also Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='9 of [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' For the resolvent GA(θ, z), we will use the following lemma for the random Gram matrix: Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4 (Anisotropic local law for random Gram matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Let Γε be the ε-neighborhood of Γ as in Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Then, for any deterministic x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' , xM), y = (y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' , yM) ∈ CM with ∥x∥ = ∥y∥ = 1, the following estimate holds uniformly on z ∈ Γε ∩ {z ∈ C+ : Im z > N − 1 2 }: ������ M � i=1 M � j=1 xiGA ij(θ, z)yj − M � i=1 si(θ, z)xiyi ������ = O(N − 1 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='61) and, for any deterministic x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' , xN), y = (y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' , yN) ∈ CN with ∥x∥ = ∥y∥ = 1, ������ N � i=1 N � j=1 xiGA ij(θ, z)yj − N � i=1 si(θ, z)xiyi ������ = O(N − 1 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='62) Proof of Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Let Ψ(z) = � 1 M Im z be the control parameter for the random gram matrix model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We then note that the bound for the entrywise local law is N −1/2 since Ψ(z) ≺ N −1/2 on Γε ∩ {z ∈ C+ : Im z > N − 1 2 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' With the entrywise local law in [4], the proof of the anisotropic law exactly follows the maximal 65 expansion argument used in [2, 17] and Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We consider the following decomposition of (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='61): M � i̸=j xiGA ij(θ, z)xj + M � i=1 (GA ii − si(θ, z))xixi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='63) From now, we drop A, θ and z-dependencies for brevity and use the linearization matrix HA(θ)(z) ≡ H and its inverse R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Then, in usual, we suffices to prove that Z ≡ � a̸=b xaRabxb ≺ N −1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' To prove the above high probability bound, we will bound the 2p-moments E[|Z|2p] ≺ N −p/2 by deriving the maximally expanded form via the resolvent identity in Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Now, we will check the representation of the maximally expanded diagonal resolvent elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' R(B\\b) bb , b ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Together with Remark C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3, we then conclude that the standard argument in [2] is valid for our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' By applying Shur’s complement lemma and (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='59), for b ∈ B, 1 R(B\\b) bb = −z − (B) � α,β HbαR(B) αβ Hβb = 1 sb(θ, z) + � β Sbβ (zsβ(θ, z)) − (B) � α,β HbαR(B) αβ Hβb = 1 sb(θ, z) − (B) � β (HbβR(B) ββ Hβb − Sbβ (zsβ(θ, z))) − (B) � α̸=β HbαR(B) αβ Hβb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='64) Then (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='64) and the analogue representation of R(B\\β) ββ for β ∈ B replace the (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2) in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' With linearization H and its inverse R, one useful by-product of the above argument is ⟨x, GA(θ)A(θ)y⟩ = � a � α xaRaαyα ≺ N −1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='65) Note that our model satisfies the closeness condition (A3) of Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2 in [3] (See also Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4 therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' On Γ\\Γε 1/2, we use the following results on the rigidity of eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5 (Rigidity of eigenvalues for the random Gram matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Denote by µA 1 (θ) ≥ µA 2 (θ) ≥ · · · ≥ µA M(θ) the eigenvalues of A(θ)A(θ)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Let γi be the classical location of the eigenvalues with respect to the Marchenko-Pastur measure defined by � ∞ γi ρMP,d0(dx) = 1 M � i − 1 2 � (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='66) for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' , M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Then, |µA i (θ) − γi| = O(M −2/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='67) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Note that the rigidity of the eigenvalues with an error of at most O(M −2/3) holds for random gram matrices at the classical location of the eigenvalues with respect to the probability measure ρ from the Stieltjes transform sρ(z) := 1 M � i si(z), see Lemma 4 in [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Moreover, since |si(θ, z) − s(z)|, |sj(θ, z) − s(z)| = 66 O(M −2φ) for all i and j, we also have the desired rigidity near the classical location of Marchenko-Pastur law ρMP,d0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Remark C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' In fact, rigorous proofs of the rigidity and anisotropic law are not given in [4, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' However, as in the proof of anisotropic local law for general Wigner-type matrix in [2], the above lemmas may be proved by using the local laws in [4] and standard methods in [2] (Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='10 in [4] and Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='7 in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=') On Γε 1/2, as a simple corollary to Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4, we obtain ��⟨x, GA(θ, z)y⟩ − s(z)⟨x, y⟩ �� = O(N − 1 2 ), (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='68) and ��⟨x, GA(θ, z)y⟩ − s(z)⟨x, y⟩ �� = O(N − 1 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='69) We have the following lemma for the difference between Tr GA(0, z) and Tr GA(1, z) on Γε 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Let GA(θ, z) be defined as in Equations (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='57) and (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='58).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Then, the following holds uniformly for z ∈ Γε 1/2: Tr GA(1, z) − Tr GA(0, z) = −λ(GH − Fg)k ∂ ∂z (zs(z) + 1) + O(N∥U∥2 ∞∥V ∥2 ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='70) We will prove Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='7 later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' From Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5, we find that | Tr GA(1, z) − Tr GA(0, z)| = ����� N � i=1 � 1 µA i (1) − z − 1 µA i (0) − z ������ = ����� N � i=1 µA i (0) − µA i (1) (µA i (1) − z)(µA i (0) − z) ����� ≤ ����� N � i=1 |µA i (0) − γi| + |γi − µA i (1)| (µA i (1) − z)(µA i (0) − z) ����� = O(N 1/3) (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='71) uniformly for z ∈ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Thus, from (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='70) and (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='71), 1 2πi � Γ f(z) Tr GA(1, z)dz − 1 2πi � Γ f(z) Tr GA(0, z)dz = 1 2πi � Γε 1/2 f(z) � Tr GA(1, z) − Tr GA(0, z) � dz + 1 2πi � Γ\\Γε 1/2 f(z) � Tr GA(1, z) − Tr GA(0, z) � dz = −λ(GH − Fg)k 2πi � Γε 1/2 f(z) ∂ ∂z (zs(z) + 1)dz + O(N∥U∥2 ∞∥V ∥2 ∞) + O(N −1/6) = −λ(GH − Fg)k 2πi � Γ f(z) ∂ ∂z (zs(z) + 1)dz + o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='72) Furthermore, using the relation (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='28), we have 1 2πi � Γ f(z) ∂ ∂z (zs(z) + 1)dz = 1 2πi � Γ f(z) 1 √d0 s′ sc(ϕ(z))ϕ′(z)dz = 1 2√d0πi � �Γ �f(ϕ)s′ sc(ϕ)dϕ = 1 √d0 τ1( �f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' 67 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3 CLT for a random Gram matrix with a spike and small perturbation Recall that A(1) = �Y − E[�Y ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Our next step in the approximation is to consider �Y = A(1) + E[�Y ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Since E[�Y ] is not a matrix of rank k, we instead consider B(θ) = A(1) + θ � λFgUV T , GB(θ, z) = (B(θ)B(θ)T − zI)−1 (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='73) To prove this part of CLT, we will adapt the strategy for the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5 with Lemmas C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4 and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We then find that, uniformly for z ∈ Γε 1/2, Tr GB(1, z) − Tr GB(0, z) = −k d dzλFg(zs(z) + 1) λFgzs(z) + λFg + 1 + O≺(N −φ), (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='74) since ∥U T U − Ik∥F , ∥V T V − Ik∥F ≺ N −φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Using the rigidity (Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5) and the eigenvalue interlacing property, we have Tr GB′(1, z) − Tr GB′(0, z) = O(1) on Γ\\Γ1/2 and so 1 2πi � Γ f(z) Tr GB(1, z)dz − 1 2πi � Γ f(z) Tr GB(0, z)dz = − k 2πi � Γ f(z) λFg d dz(zs + 1) λFg(zs + 1) + 1dz + o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='75) The remaining part is to control an effect of small perturbation (E[�Y ] − � λFgUV T )ij = CN(uivT j )3 + O(N 2(uivT j )5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' First, we let B′ = B(1) + CN(uivT j )3, GB′(z) = (B′(B′)T − zI)−1 (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='76) For 1 ≤ ℓ1, ℓ2, ℓ3 ≤ k, we consider vectors u3 and v3 such that (u3(ℓ1, ℓ2, ℓ3))i = u3 i (ℓ1, ℓ2, ℓ3) := √ Nui(ℓ1)ui(ℓ2)ui(ℓ3) and (v3(ℓ1, ℓ2, ℓ3))j = v3 j (ℓ1, ℓ2, ℓ3) := √ Nvj(ℓ1)vj(ℓ2)vj(ℓ3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We then observe that B′ contains k3 additional small spikes: B′ = B(1) + C � ℓ1,ℓ2,ℓ3 u3(ℓ1, ℓ2, ℓ3)v3(ℓ1, ℓ2, ℓ3)T where ∥u3(ℓ1, ℓ2, ℓ3)∥∞, ∥v3(ℓ1, ℓ2, ℓ3)∥∞ ≺ N 1/2−3φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' In the above point of view, we are able to consider B′ as another spiked Gram matrix model with two types of spikes u(ℓ)v(ℓ)T and u3(ℓ1, ℓ2, ℓ3)(v3(ℓ1, ℓ2, ℓ3))T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' As before, for 0 ≤ θ ≤ 1, let B′(θ) = A(1) + θ � λFg � ℓ u(ℓ)v(ℓ)T + θC � ℓ1,ℓ2,ℓ3 u3(ℓ1, ℓ2, ℓ3)(v3(ℓ1, ℓ2, ℓ3))T and GB′(θ, z) = (B′(θ)B′(θ)T − zI)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' 68 Following the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5, we have ∂ ∂θ Tr GB′(θ, z) = −2 � λFg ∂ ∂z � ℓ ⟨u(ℓ), GB′(θ, z)B′(θ)v(ℓ)⟩ − 2C ∂ ∂z � ℓ1,ℓ2,ℓ3 ⟨u3(ℓ1, ℓ2, ℓ3), GB′(θ, z)B′(θ)v3(ℓ1, ℓ2, ℓ3)⟩, and it can be observed that the first term of the right-hand side is the leading order term, since ∥u3(ℓ1, ℓ2, ℓ3)∥∞, ∥v3(ℓ1, ℓ2, ℓ3)∥∞ ≺ N 1/2−3φ < N −φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Moreover, from the definition of B′(θ), the leading order term of ⟨u(ℓ), GB′(θ, z)B′(θ)v(ℓ)⟩ is ⟨u(ℓ), GB′(θ, z)B′(0)v(ℓ)⟩+θ � λFg⟨u(ℓ), GB′(θ, z)u(ℓ)⟩, since ⟨v(ℓ1), v(ℓ2)⟩ = δℓ1ℓ2+O(N −φ), ⟨v(ℓ), v3(ℓ1, ℓ2, ℓ3)⟩ = O(N 1/2−2φ) and ⟨v3(ℓ1, ℓ2, ℓ3), v3(ℓ4, ℓ5, ℓ6)⟩ = O(N 1−4φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Carrying out the remaining procedures presented in the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5 and collecting the leading order terms, we eventually obtain Tr GB′(1, z) − Tr GB′(0, z) = −k d dzλFg(zs(z) + 1) λFgzs(z) + λFg + 1 + O≺(N 1/2−2φ) (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='77) uniformly for z ∈ Γε 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Here, we last apply Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4 and (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='65) for B′(0) = A(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Further, on Γ\\Γ1/2, from the rigidity and the interlacing property of the eigenvalues, Tr GB′(1, z) − Tr GB′(0, z) = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='78) Thus, we conclude that 1 2πi � Γ f(z) Tr GB′(z)dz − 1 2πi � Γ f(z) Tr GB′(0, z)dz = − k 2πi � Γ f(z) λFg d dz(zs + 1) λFg(zs + 1) + 1dz + o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Furthermore, we set Eij = (�Y − B′)ij = O(N 2(uivT j )5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Then ∥E∥ ≤ ∥E∥F = O(N 2∥U∥4 ∞∥V ∥4 ∞) = o(N −1) (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='79) for some φ > 3/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' This implies that 1 2πi � Γ f(z) Tr G �Y (z)dz − 1 2πi � Γ f(z) Tr GB′(1, z)dz = o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Remark C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Under the assumption that φ > 3/8, we suffices to consider E[�Yij] up to O(N 2(uivT j )5) error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' However, (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='77) and (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='78) are valid for any finite approximation of E[�Y ] as presented in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='51), even for any φ > 1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' This means that the condition φ > 3/8 can be improved by considering a higher order expansion of E[�Y ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' For example, if we consider E[�Y ] = � λFguivT j + C1N(uivT j )3 + C2N 2(uivT j )5 + O(N 3(uivT j )7), then it can be checked that the contributions of the second and third terms are negligible, and the error Eij = O(N 3(uivT j )7) is also negligible if φ > 1/3, since ∥E∥ ≤ ∥E∥F = O(N 3∥U∥6 ∞∥V ∥6 ∞) ≺ N 3−12φ = o(N −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' 69 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4 Conclusion for the proof of pre-transformed CLT We are now ready to prove pre-transformed CLT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Denote by �µ1 ≥ �µ2 ≥ · · · ≥ �µN the eigenvalues of �Y �Y T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Recall that we denoted by µA 1 (0) ≥ µA 2 (0) ≥ · · · ≥ µA N(0) the eigenvalues of A(0)A(0)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' From Cauchy’s integral formula, we have M � i=1 f(�µi) − M � d+ d− f(x) ρMP,d0(dx) = � M � i=1 f(µA i (0)) − � d+ d− f(x) ρMP,d0(dx) � + � M � i=1 f(�µi) − M � i=1 f(µA i (0)) � = � M � i=1 f(µA i (0)) − M � d+ d− f(x) ρMP,d0(dx) � − � 1 2πi � Γ f(z) Tr G �Y (z)dz − 1 2πi � Γ f(z) Tr GA(0, z)dz � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='80) Since AA∗ is a usual sample covariance matrix, the first term in the right-hand side converges to a Gaussian random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Further, as computed in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='52), E[�Y 4 ij] =: � w4 N 2 + 1 (NFg)2 � ∞ −∞ �g′(w) g(w) �4 � g � w − √ NλuivT j � − g(w) � dw, where the first term is the leading term of E[�Y 4 ij] and hence the leading term of E[A4 ij] as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' This means that the difference between �w4 and E[A4 ij] is negligible in the sense that it has no contribution in the limiting behavior of the resolvent, which can be checked from standard Green function comparison theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (Refer to [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=') Thus, the mean and the variance of the limiting Gaussian distribution are given by mA(f) = �f(2) + �f(−2) 4 − 1 2τ0( �f) + (� w4 − 3)τ2( �f) (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='81) and VA(f) = 2 ∞ � ℓ=1 ℓτℓ( �f)2 + (� w4 − 3)τ1( �f)2, (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='82) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' For the second term in the right-hand side of (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='80), by (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='75), we obtain that 1 2πi � Γ f(z) Tr G �Y (z)dz − 1 2πi � Γ f(z) Tr GA(0, z)dz = − k 2πi � Γ f(z) λFg d dz(sz + 1) λFg(sz + 1) + 1dz + o(1) (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='83) with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' From (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='80), we thus find that the CLT for the LSS holds, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=', � M � i=1 f(�µi) − M � d+ d− f(x) ρMP,d0(dx) � → N(m�Y (f), V�Y (f)), (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='84) and the variance V�Y (f) = VA(f) since the second term in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='80) converges to a deterministic value as 70 N → ∞, which corresponds to the change of the mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' In particular, m�Y (f) − mA(f) = (GH − Fg)λk 2πi � Γ f(z) ∂ ∂z (zs(z) + 1)dz + k 2πi � Γ f(z) λFg d dz(sz + 1) λFg(sz + 1) + 1dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='85) Following the computation in the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4 in [9] with the relation (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='28), we find that the right-hand side of (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='85) is given by k 2πi � Γ f(z)(zs(z) + 1)′ � λ(GH − Fg) + λFg λFg(zs(z) + 1) + 1 � dz = λk √d0 (GH − Fg)τ1( �f) + k ∞ � ℓ=1 � λFg √d0 �ℓ τℓ( �f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='86) (See also Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='7 of [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=') Thus, m�Y (f) = �f(2) + �f(−2) 4 − 1 2τ0( �f) + λk √d0 (GH − Fg)τ1( �f) + (� w4 − 3)τ2( �f) + k ∞ � ℓ=1 � λFg √d0 �ℓ τℓ( �f) (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='87) and V�Y (f) = 2 ∞ � ℓ=1 ℓτℓ( �f)2 + (� w4 − 3)τ1( �f)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='88) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3 Proof of Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='7 Notational remarks In the rest of the section, we use C order to denote a constant that is independent of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Even if the constant is different from one place to another, we may use the same notation C as long as it does not depend on N for the convenience of the presentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Now, we recall the linearization HA(θ)(z) and its inverse RA(θ, z) = HA(θ)(z)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' For simplicity, we drop the subscript A and index z of the linearization entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Proof of Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' To prove the lemma, we consider ∂ ∂θ Tr GA(θ, z) = � b � a � α ∂Aaα(θ) ∂θ ∂Gbb(θ) ∂Aaα(θ) = � b � a � α ∂Haα(θ) ∂θ ∂Rbb(θ) ∂Haα(θ) = − � b � a � α ∂Haα(θ) ∂θ [Rba(θ)Rαb(θ) + Rbα(θ)Rab(θ)] = −2 � a � α ∂Haα(θ) ∂θ (R(θ)2)aα = −2 � a � α ∂Haα(θ) ∂θ ∂ ∂z Raα(θ), (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='89) where we again used that ∂ ∂zGA(θ, z) = GA(θ, z)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We expand the right-hand side by using the definition of 71 A(θ), Haα(θ) = Aaα(θ) = � 1 − θ + θ � NSaα � Aaα(0) = � 1 − θ + θ � NSaα � Haα(0), (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='90) and so � a � α ∂Haα(θ) ∂θ Raα(θ) = � a � α � −1 + � NSaα � Haα(0)Raα(θ) = � a � α −1 + √NSaα 1 − θ + θ√NSaα Haα(θ)Raα(θ) = Nλ(GH − Fg) 2 � a � α (uavT α)2Haα(θ)Raα(θ) + O(N∥U∥2 ∞∥V ∥2 ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='91) From now, we further drop the θ-dependency for the brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Then ∂ ∂θ Tr GA(θ, z) = −Nλ(GH − Fg) ∂ ∂z � a � α (uavT α)2HaαRaα + O(N∥U∥2 ∞∥V ∥2 ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Here, we used the properties that Haα = Aab(θ) = O(N − 1 2 ), Rab = GA ba(θ) = O(N − 1 2 ) for b ̸= a, Raa = GA aa(θ) = O(1), and � a ua(ℓ1)ua(ℓ2) = δℓ1ℓ2 = � α vα(ℓ1)vα(ℓ2), which imply �����N 2 � a � α (uavT α)4HaαRaα ����� ≤ N 2∥U∥2 ∞∥V ∥2 ∞ � a � α (uavT α)2|HaαRaα| = O(N∥U∥2 ∞∥V ∥2 ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='92) Together with Remark C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='3, from the elementary equality for R and H, we have � a � α ua(ℓ)2HaαRaα = � a ua(ℓ)2 �� α HaαRaα � = � a ua(ℓ)2(1 + zRaa) = 1 + zs(z) + O(N − 1 2 ), (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='93) and � a � α vα(ℓ)2HaαRaα = � α vα(ℓ)2 �� a HaαRaα � = � α vα(ℓ)2(1 + Rαα) = d0(1 + zs(z)) + O(N − 1 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='94) 72 Plugging them into (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='91), we get 4 λ(GH − Fg) × (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='91) = N � ℓ � a � α � 1 N ua(ℓ)2HaαRaα + 1 M vα(ℓ)2HaαRaα + � ua(ℓ)2 − 1 M � vα(ℓ)2HaαRaα + ua(ℓ)2 � vα(ℓ)2 − 1 N � HaαRaα � + 2N � ℓ1̸=ℓ2 � a � α ua(ℓ1)ua(ℓ2)vα(ℓ1)vα(ℓ2)HaαRaα + O(N∥U∥2 ∞∥V ∥2 ∞) = N � ℓ � a � α � � ua(ℓ)2 − 1 M � vα(ℓ)2HaαRaα + ua(ℓ)2 � vα(ℓ)2 − 1 N � HaαRaα � + 2N � ℓ1̸=ℓ2 � a � α ua(ℓ1)ua(ℓ2)vα(ℓ1)vα(ℓ2)HaαRaα + 2k(zs(z) + 1) + O(N∥U∥2 ∞∥V ∥2 ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='95) It remains to estimate the first three terms in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='95).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Set X1 ≡ X1(θ, z, ℓ) := � a � α � ua(ℓ)2 − 1 M � vα(ℓ)2HaαRaα, (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='96) X2 ≡ X2(θ, z, ℓ) := � a � α ua(ℓ)2 � vα(ℓ)2 − 1 N � HaαRaα (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='97) and X3 ≡ X3(θ, z, ℓ1, ℓ2) := � a � α ua(ℓ1)ua(ℓ2)vα(ℓ1)vα(ℓ2)HaαRaα (ℓ1 ̸= ℓ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='98) We notice that |X1|, |X2|, |X3| = O(N −1) on Γ1/2 by a naive power counting as in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='91) after applying H¨older inequality once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' To obtain a better bound, we use a method based on a recursive moment estimate, introduced in [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We need the following lemma: Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Let X1, X2 and X3 be as in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='96), (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='97) and (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='98).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Define an event Ωε by Ωε = � a,b,α,β {|Haα|, |Raα| ≤ N − 1 2 +ε} ∩ {|Rab − s(z)δab| ≤ N − 1 2 +ε} ∩ {|Rαβ − zs(z)δαβ| ≤ N − 1 2 +ε} Then, for any fixed (large) D and (small) ε, which may depend on D, E[|X|2D|Ωε] ≤ CN − 1 2 +ε∥u∥2 ∞E[|X|2D−1|Ωε] + CN −1+4ε∥u∥4 ∞E[|X|2D−2|Ωε] + CN −2+10ε∥u∥6 ∞E[|X|2D−3|Ωε] + CN −3+14ε∥u∥8 ∞E[|X|2D−4|Ωε], (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='99) where X is X1, X2 and X3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Since the rank of the signal k is fixed, we suffices to prove the above lemma for fixed ℓ, ℓ1 and ℓ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We will prove Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='9 for X1 at the end of this section (the calculation for the X2 and X3 is almost the same).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' With Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='9, we are ready to obtain an improved bound for X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' First, note that the contribution from the exceptional event Ωc ε is negligible i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=', P(Ωc ε) < N −D2, which can be checked by applying a high-order 73 Markov inequality with the moment condition on �Y (See Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We decompose E[|X|2D] by E[|X|2D] = E[|X|2D · 1(Ωε)] + E[|X|2D · 1(Ωc ε)] = E[|X|2D|Ωε] · P(Ωε) + E[|X|2D · 1(Ωc ε)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='100) Then the second term in the right-hand side of (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='100), E[|X|2D · 1(Ωc ε)] ≤ � E[|X|4D] � 1 2 (P(Ωc ε)) 1 2 ≤ N − D2 2 � E[|X|4D] � 1 2 (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='101) and by using a trivial bound for the resolvent |Rab(z)| ≤ ∥GA(z)∥ ≤ 1 Im z E[|X|4D] ≤ E �� a � α |HaαRaα| �4D ≤ (M 2N)4D (Im z)4D max a,b,α E|HaαHbα|4D ≤ CN 14D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='102) To bound the right-hand side of (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='99), we use Young’s inequality: For any a, b > 0 and p, q > 0 with 1 p + 1 q = 1, ab ≤ ap p + bq q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We then find that the first term has the following upper bound N − 1 2 +ε∥u∥2 ∞|X|2D−1 = N (2D−1)ε 2D N − 1 2 +ε∥u∥2 ∞ · N − (2D−1)ε 2D |X|2D−1 ≤ 1 2DN (2D−1)ε(N − 1 2 +ε∥u∥2 ∞)2D + 2D − 1 2D N −ε|X|2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='103) Applying Young’s inequality for other terms in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='99), we get E[|X|2D|Ωε] ≤ CN (2D−1)ε(N − 1 2 +ε∥u∥2 ∞)2D + CN (D−1)ε(N −1+4ε∥u∥4 ∞)D + CN ( 2D 3 −1)ε(N −2+10ε∥u∥6 ∞) 2D 3 + CN ( D 2 −1)ε(N −3+14ε∥u∥8 ∞) D 2 + CN −εE[|X|2D|Ωε].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='104) Absorbing the last term in the right-hand side to the left-hand side and plugging the estimates (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='101) and (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='102) into (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='100), we now get E[|X|2D] ≤ CN (2D−1)ε(N − 1 2 +ε∥u∥2 ∞)2D + CN (D−1)ε(N −1+4ε∥u∥4 ∞)D + CN ( 2D 3 −1)ε(N −2+10ε∥u∥6 ∞) 2D 3 + CN ( D 2 −1)ε(N −3+14ε∥u∥8 ∞) D 2 + CN − D2 2 +7D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='105) From the (2D)-th order Markov inequality, for any fixed ε′ > 0 independent of D, P � |X| ≥ N ε′N − 1 2 ∥u∥2 ∞ � ≤ N −2Dε′ E[|X|2D] (N − 1 2 ∥u∥2∞)2D ≤ N −2Dε′N 8Dε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='106) By choosing ε = 1/D, for sufficiently large D, we find that |X| = O(N − 1 2 ∥u∥2 ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='107) 74 We now return to (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='89) and use (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='95) with the bound (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='107), M � j=1 N � k=1 ∂Ajk(θ) ∂θ (GA(θ)A(θ))jk = (GH − Fg)λk 2 (1 + zs(z)) + O(N∥u∥2 ∞∥v∥2 ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='108) To handle the derivative of the right-hand side, we use Cauchy’s integral formula with a rectangular contour, contained in Γε 1/2, whose perimeter is larger than ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Then, we get from (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='89) that ∂ ∂θ Tr GA(θ, z) = −λ(GH − Fg) · k ∂ ∂z (1 + zs(z)) + O(N∥u∥2 ∞∥v∥2 ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='109) After integrating over θ from 0 to 1, we conclude that (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='70) holds for a fixed z ∈ Γε 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' At last, we prove Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Proof of Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' As we mentioned above, we consider X = X1 and drop the ℓ-dependency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' E[|X|2D] = E �� a � α � u2 a − 1 M � v2 αHaαRaαXD−1X D � We use the following inequality that generalizes Stein’s lemma (see Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2 of [11]): Let Φ be a C2 function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Fix a (small) ε > 0, which may depend on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Recall that Ωε is the complement of the exceptional event on which |Haα| or |Raα| is exceptionally large for some a, α, defined by Ωε by � a,b,α,β {|Haα|, |Raα| ≤ N − 1 2 +ε} ∩ {|Rab − s(z)δab| ≤ N − 1 2 +ε} ∩ {|Rαβ − zs(z)δαβ| ≤ N − 1 2 +ε} Then, E[HaαΦ(Haα)|Ωε] = (E[H2 aα|Ωε] − E[Haα|Ωε]2)E[Φ′(Haα)|Ωε] + ε1, (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='110) where the error term ε1 admits the bound |ε1| ≤ C1E � |Haα|3 sup |t|≤1 Φ′′(tHaα) ���Ωε � (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='111) for some constant C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Note that by applying a decomposition (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='100) to E[Haα|Ωε] and E[H2 aα|Ωε], we see that E[Haα|Ωε] − E(E[Haα|Ωε]) = E[Haα|Ωε] = O(N −D0) (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='112) and E[H2 aα|Ωε] = E(E[H2 aα|Ωε]) + O(N −D0) = 1 N + O(∥u∥2 ∞∥v∥2 ∞) + O(N −D0) (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='113) for D0 = D2+1 2 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The estimate (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='110) follows from the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2 of [11] with p = 1, where we use the inequality (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='38) therein only up to second to the last line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' In the estimate (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='110), we choose Φ(Haα) = RaαXD−1X D (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='114) so that E[|X|2D|Ωε] = � a � α � u2 a − 1 M � v2 αE [HaαΦ(Haα)|Ωε] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='115) 75 Applying (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='112) and (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='113) to the equation (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='110), E [HaαΦ(Haα)|Ωε] = E � H2 aα � E[Φ′(Haα)|Ωε] + ε1 = E[H2 aα] � −E � RaaRααXD−1X D|Ωε � − E � R2 aαXD−1X D|Ωε � +(D − 1)E � Raα ∂X ∂Haα XD−2X D��Ωε � + DE � Raα ∂X ∂Haα XD−1X D−1��Ωε �� + ε1, (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='116) for sufficiently large D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We plug it into (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='115) and estimate each term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Then the term originated from the first term in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='116) can be separated by � a � α � u2 a − 1 M � v2 αE[H2 aα]E � RaaRααXD−1X D|Ωε � = � a � α � u2 a − 1 M � v2 αE[H2 aα]E � (Raa − s)RααXD−1X D|Ωε � + s � a � α � u2 a − 1 M � v2 αE[H2 aα]E � RααXD−1X D|Ωε � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='117) The first term satisfies that ����� � a � α � u2 a − 1 M � v2 αE[H2 aα]E � (Raa − s)RααXD−1X D|Ωε ������ ≤ CM∥u∥2 ∞N −1N − 1 2 +εE[|X|2D−1|Ωε] � α v2 α = CN − 1 2 +ε∥u∥2 ∞E[|X|2D−1|Ωε] (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='118) for some constant C since � α v2 α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Using (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='113) and � a � u2 a − 1 M � = 0, we also have �����s � a � α � u2 a − 1 M � v2 αE[H2 aα|Ωε]E � RααXD−1X D|Ωε ������ ≤ C∥u∥2 ∞∥v∥2 ∞|s| � a � α � u2 a + 1 M � v2 αE � |RααXD−1X D||Ωε � ≤ C∥u∥2 ∞∥v∥2 ∞E[|X|2D−1|Ωε] (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='119) for some constant C and large D > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' For the second term in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='116), we also have ����� � a � α � u2 a − 1 M � v2 αE[H2 aα|Ωε]E � R2 aαXD−1X D|Ωε ������ ≤ CN −1∥u∥2 ∞ ����� � a � α v2 αE � R2 aαXD−1X D|Ωε ������ ≤ CN −1+2ε∥u∥2 ∞E[|X|2D−1|Ωε] � α v2 α ≤ CN −1+2ε∥u∥2 ∞E[|X|2D−1|Ωε].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='120) 76 To estimate the third term and the fourth term in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='116), we notice that on Ωε ���� ∂X ∂Haα ���� = ������ − � b � β � u2 b − 1 M � v2 βHbβ[RabRαβ + RbαRaβ] + � u2 a − 1 M � v2 αRaα ������ ≤ CN − 1 2 +3ε∥u∥2 ∞ � α v2 α + CN − 1 2 +ε∥u∥2 ∞∥v∥2 ∞ ≤ CN − 1 2 +3ε∥u∥2 ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='121) for some constant C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Similarly, we can observe that ���� ∂2X ∂H2aα ���� ≤ CN − 1 2 +3ε∥u∥2 ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='122) Thus, we also obtain that ����� � a � α � u2 a − 1 M � v2 αE[H2 aα|Ωε]E � Raα ∂X ∂Haα XD−2X D��Ωε ������ ≤ CN −1+4ε∥u∥4 ∞E[|X|2D−2|Ωε] (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='123) and ����� � a � α � u2 a − 1 M � v2 αE[H2 aα|Ωε]E � Raα ∂X ∂Haα XD−1X D−1��Ωε ������ ≤ CN −1+4ε∥u∥4 ∞E[|X|2D−2|Ωε].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='124) Hence, from (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='116), (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='120), (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='123), and (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='124), ����� � a � α � u2 a − 1 M � v2 αE[H2 aα|Ωε]E[Φ′(Haα)|Ωε] ����� ≤ CN − 1 2 +ε∥u∥2 ∞E[|X|2D−1|Ωε] + CN −1+4ε∥u∥4 ∞E[|X|2D−2|Ωε] + ε1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='125) It remains to estimate |ε1| in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='111).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Proceeding as before, � a � α � u2 a − 1 M � v2 αE � |Haα|3Φ′′(Haα) ���Ωε � ≤ CN −1+4ε∥u∥2 ∞E[|X|2D−1|Ωε] + CN −2+7ε∥u∥4 ∞E[|X|2D−2|Ωε] + CN −2+10ε∥u∥6 ∞E[|X|2D−3|Ωε].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='126) Our last goal is to find the bound for the error term ε1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' To handle Φ′′(tHaα), we want to compare Φ′′(Haα) and Φ′′(tHaα) for some |t| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Let GA,t be the resolvent of A where Aaα is replaced by tAaα, and let Xt be defined as X in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='96) with the same replacement for Aaα and also GA is replaced by GA,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Correspondingly, we also consider the replacement Rt of the linearization R by substituting tHaα into Haα (also for Hαa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Then, Rt AB − RAB = (Rt(H − Ht)R)AB = (1 − t)Rt AaHaαRαB + (1 − t)Rt AαHαaRaB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='127) 77 and Xt − X = � b � β � u2 b − 1 M � v2 β(Ht bβRt bβ − HbβRbβ) = � b � β � u2 b − 1 M � v2 βHbβ(Rt bβ − Rbβ) + (t − 1) � u2 a − 1 M � v2 α(HaαRt aα) = (1 − t) � b � β � u2 b − 1 M � v2 βHbβRt baHaαRαβ + (1 − t) � b � β � u2 b − 1 M � v2 βHbβRt bαHαaRaβ + (t − 1) � u2 a − 1 M � v2 αHaαRt aα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='128) Thus, on Ωε, |Xt − X| ≤ CN −1+4ε∥u∥2 ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='129) Using the estimates (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='127) and (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='129), on Ωε, we obtain that |Φ′′(Haα) − Φ′′(tHaα)| ≤ C|Φ′′(Haα)| + N − 5 2 +11ε∥u∥6 ∞|X|2D−4 (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='130) uniformly on t ∈ (−1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Combining (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='115) and (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='125) with (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='126), (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='130), and (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='111), we finally get E[|X|2D|Ωε] ≤ CN − 1 2 +ε∥u∥2 ∞E[|X|2D−1|Ωε] + CN −1+4ε∥u∥4 ∞E[|X|2D−2|Ωε] + CN −2+10ε∥u∥6 ∞E[|X|2D−3|Ωε] + CN −3+14ε∥u∥8 ∞E[|X|2D−4|Ωε].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='131) This proves the desired lemma for X = X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' For the cases X = X2 or X = X3, the proofs are almost the same with the following changes: For X = X2, we change the role of U and V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' In other words, we will use � a ua(ℓ)2 = 1 and � α � vα(ℓ)2 − 1 N � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Then the recursive bound for E[|X2|2D|Ωε] obtained by putting v instead of u in the upper bound in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='99).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' In the same way, we use � a ua(ℓ1)ua(ℓ2) = δℓ1ℓ2 and � α |vα(ℓ1)||vα(ℓ2)| ≤ 1 instead of � a � u2 a − 1 M � = 0 and � a ua(ℓ)2 = 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We then obtain the exactly same recursive bound in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='99) for X3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='4 Computation of the test statistic In this section, we prove the second part of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='5 and also provide the details on the computation of the test statistic in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' By performing the same calculations as we will do in this section, we can obtain optimal functions for the other models, so we omit the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (Refer to [22, 33, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=') Recall that mY (f)|H1 − mY (f)|H0 = k � s=1 ∞ � ℓ=1 � ωs √d0 �ℓ τℓ( �f) (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='132) and VY (f) = 2 ∞ � ℓ=2 ℓτℓ( �f)2 + (w4 − 1)τ1( �f)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='133) 78 Assuming w2 > 0 and w4 > 1, from Cauchy’s inequality and the identity log(1 − λ) = − �∞ ℓ=1 λℓ/ℓ, ����� mY (f)|H1 − mY (f)|H0 � VY (f) ����� 2 ≤ k � p,q=1 ωpωq d0 � 1 w4 − 1 − 1 2 � − 1 2 log � 1 − ωpωq d0 � = ���� m(Ω) − m(0) √V0 ���� 2 , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='134) which proves the first part of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' The equality in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='134) holds if and only if √d0(w4 − 1)τ1( �f) � s ωs = 2ℓ(√d0)ℓτℓ( �f) � s ωℓs (ℓ = 2, 3, 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='135) We now find all functions f that satisfy (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='135).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Letting 2C be the common value in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='135), τ1( �f) = 2C √d0(w4 − 1) � s ωs, τℓ( �f) = C ℓ(√d0)ℓ � s ωℓ s (ℓ = 2, 3, 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='136) We can expand �f in terms of the Chebyshev polynomials as �f(x) = ∞ � ℓ=0 CℓTℓ �x 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='137) The orthogonality relation of the Chebyshev polynomials implies that for ℓ ≥ 1 τℓ( �f) = Cℓ π � 2 −2 Tℓ �x 2 � Tℓ �x 2 � dx √ 4 − x2 = Cℓ π � 1 −1 Tℓ (y) Tℓ (y) dy � 1 − y2 = Cℓ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='138) Thus, (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='136) holds if and only if �f(x) = c0 + 2C � s � 2ωs √d0(w4 − 1)T1 �x 2 � + ∞ � ℓ=2 1 ℓ � ωs √d0 �ℓ Tℓ �x 2 �� = c0 + 2C � s � ωs √d0 � 2 w4 − 1 − 1 � T1 �x 2 � + ∞ � ℓ=1 1 ℓ � ωs √d0 �ℓ Tℓ �x 2 �� (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='139) for some constant c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' We notice that the following identity holds for the Chebyshev polynomials: ∞ � ℓ=1 tℓ ℓ Tℓ (x) = log � 1 √ 1 − 2tx + t2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='140) (See, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=', (18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='9) of [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=') Since T1(x) = x, we find that (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='139) is equivalent to �f(x) = c0 + C � s � ωs √d0 � 2 w4 − 1 − 1 � x − log �d0 − ωs √d0x + ω2 s d0 �� , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='141) 79 or f(x) = c0 + C � s �ωs d0 � 2 w4 − 1 − 1 � x − ωs(1 + d0) d0 � 2 w4 − 1 − 1 �� − C � s log �ωs d0 �� 1 + d0 ωs � (1 + ωs) − x �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='142) This concludes the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2 with an optimal function φΩ(x) = �φΩ(ϕ(x)) (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='143) where �φΩ(x) = c0 + � s � ωs √d0 � 2 w4 − 1 − 1 � x − log �d0 − ωs √d0x + ω2 s d0 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='144) Choosing c0 = � s �(1 + d0) d0 � 2 w4 − 1 − 1 � ωs + log(ωs/d0) � , we get (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' Further, we can see that φΩ(x) = � s φωs(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='145) From this, we directly obtain that LΩ = � s Lωs, mY (φω)|H0 = −1 2 � s log � 1 − ω2 s d0 � + 1 2d0 (w4 − 3) � s ω2 s, (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='146) mY (φω)|H1 = mY (φω)|H0 + � p,q � − log � 1 − ωpωq d0 � + ωpωq d0 � 2 w4 − 1 − 1 �� (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='147) and VY (φω)|H1 = VY (φω)|H0 = 2 � p,q � − log � 1 − ωpωq d0 � + ωpωq d0 � 2 w4 − 1 − 1 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} +page_content='148) 80' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQf6Q7Z/content/2301.05331v1.pdf'} diff --git a/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf b/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf new file mode 100644 index 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b/ENAzT4oBgHgl3EQfGvvx/content/tmp_files/2301.01035v1.pdf.txt @@ -0,0 +1,1232 @@ +arXiv:2301.01035v1 [math.FA] 3 Jan 2023 +BOUNDARY REPRESENTATIONS OF INTERMEDIATE FORMS +BETWEEN A REGULAR DIRICHLET FORM AND ITS ACTIVE +MAIN PART +MATTHIAS KELLER, DANIEL LENZ, MARCEL SCHMIDT, MICHAEL SCHWARZ, +AND MELCHIOR WIRTH +Abstract. We characterize all semigroups sandwiched between the semigroup of +a Dirichlet form and the semigroup of its active main part. In case the Dirichlet +form is regular, we give a more explicit description of the quadratic forms of the +sandwiched semigroups in terms of pairs consisting of an open set and a measure +on an abstract boundary. +Introduction +One prime example of different self-adjoint realizations of the same differential +expression are the Dirichlet and Neumann Laplacian on a bounded domain, i.e. two +operators that only differ by the choice of boundary conditions. +More generally +one may ask which self-adjoint realizations of a differential expression arise from +choosing boundary conditions. +For the Laplacian, one possible answer was given by Arendt and Warma in [AW03]: +If Ω is a domain with Lipschitz boundary, a self-adjoint positive operator L on L2(Ω) +is a Laplacian with Robin-type boundary conditions if and only if the associated +semigroup (e−tL) is sandwiched between the Dirichlet and Neumann heat semigroup +in the sense that +et∆(D)f ≤ e−tLf ≤ et∆(N)f +for all f ≥ 0 and t > 0. Here the Laplacians with Robin-type boundary conditions +can best be described in terms of associated quadratic forms: The Dirichlet form Q +associated with L satisfies D(Q) = {f ∈ H1(Ω) | f = 0 quasi everywhere on Ω \ O} +and +Q(f) = +� +Ω |∇f|2dx + +� +∂Ω | ˜f|2dµ +for some open O ⊆ ∂Ω and a measure µ on ∂Ω not charging sets of capacity zero. +Here ˜f denotes a quasi-continuous modification of f. +Note that in the original work of Arendt and Warma there was an additional +condition that L be local, but this was later shown to be superfluous by Akhlil +[Akh18]. +This result has been generalized in several directions. Chill and Warma [CW12] +gave a similar characterization of (nonlinear) semigroups sandwiched between the +1 + +2 +M. KELLER, D. LENZ, M. SCHMIDT, M. SCHWARZ, AND M. WIRTH +semigroup generated by the p-Laplacian with Dirichlet boundary conditions and the +p-Laplacian with Neumann boundary conditions. Later this characterization was +extended to semigroups associated with local nonlinear Dirichlet forms by Claus +[Cla21]. In [ACD21] Arora, Chill and Djida do not study sandwiched semigroups +but give a characterization of all semigroups dominating a semigroup induced by a +regular form. +A related problem was studied by Posilicano in [Pos14]. For a bounded domain +with smooth boundary he characterizes all self-adjoint realizations of the Laplacian +that generate Markovian semigroups via certain Dirichlet forms on the boundary of +the domain. Applying his findings to realizations with sandwiched semigroups one +obtains the result of Arendt and Warma under higher regularity assumptions on +the boundary of the domain. For discrete Laplacians associated with infinite graphs +similar characterizations of Markovian realizations were obtained by the first four +authors in [KLSS19]. In this case, the employed boundary is the Royden boundary +of the graph, which is defined using Gelfand theory. +In this article we treat the question of sandwiched semigroups in the abstract +context of Dirichlet forms. We start with a regular Dirichlet form without killing +whose generator we take as an abstract analogue of the Dirichlet Laplacian. +In +this setting there is a natural analogue of the Neumann Laplacian, namely the +generator of the active main part of our given regular Dirichlet form, which was +introduced in [Sch17, Sch20a]. Our framework includes not only the Laplacian on +domains treated by Arendt and Warma, but also various Laplace-like operators +like fractional Laplacians, Laplacians on manifolds and metric measure spaces or +Laplacians on weighted graphs and quantum graphs. +We first give an abstract characterization of the generators of semigroups that are +sandwiched between the semigroup associated with a regular Dirichlet form and the +semigroup associated with its active main part in terms of order properties. +To connect these sandwiched semigroups to boundary conditions, the first problem +is to find a good notion of boundary in this setting. As all the quadratic forms +involved are defined on the L2-space of some abstract topological measure space, +there is no immediate geometric notion of boundary available. As in [KLSS19] and +[ACD21] we introduce a notion of boundary that is defined using Gelfand theory +and depends on the given regular Dirichlet form. +With this notion of boundary, we can prove an abstract version of the main result +of Arendt and Warma (Theorem 4.6): +Theorem. Let Q be a regular Dirichlet form on L2(X, m) without killing and Q(M) +its active main part. For a Dirichlet form Q′ on L2(X, m), the following assertions +are equivalent: +(i) There exists an open subset O of X ∪ ∂X and a measure µ on O ∩ ∂X that +does not charge polar sets such that Q′ is the closure of the quadratic form Qc +O,µ + +INTERMEDIATE DIRICHLET FORMS +3 +given by D(Qc +O,µ) = D(Q) ∩ Cc(O) and +Qc +O,µ(f) = Q(f) + +� +O∩∂X f 2 dµ. +(ii) The semigroup associated with Q′ is sandwiched between the semigroup associ- +ated with Q and the semigroup associated with Q(M), and D(Q′) ∩ Cc(X ∪ ∂X) +is a form core for Q′. +In other words, the Dirichlet forms sandwiched between Q and Q(M) (in the sense +of domination of semigroups) are parametrized by measures on open subsets of an +abstract boundary. +In spirit our main result for regular Dirichlet forms is similar to the one of +[ACD21], which treats an even more general setting without assuming the Markov +property. The main differences are that for the first abstract part we need not as- +sume any regularity of the forms and when we assume regularity, our results are +more explicit. +The article is organized as follows: In Section 1 we introduce the notation used +throughout this article and recall some basic facts about Dirichlet forms and domina- +tion of semigroups. In Section 2 we review the active main part of a regular Dirichlet +form and give an abstract characterization of the Dirichlet forms sandwiched between +the given regular Dirichlet form and its active main part (Theorem 2.8). In Section +3 we study some properties of the forms Qc +O,µ in the main theorem stated above, +in particular their closability. In Section 4 we introduce our notion of boundary +and show how sandwiched Dirichlet forms can be represented by measures on open +subsets of the boundary (Theorem 4.6). Finally, in the appendix we collect some +facts about bilinear forms on spaces of compactly supported continuous functions. +Parts of this paper are based on the PhD thesis of the fourth-named author +[Sch20b]. +Acknowledgments. The first three authors acknowledge financial support of the +DFG within the priority programme Geometry at Infinity. M.W. acknowledges fi- +nancial support by the German Academic Scholarship Foundation, by the Austrian +Science Fund (FWF) through grant number F65 and the Esprit Programme [ESP +156], and by the European Research Council (ERC) under the European Union’s +Horizon 2020 research and innovation programme (grant agreement No 716117). +For the purpose of Open Access, the authors have applied a CC BY public copy- +right licence to any Author Accepted Manuscript (AAM) version arising from this +submission. +1. Dirichlet forms and domination of associated semigroups +In this section we introduce notation and review some basic definitions and results +about Dirichlet forms and domination of the associated semigroups. Unless stated +otherwise, all functions are real-valued. Throughout (X, A, m) is a σ-finite measure + +4 +M. KELLER, D. LENZ, M. SCHMIDT, M. SCHWARZ, AND M. WIRTH +space and Q denotes a nonnegative quadratic form with domain D(Q) ⊆ L2(X, m). +We tacitly identify Q and the bilinear form it induces by polarization. In particular, +we have the convention Q(f) = Q(f, f) for f ∈ D(Q). The form norm ∥·∥Q is the +norm on D(Q) defined by +∥f∥2 +Q = Q(f) + ∥f∥2, +where ∥·∥ is the norm on L2(X, m). If Q′ is another quadratic form we write Q ⊑ Q′ +if D(Q) ⊆ D(Q′) and Q(f) ≥ Q′(f) for all f ∈ D(Q). The induced order relation +⊑ on all quadratic forms is called the natural order. +We say that a quadratic form is positive if Q(f, g) ≥ 0 for all nonnegative f, g ∈ +D(Q). It is called local if fg = 0 implies Q(f, g) = 0 for all f, g ∈ D(Q). Moreover, +Q is called monotone if |f| ≤ |g| implies Q(f) ≤ Q(g) whenever f, g ∈ D(Q). In this +case, Q(f) only depends on the absolute value of f and not on its sign. We discuss +these properties for forms whose domains are continuous functions in Appendix A. +1.1. (Regular) Dirichlet forms. A densely defined closed quadratic form Q on +L2(X, m) is called Dirichlet form if f ∈ D(Q) implies f+∧1 ∈ D(Q) and Q(f+∧1) ≤ +Q(f). The second Beurling-Deny criterion [RS78, Theorem XIII.51] asserts that Q +is a Dirichlet form if and only if the semigroup (e−tL) generated by the positive +self-adjoint operator L associated with Q is Markovian, i.e., 0 ≤ f ≤ 1 implies +0 ≤ e−tLf ≤ 1 for all t ≥ 0. +If Q is a Dirichlet form, then D(Q) ∩ L∞(X, m) is an algebra with respect to +pointwise multiplication and +Q(fg)1/2 ≤ ∥g∥∞Q(f)1/2 + ∥f∥∞Q(g)1/2 +for all f, g ∈ D(Q) ∩ L∞(X, m), see [FOT11, Theorem 1.4.2]. +A Dirichlet form Q is called regular if the following are satisfied: +• X is a locally compact separable metric space and m is a Radon measure of +full support. +• D(Q)∩Cc(X) is uniformly dense in Cc(X) and in D(Q) with respect to ∥·∥Q. +In this case, the Q-capacity (or simply capacity if Q is fixed) of an open set O ⊆ X +is defined by +cap(O) = inf{∥f∥2 +Q | f ∈ D(Q) with f ≥ 1 m-a.e. on O}. +Here we use the convention cap(O) = ∞ if there does not exist f ∈ D(Q) with +f ≥ 1 on O. For an arbitrary set A ⊆ X, the capacity is defined by +cap(A) = inf{cap(O) | O open with A ⊆ O}. +The capacity is inner regular, i.e., for any Borel set A ⊆ X it satisfies +cap(A) = sup{cap(K) | K compact with K ⊆ A}, +see [FOT11, Theorem 2.1.1]. Moreover, by [FOT11, Lemma 2.2.7], the capacity for +compact K ⊆ X can alternatively be described as +cap(K) = inf{∥f∥2 +Q | f ∈ D(Q) ∩ Cc(X) with f ≥ 1 on K}. + +INTERMEDIATE DIRICHLET FORMS +5 +A subset A of X is called polar if Cap(A) = 0 holds. A property is said to hold +quasi everywhere, abbreviated q.e., if it holds on the complement of a polar set. +A measurable function f : X → [−∞, ∞] is said to be quasi continuous if for every +ε > 0 there is an open set O with Cap(O) < ε such that f|X\O is finite-valued and +continuous. If Q is a regular Dirichlet form, then every f in D(Q) has a unique (up +to equality quasi everywhere) quasi continuous representative ˜f, cf. [CF12, Theorem +2.3.4]. +1.2. Domination of Dirichlet forms and semigroups. If U, V are sublattices +of L2(X, m), we say that U is an order ideal in V if f ∈ U, g ∈ V and |g| ≤ |f| +implies g ∈ U. +If U, V are subalgebras of L∞(X, m) we say U is an algebraic ideal in V if f ∈ U +and g ∈ V implies fg ∈ U. +We will frequently use the following characterization. The equivalence of (i) and +(ii) is Ouhabaz’ domination criterion [Ouh96, Theorem 3.7], whereas the equivalence +with (iii) is taken from [Sch20a, Lemma 2.2]. +Proposition 1.1 (Characterization of Domination). Let Q, Q′ be Dirichlet forms +with associated self-adjoint operators L, L′. The following assertions are equivalent. +(i) For all nonnegative f ∈ L2(X, m) and all t ≥ 0 we have +e−tLf ≤ e−tL′f. +(ii) D(Q) ⊆ D(Q′), D(Q) is an order ideal in D(Q′) and +Q(f, g) ≥ Q′(f, g) +for all non-negative f, g ∈ D(Q). +(iii) D(Q) ⊆ D(Q′), D(Q) ∩ L∞(X, m) is an algebraic ideal in D(Q′) ∩ L∞(X, m) +and +Q(f, g) ≥ Q′(f, g) +for all non-negative f, g ∈ D(Q). +If Q and Q′ satisfy one of the conditions of this proposition, we say that Q′ +dominates Q and write Q ⪯ Q′. Similarly, in this situation we write (e−tL) ⪯ (e−tL′) +and say that the semigroup (e−tL′) dominates the semigroup (e−tL). +Domination also induces an order relation on the set of all Dirichlet forms on +L2(X, m). Note that in general Q ⪯ Q′ does not imply Q ⊑ Q′ nor the other way +round. +2. The maximal dominating form and an abstract characterization +of sandwiched semigroups +For every Dirichlet form Q there is a maximal Dirichlet form Q(M) (with respect +to the natural order) that dominates the given Dirichlet form Q. In this section we +describe the construction of this maximal form and give an abstract characterization + +6 +M. KELLER, D. LENZ, M. SCHMIDT, M. SCHWARZ, AND M. WIRTH +of all Dirichlet forms Q′ that satisfy Q ⪯ Q′ ⪯ Q(M) provided that Q(M) is an +extension of Q. +We denote by (Tt) the semigroup generated by Q and by (T (M) +t +) the semigroup +generated by Q(M). According to the discussion in Subsection 1.2, any self-adjoint +C0-semigroup (St) with +(Tt) ⪯ (St) ⪯ (T (M) +t +) +corresponds to a Dirichlet form Q′ with Q ⪯ Q′ ⪯ Q(M). Hence, our result can be +seen as an abstract characterization of all semigroups sandwiched between (Tt) and +(T (M) +t +). +2.1. The active main part and the killing part. We will next recall the def- +inition of the active main part and the killing part of a Dirichlet form. For two +concrete examples see Examples 2.5, 2.6 below. +Let Q be a Dirichlet form on L2(X, m). For ϕ ∈ D(Q) with 0 ≤ ϕ ≤ 1 we define +the domain of the quadratic form ˜Qϕ on L2(X, m) by +D( ˜Qϕ) = {f ∈ L2(X, m) ∩ L∞(X, m) | fϕ, f 2ϕ ∈ D(Q)}, +on which it acts by +˜Qϕ(f) = Q(ϕf) − Q(ϕf 2, ϕ). +Since D(Q) ∩ L∞(X, m) is an algebra, we have D(Q) ∩ L∞(X, m) ⊆ D( ˜Qϕ). The +form ˜Qϕ is closable on L2(X, m). Indeed, [Sch20a, Theorem 3.1] shows that ˜Qϕ is +lower semicontinuous on its domain with respect to local convergence in measure +and hence it is lower semicontinuous on its domain with respect to L2-convergence. +We denote its closure by Qϕ. The next proposition summarizes further important +properties of Qϕ. +Proposition 2.1. Let ϕ, ψ ∈ D(Q) with 0 ≤ ϕ ≤ ψ ≤ 1. +(a) Qϕ is a Dirichlet form and its domain satisfies +D(Qϕ) ∩ L∞(X, m) = D( ˜Qϕ) = {f ∈ L2(X, m) ∩ L∞(X, m) | fϕ ∈ D(Q)}. +(b) D(Q) ⊆ D(Qϕ) and +Qϕ(f) ≤ Q(f), +f ∈ D(Q). +(c) D(Qψ) ⊆ D(Qϕ) and +Qϕ(f) ≤ Qψ(f), +f ∈ D(Qψ). +Proof. This follows from [Sch20a, Theorem 3.18]. The proofs given there treat an +extension of Qϕ to all measurable m-a.e. defined functions that is lower semicontin- +uous with respect to local convergence in measure. Restricting this form with larger +domain to L2(X, m) yields all the claims. +□ + +INTERMEDIATE DIRICHLET FORMS +7 +Remark 2.2. Part (a) of this proposition is important because it yields a formula +for Qϕ for bounded functions in its domain. Namely, for f, g ∈ D(Qϕ) ∩ L∞(X, m) +we have f, g ∈ D( ˜Qϕ) and hence +Qϕ(f, g) = ˜Qϕ(f, g) = Q(ϕf, ϕg) − Q(ϕfg, ϕ). +For the last equality, we used the definition of ˜Qϕ and polarization. +Definition 2.3 (Active main part). The active main part Q(M) of Q is defined as +follows: Its domain D(Q(M)) consists of all f ∈ L2(X, m) that satisfy f ∈ D(Qϕ) +for all ϕ ∈ D(Q) with 0 ≤ ϕ ≤ 1 such that +{ϕ ∈ D(Q) | 0 ≤ ϕ ≤ 1} → [0, ∞), +ϕ �→ Qϕ(f) +is bounded. On it Q(M) acts by +Q(M)(f) = sup{Qϕ(f) | ϕ ∈ D(Q) with 0 ≤ ϕ ≤ 1}. +Since ϕ �→ Qϕ(f) is monotone increasing, the form Q(M) is indeed a Dirichlet +form, see [Sch20a, Theorem 3.6]. It turns out that Q(M) is the maximal Dirichlet +form with respect to the natural order that dominates Q, i.e., Q ⪯ Q(M) and for all +Dirichlet forms Q′ with Q ⪯ Q′ we have Q′ ⊑ Q(M), see [Sch20a, Theorem 3.19]. +However, Q(M) need not be an extension of Q and hence we introduce the following +definition. +Definition 2.4 (Killing part). The difference +Q(k) = Q − Q(M) +with domain D(Q(k)) = D(Q) is called the killing part of Q. +The killing part is a local and positive quadratic form. Both properties are a +consequence of Q(k) being monotone, see [Sch20a, Lemma 3.11] for monotonicity +and [Sch20a, Lemma B.1] for how monotonicity implies the other properties. In +particular, the value of Q(k)(f) only depends on |f| and not on the sign of f. +We illustrate these objects with an example. It shows that the active main part is +an abstract way of constructing operators with Neumann boundary conditions from +the quadratic forms leading to Dirichlet boundary conditions. +Example 2.5 (Dirichlet and Neumann Laplacian on domains). Let Ω ⊆ Rn be +open (or more generally let Ω be a Riemannian manifold) and let V ∈ L1 +loc(Ω) be +nonnegative. We consider the Dirichlet form E(N) +V +with domain D(E(N) +V +) = {f ∈ +H1(Ω) | V 1/2f ∈ L2(Ω)}, on which it acts by +E(N) +V +(f) = +� +Ω |∇f|2dx + +� +Ω |f|2V dx. +The associated operator is the self-adjoint realization of the Schrödinger operator +H = −∆ + V with (abstract) Neumann boundary conditions, which we denote by +H(N). Moreover, we let E(D) +V +be the restriction of E(N) +V +to D(E(D) +V +) = {f ∈ H1 +0(Ω) | + +8 +M. KELLER, D. LENZ, M. SCHMIDT, M. SCHWARZ, AND M. WIRTH +V 1/2f ∈ L2(Ω)}. This is a regular Dirichlet form and the associated operator is the +self-adjoint realization of the Schrödinger operator H = −∆ + V with (abstract) +Dirichlet boundary conditions, which we denote by H(D). +The active main part of E(D) +V +is given by E(N) +0 +. Hence, the self-adjoint operator +associated to the active main part is −∆(N). For f ∈ D(E(D) +V +), the killing part of +E(D) +V +is given by +(E(D) +V +)(k)(f) = +� +Ω |f|2V dx. +In particular, if V = 0, this discussion shows that a Dirichlet form Q satisfies E(D) +0 +⪯ +Q ⪯ (E(D) +0 +)(M) if and only if the associated semigroup (St) satisfies (et∆(D)) ⪯ (St) ⪯ +(et∆(N)). Hence, forms sandwiched between E(D) +0 +and (E(D) +0 +)(M) = E(N) +0 +correspond +to semigroups sandwiched between the Dirichlet and the Neumann semigroup of the +Laplacian. This is precisely the situation studied in [AW03]. +Proof. Here we only sketch the main ideas of the proof. For the details we refer to +[Sch20a, Example 3.9]. We only consider bounded functions, the general case can +be treated through approximations. +Let f ∈ H1(Ω)∩L∞(Ω) and let ϕ ∈ C∞ +c (Ω) with 0 ≤ ϕ ≤ 1. A direct computation +using the product rule for ∇ shows f ∈ D((E(D) +V +)ϕ) and +(E(D) +V +)ϕ(f) = +� +Ω ϕ2|∇f|2dx. +Letting ϕ ր 1 and taking into account that C∞ +c (Ω) is dense in D(E(D) +V +) yields +f ∈ D((E(D) +V +)(M)) and the formula for the action of (E(D) +V +)(M). +Similarly, if f ∈ D((E(D) +V +)(M))∩L∞(Ω), by the definition of (E(D) +V +)ϕ and the active +main part, we have ϕf ∈ D(E(D) +V +) = H1 +0(Ω) ∩ L2(Ω, V · dx) for every ϕ ∈ C∞ +c (Ω). +This yields ∇f ∈ ⃗L2 +loc(Ω). With this at hand, an application of the product rule for +∇ as above shows (E(D) +V +)ϕ(f) = +� +Ω ϕ2|∇f|2dx. Since (E(D) +V +)ϕ(f) ≤ (E(D) +V +)(M)(f) and +ϕ is arbitrary, we conclude ∇f ∈ ⃗L2(Ω) so that f ∈ H1(Ω). +The statement on the killing part is an immediate consequence. +□ +Example 2.6 (Fractional Laplacians). As above we let Ω ⊆ Rn be open. For a +background on fractional Sobolev spaces we refer to [DNPV12]. For 0 < s < 1, we +denote by Qs,(N) the Dirichlet form with domain D(Qs,(N)) = W s(Ω) on which it +acts by +Qs,(N)(f) = 1 +2 +� +Ω×Ω +|f(x) − f(y)|2 +|x − y|n+2s +dx dy. +The restriction of this form to W s +0 (Ω) is denoted by Qs,(D), it is a regular Dirichlet +form. Note that at least if Ω is bounded and has C∞-boundary, the spaces W s +0 (Ω) +and W s(Ω) coincide for 0 < s ≤ +1 +2 by [LM72, Theorem 11.1], which makes the +problem of finding the Dirichlet forms sandwiched between Qs,(D) and Qs,(N) trivial. + +INTERMEDIATE DIRICHLET FORMS +9 +It is well-known that the associated self-adjoint operators H(N) +s +and H(D) +s +are +restrictions of the restricted fractional Laplacian Hs given by +Hsf(x) = P.V. +� +Ω +f(x) − f(y) +|x − y|n+2s dy = lim +ε→0+ +� +Ω\Bε(x) +f(x) − f(y) +|x − y|n+2s dy. +Hence, they can be viewed as realizations of Hs with abstract Neumann and Dirich- +let boundary conditions. +Note that we ignore a constant so that our fractional +Laplacian is only a constant multiple of the ’usual’ restricted fractional Laplacian, +cf. [DNPV12, Section 3]. Similar as in the previous example the active main part +of Qs,(D) is Qs,(N). +Proof. Here we only show the statement on the active main part of Qs,(D), the rest +is well-known. Since Qs,(N) and (Qs,(D))(M) are Dirichlet forms, it suffices to prove +D(Qs,(N)) ∩ L∞(Ω) = D((Qs,(D))(M)) ∩ L∞(Ω) and that Qs,(N) and (Qs,(D))(M) agree +on these sets (use that bounded functions are dense in the domains of Dirichlet +forms, see [FOT11, Theorem 1.4.2]). +We first proof that Qs,(N) is a restriction of (Qs,(D))(M) (on L∞(Ω)). Let f ∈ +W s(Ω) ∩ L∞(Ω) and let ϕ ∈ W s +0 (Ω) with 0 ≤ ϕ ≤ 1. Then fϕ ∈ W s +0 (Ω). We infer +Qs,(D)(ϕf) − Qs,(D)(ϕf 2, ϕ) = 1 +2 +� +Ω×Ω +(ϕ(x)f(x) − ϕ(y)f(y))2 +|x − y|n+2s +dx dy +− 1 +2 +� +Ω×Ω +(ϕ(x)f(x)2 − ϕ(y)f(y)2)(ϕ(x) − ϕ(y)) +|x − y|n+2s +dx dy += 1 +2 +� +Ω×Ω ϕ(x)ϕ(y)|f(x) − f(y)|2 +|x − y|n+2s +dx dy. +Taking the supremum over such ϕ yields f ∈ D((Qs,(D))(M)) and (Qs,(D))(M)(f) = +Qs,(N)(f). +It remains to prove D((Qs,(D))(M)) ∩ L∞(Ω) ⊆ W s(Ω). Let f ∈ D((Qs,(D))(M)) ∩ +L∞(Ω). For ϕ ∈ W s +0(Ω) with 0 ≤ ϕ ≤ 1, we have by definition of the main part +ϕf, ϕf 2 ∈ W s +0 (Ω) and +(Qs,(D))(M)(f) ≥ Qs,(D)(ϕf) − Qs,(D)(ϕf 2, ϕ) += 1 +2 +� +Ω×Ω ϕ(x)ϕ(y)|f(x) − f(y)|2 +|x − y|n+2s +dx dy. +For the last equality we used the same computation as above. Since ϕ was arbitrary, +this shows f ∈ W s(Ω). +□ +Remark 2.7. These examples show that it is a good intuition to think of a regular +Dirichlet form Q with Q(k) = 0 as being a form with ‘Dirichlet type’ boundary con- +ditions and Q(M) being the ’same’ form with ‘Neumann type’ boundary conditions. + +10 +M. KELLER, D. LENZ, M. SCHMIDT, M. SCHWARZ, AND M. WIRTH +2.2. An abstract characterization of sandwiched semigroups and forms. +The main abstract result of this paper is the following characterization of Dirichlet +forms sandwiched between a Dirichlet form without killing and its active main part. +Theorem 2.8. Let Q, Q′ be Dirichlet forms on L2(X, m) with Q(k) = 0. The fol- +lowing assertions are equivalent. +(i) Q ⪯ Q′ ⪯ Q(M). +(ii) (a) D(Q′) ⊆ D(Q(M)) and D(Q′) is an order ideal in D(Q(M)). +(b) Q′ − Q(M) is a positive and local form on D(Q′). +(c) Q′ is an extension of Q. +Proof. (i) =⇒ (ii): (a) This is a consequence of Proposition 1.1. +(b) The positivity of Q′ − Q(M) follows directly from Q′ ⪯ Q(M), cf. Proposi- +tion 1.1. In order to see that Q′ − Q(M) is local, we let f, g ∈ D(Q′) with fg = 0. +Without loss of generality we may assume f, g ≥ 0, for otherwise we can decom- +pose f, g into positive and negative parts and use f±g± = 0. Since Q′ and Q(M) +are Dirichlet forms, we can further assume that f, g are bounded. As we already +established positivity, it remains to prove Q′(f, g) − Q(M)(f, g) ≤ 0. +Let ϕ ∈ D(Q) with 0 ≤ ϕ ≤ 1. According to Proposition 1.1 we have fϕ, gϕ ∈ +D(Q), so that by Proposition 2.1 f, g ∈ D(Qϕ). Using fg = 0 and Q ⪯ Q′ we obtain +Q′(f, g) − Qϕ(f, g) = Q′(f, g) − Q(ϕf, ϕg) + Q(ϕfg, ϕ) += Q′(f, g) − Q′(ϕf, ϕg) += Q′((1 − ϕ)f, g) + Q′(ϕf, (1 − ϕ)g). +The functions η = (1 − ϕ)f and ζ = g are nonnegative and satisfy ηζ = 0. The +Dirichlet form property of Q′ implies +Q′(η + ζ) = Q′(|η + ζ|) = Q′(|η − ζ|) ≤ Q′(η − ζ), +from which we deduce Q′(η, ζ) ≤ 0 by bilinearity. The same argument applies to +η = ϕf and ζ = (1 − ϕ)g so that we obtain +Q′(f, g) − Qϕ(f, g) = Q′((1 − ϕ)f, g) + Q′(ϕf, (1 − ϕ)g) ≤ 0. +By the definition of Q(M) we can choose ϕ such that Qϕ(f, g) is arbitrarily close to +Q(M)(f, g) and hence obtain locality. +(c) The domination Q ⪯ Q′ ⪯ Q(M) and Q(k) = 0 yield for all nonnegative +f, g ∈ D(Q) the inequality +Q(f, g) = Q(M)(f, g) ≤ Q′(f, g) ≤ Q(f, g). +By splitting functions into positive and negative parts this shows Q = Q′ on D(Q). +(ii) =⇒ (i): Q′ ⪯ Q(M) follows directly from (a) and (b) and the characterization +of domination Proposition 1.1. + +INTERMEDIATE DIRICHLET FORMS +11 +Q ⪯ Q′: Since D(Q′) is contained in D(Q(M)) and D(Q) is an order ideal in +D(Q(M)) we obtain that D(Q) is also an order ideal in D(Q′). +Since Q′ is an +extension of Q, this already implies domination. +□ +We can rephrase this theorem slightly. Let Q be a Dirichlet form with Q(k) = 0. +We say that a pair (F, q) consisting of a vector lattice F ⊆ D(Q(M)) that is an order +ideal in D(Q(M)) and a quadratic form q with D(q) = F is an abstract admissible +pair for Q, if it satisfies the following properties: +• D(Q) ⊆ D(q) and q(f) = 0 for f ∈ D(Q), +• q is local and positive, +• the form QF,q = Q(M)|F + q is closed. +Corollary 2.9. Let Q be Dirichlet forms with Q(k) = 0. The following assertions +are equivalent. +(i) Q′ is a Dirichlet form with Q ⪯ Q′ ⪯ Q(M). +(ii) There exists an abstract admissible pair (F, q) such that Q′ = QF,q. +Proof. (i) =⇒ (ii): This is a reformulation of the previous theorem. +(ii) =⇒ (i): Using the previous theorem it suffices to show that QF,q is a Dirichlet +form. Since closedness and density of D(QF,q) = F are part of the definition of +abstract admissible pairs, it suffices to prove the Markov property. By assumption +F is an order ideal in D(Q(M)) and for f ∈ F we have f+ ∧ 1 ∈ D(Q(M)) and +|f+ ∧ 1| ≤ |f|. This shows f+ ∧ 1 ∈ F whenever f ∈ F. Moreover, as already +discussed after introducing the killing part, q being local and positive yields that q is +monotone, see [Sch20a, Lemma B.1]. These observations and Q(M) being Markovian +imply +QF,q(f+ ∧ 1) = Q(M)(f+ ∧ 1) + q(f+ ∧ 1) ≤ Q(M)(f) + q(f) = QF,q(f). +□ +Remark 2.10. This corollary shows that in order to determine all sandwiched forms +between Q and Q(M) we need to characterize all abstract admissible pairs. This is +possible when Q(M) is a regular Dirichlet form on a metric space K containing X as +a dense open subset. In the next section we will prove that in this case: +(a) Positive and local forms correspond to measures if their domain contains suf- +ficiently many continuous functions, see Appendix A. If these forms satisfy +q(f) = 0 for f ∈ D(Q), the corresponding measure is supported on the boundary +K \ X. +(b) Closed order ideals in D(Q(M)) correspond to functions vanishing outside an +open set (under some additional density assumption for continuous functions). +This then allows us to identify abstract admissible pairs with pairs of open subsets +of the boundary and certain measures on them. + +12 +M. KELLER, D. LENZ, M. SCHMIDT, M. SCHWARZ, AND M. WIRTH +3. Domination for parts of regular Dirichlet forms +Let Q be a regular Dirichlet form on L2(X, m). Let O ⊆ X be an open set and let +µ be a Radon measure on the Borel σ-algebra of O. We define the quadratic form +Qc +O,µ by letting D(Qc +O,µ) = D(Q) ∩ Cc(O) and +Qc +O,µ(f) = Q(f) + +� +O f 2dµ. +Here Cc(O) is tacitly identified with {ϕ ∈ Cc(X) | supp ϕ ⊆ O}. +Proposition 3.1. The following assertions are equivalent. +(i) µ charges no sets of Q-capacity zero. +(ii) The quadratic form Qc +O,µ is closable. +In this case, the closure QO,µ of Qc +O,µ is given by +D(QO,µ) = {f ∈ D(Q) | ˜f = 0 q.e. on X \ O and +� +O +˜f 2dµ < ∞}, +QO,µ(f) = Q(f) + +� +O +˜f 2dµ. +Proof. (i) =⇒ (ii): This follows as in [Sto92, Theorem 1.2]. +(ii) =⇒ (i): By the inner regularity of the capacity and the inner regularity of +the Radon measure µ it suffices to show for compact sets K ⊆ O that cap(K) = 0 +implies µ(K) = 0. +Let now K ⊆ O be compact with cap(K) = 0. Since Q is regular, there exists a +sequence (ϕn) in D(Q) ∩ Cc(X) such that ∥ϕn∥Q → 0, 0 ≤ ϕn ≤ 1 and ϕn ≥ 1 on +K. Let G be open and relatively compact with K ⊆ G ⊆ O. Using regularity of Q +again yields the existence of a function ψ ∈ D(Q) ∩ Cc(X) with 0 ≤ ψ ≤ 1, ψ = 1 +on K and supp ψ ⊆ G. +We now consider fn := ψ · ϕn. +Since Q is a Dirichlet form, it satisfies fn ∈ +D(Q) ∩ Cc(O) = D(Qc +O,µ) and +Q(fn)1/2 ≤ Q(ψ)1/2 + Q(ϕn)1/2. +The inequality 0 ≤ fn ≤ 1 and supp fn ⊆ G imply ∥fn∥ → 0 as n → ∞ and +� +O |fn|2dµ ≤ µ(G), +n ≥ 1. +In particular, these estimates show that (fn) is bounded with respect to the form +norm ∥·∥QO,µ. Let QO,µ be the closure of Qc +O,µ, which exists by (ii). The Banach–Saks +theorem implies that for some subsequence (fnk) the sequence of Césaro means +gN := 1 +N +N +� +k=1 +fnk +converges to some g ∈ D(QO,µ) with respect to ∥·∥QO,µ. The form norm of QO,µ +is larger than ∥·∥ and hence we obtain gn → g with respect to ∥·∥. +But since + +INTERMEDIATE DIRICHLET FORMS +13 +∥fn∥ → 0, we conclude g = 0. By the choice of (fn) we also have gN ∈ D(Q)∩Cc(O), +0 ≤ gN ≤ 1 and gN ≥ 1 on K. Putting everything together we obtain +µ(K) ≤ +� +O |gN|2dµ ≤ QO,µ(gN) → 0 as N → ∞. +This yields the desired µ(K) = 0. +The proof of the formula for QO,µ follows as in [SV96, Proposition 1.1]. +□ +Definition 3.2. A pair (O, µ) satisfying one of the conditions of the previous the- +orem is called an admissible pair for the form Q. In this case, we write QO,µ for the +closure of the form Qc +O,µ above. +Proposition 3.3. Let (Oi, µi), i = 1, 2, be admissible pairs for Q. The following +assertions are equivalent: +(i) QO1,µ1 ⪯ QO2,µ2. +(ii) cap(O1 \ O2) = 0 and µ2(A) ≤ µ1(A) for every Borel set A ⊆ O1 ∩ O2. +Proof. (ii) =⇒ (i): This follows immediately from (ii) and the formula for QOi,µi +given in Proposition 3.1. +(i) =⇒ (ii): Since D(QO1,µ1) is a lattice and an order ideal in D(QO2,µ2), we have +D(QO1,µ1) ⊆ D(QO2,µ2). Hence, every f ∈ D(QO1,µ1) satisfies ˜f = 0 q.e. on X \ O2. +Now, suppose cap(O1 \ O2) > 0. +By [FOT11, Theorem 2.1.1] there exists a +compact set K ⊆ O1 \ O2 with cap(K) > 0 and by [FOT11, Theorem 2.1.5] there +exists f ∈ D(Q) with 0 ≤ f ≤ 1 and ˜f = 1 q.e. on K. By the regularity of Q there +exists ϕ ∈ D(Q) ∩ Cc(O1) with ϕ ≥ 1 on K. We obtain ϕf ∈ D(QO1,µ1) as ϕ ˜f = 0 +q.e. on X \ O1 and +� +O1 +|ϕ ˜f|2dµ1 ≤ +� +O1 +|ϕ|2dµ1 < ∞. +Furthermore, ϕ ˜f ≥ 1 q.e. on K ⊆ X \ O2. This and cap(K) > 0 are a contradiction +to the fact that functions in D(QO1,µ1) vanish q.e. on X \ O2. +It remains to prove the inequality for the measures. Domination implies +Q(ϕ) + +� +O2 +|ϕ|2dµ2 ≤ Q(ϕ) + +� +O1 +|ϕ|2dµ1 +for all nonnegative ϕ ∈ D(Q) ∩ Cc(O1). For any compact set K ⊆ O1 ∩ O2 and any +open neighborhood G of K with G ⊆ O1 ∩ O2 there exists ψ ∈ Cc(X) ∩ D(Q) with +supp ψ ⊆ G, 0 ≤ ψ ≤ 1 and ψ ≥ 1 on K. Plugging this into the last inequality +yields +µ2(K) ≤ +� +O2 +|ψ|2dµ2 ≤ +� +O1 +|ψ|2dµ1 ≤ µ1(G). +Thus we obtain µ2(K) ≤ µ1(K) from the outer regularity of the Radon measure +µ1. By inner regularity of Radon measures this implies the statement for all Borel +sets. +□ + +14 +M. KELLER, D. LENZ, M. SCHMIDT, M. SCHWARZ, AND M. WIRTH +4. A boundary for regular Dirichlet forms and a characterization +of sandwiched semigroups +4.1. A boundary for regular Dirichlet forms. Let Q be a regular Dirichlet form +on L2(X, m). In this subsection we introduce a locally compact separable metric +space K that contains X as an open subset and extend m to a Radon measure ˆm +on K such that Q(M) can be considered to be a regular form on L2(K, ˆm). +The spaces Cc(X) and L2(X, m) are separable because X is locally compact sep- +arable metric space and m is a Radon measure. The map +L2(X, m) → D(Q(M)), +f �→ (L(M) + 1)−1f +is continuous with respect to the form norm ∥·∥Q(M) (here L(M) denotes the positive +self-adjoint operator associated with Q(M)). It has dense image D(L(M)) in D(Q(M)) +with respect to ∥·∥Q(M), showing that (D(Q(M)), ∥·∥Q(M)) is also separable. Moreover, +[Sch20a, Theorem 4.3] asserts that for a regular Dirichlet form Q the space D(Q(M))∩ +Cb(X) is dense in D(Q(M)) with respect to ∥·∥Q(M) (this is an abstract version of the +Meyers-Serrin theorem). +Combining these observations yields the existence of a subalgebra C of D(Q(M))∩ +Cb(X) with the following three properties: +• C is countably generated. +• C is ∥·∥Q(M)-dense in D(Q(M)). +• C ∩ Cc(X) is uniformly dense in Cc(X). +Let A be the uniform closure of C. Its complexification AC = {f + ig | f, g ∈ A} +is a commutative C∗-algebra that satisfies C0(X; C) ⊆ AC ⊆ Cb(X; C). By Gelfand +theory there exists a unique (up to homeomorphism) locally compact, separable +Hausdorff space K with the following properties: +• X is a dense and open subset of K. +• Every f ∈ AC can be extended to a function ˆf ∈ C0(K; C) and +C0(K; C) = { ˆf | f ∈ AC}. +As C is countably generated, the space K is metrizable. Hence, K is Polish, that is, +separable and completely metrizable, since every locally compact, separable, second +countable space is Polish. Since X is dense in K, the continuous extension of a +function from A to K is unique and we will therefore not distinguish between ele- +ments of A and their extension. For real-valued functions this interpretation leads +to A = C0(K)(= C0(K; R)). +The measure m on X can be extended to a Borel measure ˆm on K by setting +ˆm(A) = m(A ∩ X), +A ∈ B(K). +The measure ˆm is again a Radon measure of full support. By this definition the +space L2(K, ˆm) can be naturally identified with L2(X, m) via the unitary map +R: L2(K, ˆm) → L2(X, m), +f �→ f|X. + +INTERMEDIATE DIRICHLET FORMS +15 +Our discussion shows R−1(A∩L2(X, m)) = C0(K)∩L2(K, ˆm). Since R also preserves +the order relation, any Dirichlet form on L2(X, m) can be viewed as a Dirichlet form +on L2(K, ˆm) under this transformation. In particular, the form Q(M) is a regular +Dirichlet form on L2(K, ˆm), see [Sch20a, Theorem 4.4]. +The following remark sketches the uniqueness of the space K. We leave details +(especially the involved definitions, which can be found in [FOT11, Appendix A.4]) +to the reader. +Remark 4.1 (Uniqueness of K). The space K depends on the choice of the algebra C. +However, given two algebras C, C′ with the required properties and the corresponding +spaces K, K′, there exists a unitary order isomorphism +U : L2(K, ˆm) → L2(K′, ˆm′) +such that 0 ≤ fn ≤ 1, fn ր 1 implies Ufn ր 1 and U intertwines Q(M) and Q(M) +(when considererd as a form on the corresponding space). This implies that both +forms are equivalent in the sense of [FOT11, Appendix A.4]. Since they are also +regular, [FOT11, Theorem A.4.2] yields that K and K′ are quasi-homeomorphic (and +establishes further properties of a corresponding quasi-homeomorphism). +In view of the previous remark we make the following definition. +Definition 4.2. The set ∂X = K \ X is called the boundary of X relative to the +form Q. +Example 4.3 (Dirichlet and Neumann Laplacian – continued). We use the situation +and notation of Example 2.5 and assume that the potential vanishes, i.e., V = 0. +As discussed in Example 2.5 we have (E(D) +0 +)(M) = E(N) +0 +so that D((E(D) +0 +)(M)) = +H1(Ω) and the standard Sobolev norm on H1(Ω) coincides with the form norm of +(E(D) +0 +)(M). +If Ω ⊆ Rn has continuous boundary (for a precise definition see e.g. +[EE18, Definition 4.1]), the space {f|Ω | f ∈ C∞ +c (Rn)} is dense in H1(Ω) with +respect to the standard Sobolev norm. Hence, in this case we can choose the algebra +C to be a subset of {f|Ω | f ∈ C∞ +c (Rn)} ⊆ Cc(Ω). +Since by Stone-Weierstraß +{f|Ω | f ∈ C∞ +c (Rn)} is dense in C0(Ω), we can further assume that C is dense +in C0(Ω), so that the algebra A, the uniform closure of C, equals C0(Ω). +Our +construction then yields K = Ω (up to homeomorphism) and that the boundary of +Ω relative to E(D) +0 +coincides with the metric boundary ∂Ω = Ω \ Ω in Rn. +Example 4.4 (Fractional Laplacian – continued). We use the situation and notation +of Example 2.6. As discussed above we have D((Qs,(D))(M)) = W s(Ω). Moreover, if +Ω has Lipschitz boundary, then {f|Ω | f ∈ C∞ +c (Rn)} is dense in W s(Ω) with respect +to the form norm of Qs,(N), which coincides with the ususal norm on W s(Ω), see +[DNPV12, Corollary 5.5]. With this at hand the same argument as in the previous +example yields that we can choose K = Ω such that the boundary of Ω relative to +Qs,(D) coincides with the metric boundary ∂Ω = Ω \ Ω in Rn. + +16 +M. KELLER, D. LENZ, M. SCHMIDT, M. SCHWARZ, AND M. WIRTH +4.2. A characterization of sandwiched semigroups for regular Dirichlet +forms. In this subsection we prove the main result of this paper. +Let Q be a +regular Dirichlet form on L2(X, m). We apply the theory developed in Section 3 to +the form Q(M) when considered as a regular Dirichlet form on L2(K, ˆm). We start +with a simple observation that follows from the previous discussion. +Proposition 4.5. Let Q be a regular Dirichlet form. Then (K, 0) and (X, 0) are +admissible pairs for the regular Dirichlet form Q(M) on L2(K, ˆm) and we have Q(M) = +(Q(M))K, 0 and Q = (Q(M))X, 0. In particular, +D(Q) = {f ∈ D(Q(M)) | ˜f = 0 q.e. on ∂X}. +The following is the main result of the paper. +Theorem 4.6. Let Q be a regular Dirichlet form with Q(k) = 0. For a Dirichlet +form Q′ the following assertions are equivalent: +(i) There exists an admissible pair (O, µ) for Q(M) with X ⊆ O and µ(X) = 0 +such that Q′ = (Q(M))O,µ. +(ii) (a) Q ⪯ Q′ ⪯ Q(M) +(b) D(Q′) ∩ Cc(K) is dense in D(Q′) with respect to ∥·∥Q′. +Proof. (i) =⇒ (ii): (a) follows from Proposition 3.3 and the identities discussed in +Propostion 4.5. The density of D((Q(M))O,µ) ∩ Cc(O) in D((Q(M))O,µ) with respect +to ∥·∥(Q(M))O,µ is part of the definition of the form (Q(M))O,µ. +(ii) =⇒ (i): Let D be the uniform closure of the algebra D(Q′) ∩ Cc(K). Since +D(Q′) ∩ Cc(K) is an algebraic ideal in D(Q(M)) ∩ Cc(K) (here we use domination +and Proposition 3.3), D is a uniformly closed ideal in +D(Q(M)) ∩ Cc(K) +∥·∥∞ = C0(K) +(here we use the regularity of Q(M)). Moreover, by Theorem 2.8 we have D(Q) ⊆ +D(Q′) so that D(Q)∩Cc(X) ⊆ D(Q′)∩Cc(K). Since Q is regular on L2(X, m), this +yields C0(X) ⊆ D. By the characterization of closed ideals in C0(K) there exists an +open set X ⊆ O ⊆ K such that +D = {f ∈ C0(K) | f = 0 on K \ O}. +Altogether this discussion shows that D(Q′) ∩ Cc(O) is ∥·∥Q′ dense in D(Q′) and +uniformly dense in Cc(O). +Next, we show D(Q′) ∩ Cc(O) = D(Q(M)) ∩ Cc(O). Let ϕ ∈ D(Q(M)) ∩ Cc(O) +and let K = supp ϕ ⊆ O. +Since Q′ is a Dirichlet form and D(Q′) ∩ Cc(O) is +uniformly dense in Cc(O), there exists ψ ∈ D(Q′) ∩ Cc(O) with ψ = 1 on K. +We obtain ϕ = ψϕ ∈ D(Q′) ∩ Cc(O) since D(Q′) ∩ Cc(O) is an algebraic ideal in +D(Q(M)) ∩ Cc(O) (here we use domination and Proposition 3.3). +According to Theorem 2.8 the domination (a) implies that the form q = Q′−Q(M) +with domain D(q) = D(Q(M)) ∩ Cc(O) is positive, local and satisfies q(f) = 0 for + +INTERMEDIATE DIRICHLET FORMS +17 +all f ∈ D(Q) ∩ Cc(X). By Corollary A.4 there exists a Radon measure µ on O such +that +q(f) = +� +O |f|2dµ, +f ∈ D(Q(M)) ∩ Cc(O). +Since D(Q) ∩ Cc(X) is uniformly dense in Cc(X), the property q(f) = 0 for f ∈ +D(Q) ∩ Cc(X) implies µ(X) = 0. +For f ∈ D(Q(M)) ∩ Cc(O), we have by definition of q that +Q′(f) = Q(M)(f) + +� +O |f|2dµ = (Q(M))c +O,µ(f). +Since Q′ is closed and D(Q(M)) ∩ Cc(O) is ∥·∥Q′-dense in D(Q′), this implies that +(O, µ) is an admissible pair for Q(M) and Q′ = (Q(M))O,µ. +□ +We can reformulate this theorem as follows. +Corollary 4.7. Let Q be a regular Dirichlet form with Q(k) = 0. For a Dirichlet +form Q′, the following assertions are equivalent. +(i) There exists an open subset ∂µX ⊆ ∂X and a Radon measure µ on ∂µX that +does not charge sets of Q(M)-capacity zero such that +D(Q′) = {f ∈ D(Q(M)) | ˜f = 0 q.e. on ∂X \ ∂µX and +� +∂µX | ˜f|2dµ < ∞} +and +Q′(f) = Q(M)(f) + +� +∂µX | ˜f|2dµ. +(ii) (a) Q ⪯ Q′ ⪯ Q(M) +(b) D(Q′) ∩ Cc(K) is dense in D(Q′) with respect to ∥·∥Q′. +As an application of this result and our examples we obtain one of the main results +of [AW03] under slightly less restrictive assumptions. +Example 4.8. Again we use the situation of Schrödinger operators on Ω of Exam- +ple 2.5 with V = 0. Assume further that Ω ⊆ Rn has continuous boundary and +let Q be a Dirichlet form on L2(Ω) with associated Markovian semigroup (St). Let +∂Ω = Ω \ Ω be the metric boundary of Ω. The disscusion in Example 2.5 and Ex- +ample 4.3 combined with the previous corollary yield that the following assertions +are equivalent. +(i) There exists an open subset ∂µΩ ⊆ ∂Ω and a Radon measure µ on ∂µΩ that +does not charge sets of E(N) +0 +-capacity zero such that +D(Q) = {f ∈ H1(Ω) | ˜f = 0 q.e. on ∂Ω \ ∂µΩ and +� +∂µΩ | ˜f|2dµ < ∞} +and +Q(f) = +� +Ω |∇f|2dx + +� +∂µΩ | ˜f|2dµ. +(ii) (a) (et∆(D)) ⪯ (St) ⪯ (et∆(N)) + +18 +M. KELLER, D. LENZ, M. SCHMIDT, M. SCHWARZ, AND M. WIRTH +(b) D(Q) ∩ Cc(Ω) is dense in D(Q) with respect to ∥·∥Q. +This is precisely the statement of [AW03, Theorem 4.1] under the slightly less restric- +tive assumption of Ω having continuous boundary instead of Lipschitz boundary. +Example 4.9. We use the situation of fractional Laplacians of Example 2.6. Assume +further that Ω ⊆ Rn has Lipschitz boundary and let Q be a Dirichlet form on L2(Ω) +with associated Markovian semigroup (St). Let ∂Ω = Ω \ Ω be the metric boundary +of Ω. The disscusion in Example 2.6 and Example 4.4 combined with the previous +corollary yield that the following assertions are equivalent. +(i) There exists an open subset ∂µΩ ⊆ ∂Ω and a Radon measure µ on ∂µΩ that +does not charge sets of Qs,(N)-capacity zero such that +D(Q) = {f ∈ W s(Ω) | ˜f = 0 q.e. on ∂Ω \ ∂µΩ and +� +∂µΩ | ˜f|2dµ < ∞} +and +Q(f) = 1 +2 +� +Ω×Ω +|f(x) − f(y)|2 +|x − y|n+2s +dx dy + +� +∂µΩ | ˜f|2dµ. +(ii) (a) (e−tH(D) +s +) ⪯ (St) ⪯ (e−tH(N) +s +) +(b) D(Q) ∩ Cc(Ω) is dense in D(Q) with respect to ∥·∥Q. +The implication (i) =⇒ (ii) was also proved for Dirichlet forms associated with a re- +lated, but different fractional Laplacian by Claus and Warma [CW20, Theorem 4.2]. +As mentioned in the introducion we wanted to provide a version of the results of +[AW03] for general Dirichlet forms. In the abstract framework we were as general +as possible but held back in generality for regular Dirichlet forms. In the following +remarks we collect what else can be deduced from our general framework (at the +cost of brevity and technical simplicity). +Remark 4.10. (a) Another result of [AW03] is the descritption of the operators +corresponding to semigroups (et∆(D)) ⪯ (St) ⪯ (et∆(N)) as Laplacians with Robin +type boundary conditions. Something similar is possible here after equipping +the abstract boundary ∂X with so-called harmonic measures. This allows for +the definition of densities of normal derivatives and leads to abstract Robin +boundary conditions. In the Euclidean setting with Ω having Lipschitz boundary +the harmonic measures are mutually absolutely continuous with respect to the +surface measure on ∂Ω and the abstract normal derivatives are given by the +usual normal derivative. +(b) In Theorem 4.6 we used that D(Q′) ∩ Cc(K) is dense in D(Q′) because we +constructed the set O as the complement of the zero set of the closed ideal +D(Q′) ∩ Cc(K) +∥·∥∞ +in C0(K). One can drop the density assumption and replace this argument by +the characterization of closed ideals in regular Dirichlet spaces given in [Sto93]. + +INTERMEDIATE DIRICHLET FORMS +19 +In this case, Theorem 4.6 remains true without assertion (ii)(b) but with O open +replaced by O quasi-open. +(c) We always assumed that the killing part vanishes. If Q(k) ̸= 0, then there are +two possible choices of reference for the maximal form: +(1) One can characterize all Dirichlet forms Q′ with Q ⪯ Q′ ⪯ Q(M) via abstract +admissible pairs. Since in this case Q(M) is not an extension of Q, the form q +of the abstract admissible pair corresponding to Q′ does not vanish on D(Q) +but is bounded above by Q(k). In the regular setting this implies that the +measure µ from the admissible pair corresponding to Q′ is not necessarily +supported only on ∂X. It satisfies µ ≤ k on X, where k is the measure +corresponding to the local and positive form Q(k) (cf. Appendix A). +(2) Instead of comparing Q′ with Q(M) one can characterize Q ⪯ Q′ ⪯ Qref, +where Qref is the active reflected Dirichlet form of Q. It arises by adding a +suitable extension of Q(k) to Q(M), cf. [Sch20a, Section 3.3]. In this case, our +main theorems still hold true but the proofs become substantially longer. +Appendix A. Bilinear forms on Cc(X) +Let X be a locally compact metric space. In this section we provide a character- +ization of positive and local forms defined on Cc(X). First we show that densely +defined positive forms on Cc(X) can be extended to the whole of Cc(X) if their +domain is a lattice. In a second step we prove a representation theorem. Certainly +both results are well-known to experts. Since we could not find a proper reference, +we include the proofs for the convenience of the reader. +In the following lemma we write C(K) for the subspace {f ∈ Cc(X) | supp f ⊆ +K}. +Lemma A.1. Let q be a densely defined (with respect to the uniform norm) quadratic +form on D(q) ⊆ Cc(X). Suppose q is positive and D(q) is a lattice. +(a) For any compact K ⊆ X the restriction of q to D(q) ∩ C(K) is continuous. +(b) q can be uniquely extended to a positive quadratic form on Cc(X). +Proof. We first show that for any compact set K ⊆ X the restriction of q to D(q) ∩ +C(K) is continuous with respect to the supremum norm. +Let f, g ∈ D(q) ∩ C(K) be nonnegative. Let θK ∈ D(q) be such that θK ≥ 0 and +θK ≥ 1 on K. Such a functions exists because D(q) is a dense lattice in Cc(X). +Without loss of generality we assume +q(∥f∥∞θK, g) − q(∥g∥∞θK, f) ≤ 0, +for otherwise we could interchange f and g. Then, using the positivity of q, we get +0 ≤ q(∥f∥∞θK − f, ∥g∥∞θK + g) += −q(f, g) + ∥f∥∞∥g∥∞q(θK, θK) + q(∥f∥∞θK, g) − q(∥g∥∞θK, f) +≤ −q(f, g) + ∥f∥∞∥g∥∞q(θK, θK). + +20 +M. KELLER, D. LENZ, M. SCHMIDT, M. SCHWARZ, AND M. WIRTH +This implies +0 ≤ q(f, g) ≤ ∥f∥∞∥g∥q(θK, θK). +For arbitrary f, g ∈ D(q) ∩ C(K) we have f +, f −, g+, g− ∈ D(q) ∩ C(K) because +D(q) is a lattice. We obtain +|q(f, g)| ≤ q(f +, g+) + q(f +, g−) + q(f −, g+) + q(f −, g−) +≤ 4∥f∥∞∥g∥∞q(θK, θK). +Using this continuity in order to prove that q can be uniquely extended to a posi- +tive quadratic form on Cc(X) it suffices to show the following: For every nonnegative +ϕ ∈ Cc(X) there exists a compact K ⊆ X with supp ϕ ⊆ K such that ϕ can be +approximated by nonnegative functions in D(q) ∩ C(K). +To this end, we choose a nonnegative θ ∈ D(q) with θ ≥ ∥ϕ∥∞ on supp ϕ. Such a +function exists because D(q) is a dense lattice. Let K = supp θ. Since q is densely +defined, there exists ( ˜ϕn) in D(q) with ˜ϕn → ϕ uniformly, as n → ∞. Since D(q) is +a lattice, the sequence +ϕn = ( ˜ϕn)+ ∧ θ +belongs to D(q). +It is nonnegative and supp ϕn ⊆ supp θ = K for all n ≥ 1. +Moreover, using that 0 ≤ ϕ ≤ θ, we obtain ϕn → ϕ uniformly, as n → ∞. +□ +The following theorem provides a characterization of monotone quadratic forms +on Cc(X). +Theorem A.2. Let q: Cc(X) → [0, ∞) be a quadratic form. The following asser- +tions are equivalent: +(i) q is positive and local. +(ii) For all f, g ∈ Cc(X) the inequality fg ≥ 0 implies q(f, g) ≥ 0. +(iii) For all f, f ′, g, g′ ∈ Cc(X) the inequality fg ≥ f ′g′ implies q(f, g) ≥ q(f ′, g′). +(iv) q is monotone. +(v) There exists a Radon measure µ on X such that +q(u) = +� +X f 2dµ, +f ∈ Cc(X). +In this case, the measure µ is unique. +Proof. Clearly, (ii) implies (i), (iii) implies (ii) and (v) implies all other assertions. +(i) =⇒ (iv): Let f, g ∈ Cc(X) with |g| ≤ |f|. The positivity of q yields +q(|f|) = q(|g|, |f|) + q(|f| − |g|, |f|) ≥ q(|g|, |f|) = q(|g|) + q(|g|, |f| − |g|) ≥ q(|g|). +It is left to show q(f) = q(|f|) for every f ∈ Cc(X). Since f +, f − ∈ Cc(X) and +f +f − = 0, the locality of q implies q(f +, f −) = 0 and hence +q(f) = q(f +) − 2q(f +, f −) + q(f −) = q(f +) + 2q(f +, f −) + q(f −) = q(|f|). + +INTERMEDIATE DIRICHLET FORMS +21 +(iv) =⇒ (ii): Let f, g ∈ Cc(X) with fg ≥ 0. Then |f + g| ≥ |f − g| so that by +monotonicty +q(f) + q(g) − 2q(f, g) = q(f − g) ≤ q(f + g) ≤ q(f) + q(g) + 2q(f, g). +This shows (ii). +We already proved the equivalence of (i),(ii) and (iv) and that these assertions +are implied by (iii). Next we prove that they imply (iii). +Let f, f ′, g, g′ ∈ Cc(X) with fg ≥ f ′g′. If f = f ′, the inequality q(f, g) ≥ q(f ′, g′) +directly follows from (ii). With the help of an approximation we reduce the case +f ̸= f ′ to this one. +We start with the following observation: Locality of q implies that for ϕ, χ ∈ +Cc(X) the value q(ϕ, χ) is independent of χ as long as χ = 1 on supp ϕ. In this +case, we write I(ϕ) := q(ϕ, χ). +Let ε > 0. By compacteness of the supports we can choose finitely many rela- +tively compact open sets Gj, j = 1, . . . , N, that cover the union of the supports of +f, f ′, g, g′, and choose ξj ∈ Gj, such that +sup +x∈Gj +|f(x) − f(ξj)| < ε and sup +x∈Gj +|f ′(x) − f ′(ξj)| < ε. +We let χj ∈ Cc(X), j = 1, . . ., N, be a subordinate partition of unity, i.e. 0 ≤ χj ≤ 1, +suppχj ⊆ Gj and +N +� +j=1 +χj = 1 on +N +� +j=1 +Gj. +Such a partition of unity exists because metric spaces are normal. We define +˜f = +N +� +j=1 +f(ξj)χj and ˜f ′ = +N +� +j=1 +f ′(ξj)χj. +Then, using +� +j χj = 1 on the supports of f, g, we obtain +|q( ˜f, g) − q(f, g)| ≤ +N +� +j=1 +|q(χj(f − f(ξj)), g)| ≤ εq( +N +� +j=1 +χj, |g|) = εI(|g|). +For the second inequality we used |q(ϕ, ψ)| ≤ q(|ϕ|, |ψ|), which directly follows from +the positivity of q, and the fact that |χj(f − f(ξj))| ≤ εχj. +Similarly, we have + +22 +M. KELLER, D. LENZ, M. SCHMIDT, M. SCHWARZ, AND M. WIRTH +|q( ˜f ′, g′) − q(f ′, g′)| ≤ εI(|g′|). Moreover, +q( ˜f, g) − q( ˜f ′, g′) = +N +� +j=1 +(q(f(ξj)χj, g) − q(f ′(ξj)χj, g′)) += +N +� +j=1 +q(χj, f(ξj)g − f ′(ξj)g′) +≥ +N +� +j=1 +q(χj, fg − f ′g′) − εq( +N +� +j=1 +χj, |g| + |g|′) +≥ −εI(|g| + |g′|). +For the first inequality we used (ii) and the estimate +χj(f(ξj)g − f ′(ξj)g′) ≥ χj(fg − f ′g′ − ε(|g| + |g′|)). +The last inequality follows from χj(fg − f ′g′) ≥ 0 and +� +j χj = 1 on the support of +|g| + |g′|. Since ε > 0 was arbitrary, these estimates show (iii). +(iii) =⇒ (v): As above we define I : Cc(X) → R by letting +I(ϕ) = q(χ, ϕ) +for some χ ∈ Cc(X) with χ = 1 on the support of ϕ. It follows from (iii) that this +is well-defined and positive. Moreover, I is linear. By the Riesz-Markov-Kakutani +representation theorem there exists a unique Radon measure µ such that +I(ϕ) = +� +X ϕdµ +for all ϕ ∈ Cc(X). Let now f, g ∈ Cc(X) an let χ ∈ Cc(X) such that χ = 1 on the +supports of f and g. Since fg = χ(fg), property (iii) yields +q(f, g) = q(χ, fg) = I(fg) = +� +X fgdµ. +Thus, µ is the desired measure. +□ +Remark A.3. The statement of the theorem is not only valid for quadratic forms +on continuous functions. The equivalence of (ii) and (iv) was observed in [Sch20a, +Appendix B] for quadratic forms on sublattices of L0(Y, m), where Y is an arbitrary +set and m is a measure on Y . Indeed, the above proof yields the equivalence of (i),(ii) +and (iv) in this situation. The equivalence with (iii) requires the existence of suitable +partitions of unity in the domain of q and the equivalence with (v) requires that the +domain of q is an algebra and a representation theorem for positive functionals. +Corollary A.4. Let q be a densely defined positive and local quadratic form on +Cc(X) such that D(q) is a lattice. Then there exists a unique Radon measure µ on +X such that +q(f) = +� +X f 2dµ, +f ∈ D(q). + +INTERMEDIATE DIRICHLET FORMS +23 +Proof. As noted in the previous remark the form q is also monotone. By Lemma A.1 +it can be uniquely extended to a positive quadratic form on Cc(X) and by the +continuity of restrictions to compact sets this extension is also monotone. Hence, +the statement follows from the previous theorem. +□ +References +[ACD21] Sahiba Arora, Ralph Chill, and Jean-Daniel Djida. Domination of semigroups generated +by regular forms. 2021. +[Akh18] Khalid Akhlil. Locality and domination of semigroups. Results Math., 73(2):Art. 59, 11, +2018. +[AW03] +Wolfgang Arendt and Mahamadi Warma. Dirichlet and Neumann boundary conditions: +What is in between? J. Evol. Equ., 3(1):119–135, 2003. Dedicated to Philippe Bénilan. +[CF12] +Zhen-Qing Chen and Masatoshi Fukushima. Symmetric Markov processes, time change, +and boundary theory, volume 35 of London Mathematical Society Monographs Series. +Princeton University Press, Princeton, NJ, 2012. +[Cla21] +Burkhard Claus. Non-linear Dirichlet forms. PhD thesis, TU Dresden, Dresden, 2021. +[CW12] +Ralph Chill and Mahamadi Warma. Dirichlet and Neumann boundary conditions for the +p-Laplace operator: what is in between? Proceedings of the Royal Society of Ediburgh: +Section A Mathematics, 142(5):975–1002, 2012. +[CW20] +Burkhard Claus and Mahamadi Warma. Realization of the fractional Laplacian with +nonlocal exterior conditions via form methods. J. Evol. Equ., 20(4):1597–1631, 2020. +[DNPV12] Eleonora Di Nezza, Giampiero Palatucci, and Enrico Valdinoci. Hitchhiker’s guide to +the fractional Sobolev spaces. Bull. Sci. Math., 136(5):521–573, 2012. +[EE18] +D. E. Edmunds and W. D. Evans. Spectral theory and differential operators. Oxford +Mathematical Monographs. Oxford University Press, Oxford, 2018. +[FOT11] Masatoshi Fukushima, Yoichi Oshima, and Masayoshi Takeda. Dirichlet forms and sym- +metric Markov processes, volume 19 of De Gruyter Studies in Mathematics. Walter de +Gruyter & Co., Berlin, extended edition, 2011. +[KLSS19] Matthias Keller, Daniel Lenz, Marcel Schmidt, and Michael Schwarz. Boundary repre- +sentation of Dirichlet forms on discrete spaces. J. Math. Pures Appl. (9), 126:109–143, +2019. +[LM72] +J.-L. Lions and E. Magenes. Non-homogeneous boundary value problems and applications. +Vol. I. Die Grundlehren der mathematischen Wissenschaften, Band 181. Springer-Verlag, +New York-Heidelberg, 1972. Translated from the French by P. Kenneth. +[Ouh96] E. Ouhabaz. Invariance of closed convex sets and domination criteria for semigroups. +Potential Analysis, 5(6):611–625, 1996. +[Pos14] +Andrea Posilicano. Markovian extensions of symmetric second order elliptic differential +operators. Math. Nachr., 287(16):1848–1885, 2014. +[RS78] +Michael Reed and Barry Simon. Methods of modern mathematical physics. IV. Analysis of +operators. Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, +1978. +[Sch17] +Marcel Schmidt. Energy forms. PhD thesis, Mar 2017. +[Sch20a] Marcel Schmidt. A note on reflected Dirichlet forms. Potential Anal., 52(2):245–279, 2020. +[Sch20b] Michael Schwarz. Nodal Domains and Boundary Representation for DirichletForms. PhD +thesis, Jan 2020. +[Sto92] +Peter Stollmann. Smooth perturbations of regular Dirichlet forms. Proc. Amer. Math. +Soc., 116(3):747–752, 1992. +[Sto93] +Peter Stollmann. Closed ideals in Dirichlet spaces. Potential Anal., 2(3):263–268, 1993. + +24 +M. KELLER, D. LENZ, M. SCHMIDT, M. SCHWARZ, AND M. WIRTH +[SV96] +Peter Stollmann and Jürgen Voigt. Perturbation of Dirichlet forms by measures. Potential +Anal., 5(2):109–138, 1996. +M.Keller, Institut für Mathematik, Universität Potsdam, Campus Golm, Haus 9, +Karl-Liebknecht-Str. 24-25, 14476 Potsdam OT Golm, Germany +Email address: matthias.keller@uni-potsdam.de +D.Lenz, Institut für Mathematik, Friedrich-Schiller-Universität Jena, 07737 Jena, +Germany +Email address: daniel.lenz@uni-jena.de +M. Schmidt, Mathematisches Institut, Universität Leipzig, Augustusplatz 10, 04109 +Leipzig, Germany +Email address: marcel.schmidt@math.uni-leipzig.de +M. Schwarz, dotSource GmbH, Goethestr. 1, 07743 Jena, Germany +Email address: m.schwarz@dotSource.de +M. Wirth, Institute of Science and Technology Austria (ISTA), Am Campus 1, +3400 Klosterneuburg, Austria +Email address: melchior.wirth@ist.ac.at + diff --git a/ENAzT4oBgHgl3EQfGvvx/content/tmp_files/load_file.txt b/ENAzT4oBgHgl3EQfGvvx/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b910d2ca1d304efbf0149de64e952c4ac41d3507 --- /dev/null +++ b/ENAzT4oBgHgl3EQfGvvx/content/tmp_files/load_file.txt @@ -0,0 +1,846 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf,len=845 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='01035v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='FA] 3 Jan 2023 BOUNDARY REPRESENTATIONS OF INTERMEDIATE FORMS BETWEEN A REGULAR DIRICHLET FORM AND ITS ACTIVE MAIN PART MATTHIAS KELLER, DANIEL LENZ, MARCEL SCHMIDT, MICHAEL SCHWARZ, AND MELCHIOR WIRTH Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' We characterize all semigroups sandwiched between the semigroup of a Dirichlet form and the semigroup of its active main part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' In case the Dirichlet form is regular, we give a more explicit description of the quadratic forms of the sandwiched semigroups in terms of pairs consisting of an open set and a measure on an abstract boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Introduction One prime example of different self-adjoint realizations of the same differential expression are the Dirichlet and Neumann Laplacian on a bounded domain, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' two operators that only differ by the choice of boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' More generally one may ask which self-adjoint realizations of a differential expression arise from choosing boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' For the Laplacian, one possible answer was given by Arendt and Warma in [AW03]: If Ω is a domain with Lipschitz boundary, a self-adjoint positive operator L on L2(Ω) is a Laplacian with Robin-type boundary conditions if and only if the associated semigroup (e−tL) is sandwiched between the Dirichlet and Neumann heat semigroup in the sense that et∆(D)f ≤ e−tLf ≤ et∆(N)f for all f ≥ 0 and t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Here the Laplacians with Robin-type boundary conditions can best be described in terms of associated quadratic forms: The Dirichlet form Q associated with L satisfies D(Q) = {f ∈ H1(Ω) | f = 0 quasi everywhere on Ω \\ O} and Q(f) = � Ω |∇f|2dx + � ∂Ω | ˜f|2dµ for some open O ⊆ ∂Ω and a measure µ on ∂Ω not charging sets of capacity zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Here ˜f denotes a quasi-continuous modification of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Note that in the original work of Arendt and Warma there was an additional condition that L be local, but this was later shown to be superfluous by Akhlil [Akh18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' This result has been generalized in several directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Chill and Warma [CW12] gave a similar characterization of (nonlinear) semigroups sandwiched between the 1 2 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' KELLER, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' LENZ, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' SCHMIDT, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' SCHWARZ, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' WIRTH semigroup generated by the p-Laplacian with Dirichlet boundary conditions and the p-Laplacian with Neumann boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Later this characterization was extended to semigroups associated with local nonlinear Dirichlet forms by Claus [Cla21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' In [ACD21] Arora, Chill and Djida do not study sandwiched semigroups but give a characterization of all semigroups dominating a semigroup induced by a regular form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' A related problem was studied by Posilicano in [Pos14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' For a bounded domain with smooth boundary he characterizes all self-adjoint realizations of the Laplacian that generate Markovian semigroups via certain Dirichlet forms on the boundary of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Applying his findings to realizations with sandwiched semigroups one obtains the result of Arendt and Warma under higher regularity assumptions on the boundary of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' For discrete Laplacians associated with infinite graphs similar characterizations of Markovian realizations were obtained by the first four authors in [KLSS19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' In this case, the employed boundary is the Royden boundary of the graph, which is defined using Gelfand theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' In this article we treat the question of sandwiched semigroups in the abstract context of Dirichlet forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' We start with a regular Dirichlet form without killing whose generator we take as an abstract analogue of the Dirichlet Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' In this setting there is a natural analogue of the Neumann Laplacian, namely the generator of the active main part of our given regular Dirichlet form, which was introduced in [Sch17, Sch20a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Our framework includes not only the Laplacian on domains treated by Arendt and Warma, but also various Laplace-like operators like fractional Laplacians, Laplacians on manifolds and metric measure spaces or Laplacians on weighted graphs and quantum graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' We first give an abstract characterization of the generators of semigroups that are sandwiched between the semigroup associated with a regular Dirichlet form and the semigroup associated with its active main part in terms of order properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' To connect these sandwiched semigroups to boundary conditions, the first problem is to find a good notion of boundary in this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' As all the quadratic forms involved are defined on the L2-space of some abstract topological measure space, there is no immediate geometric notion of boundary available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' As in [KLSS19] and [ACD21] we introduce a notion of boundary that is defined using Gelfand theory and depends on the given regular Dirichlet form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' With this notion of boundary, we can prove an abstract version of the main result of Arendt and Warma (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='6): Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Let Q be a regular Dirichlet form on L2(X, m) without killing and Q(M) its active main part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' For a Dirichlet form Q′ on L2(X, m), the following assertions are equivalent: (i) There exists an open subset O of X ∪ ∂X and a measure µ on O ∩ ∂X that does not charge polar sets such that Q′ is the closure of the quadratic form Qc O,µ INTERMEDIATE DIRICHLET FORMS 3 given by D(Qc O,µ) = D(Q) ∩ Cc(O) and Qc O,µ(f) = Q(f) + � O∩∂X f 2 dµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' (ii) The semigroup associated with Q′ is sandwiched between the semigroup associ- ated with Q and the semigroup associated with Q(M), and D(Q′) ∩ Cc(X ∪ ∂X) is a form core for Q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' In other words, the Dirichlet forms sandwiched between Q and Q(M) (in the sense of domination of semigroups) are parametrized by measures on open subsets of an abstract boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' In spirit our main result for regular Dirichlet forms is similar to the one of [ACD21], which treats an even more general setting without assuming the Markov property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' The main differences are that for the first abstract part we need not as- sume any regularity of the forms and when we assume regularity, our results are more explicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' The article is organized as follows: In Section 1 we introduce the notation used throughout this article and recall some basic facts about Dirichlet forms and domina- tion of semigroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' In Section 2 we review the active main part of a regular Dirichlet form and give an abstract characterization of the Dirichlet forms sandwiched between the given regular Dirichlet form and its active main part (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' In Section 3 we study some properties of the forms Qc O,µ in the main theorem stated above, in particular their closability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' In Section 4 we introduce our notion of boundary and show how sandwiched Dirichlet forms can be represented by measures on open subsets of the boundary (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Finally, in the appendix we collect some facts about bilinear forms on spaces of compactly supported continuous functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Parts of this paper are based on the PhD thesis of the fourth-named author [Sch20b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' The first three authors acknowledge financial support of the DFG within the priority programme Geometry at Infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' acknowledges fi- nancial support by the German Academic Scholarship Foundation, by the Austrian Science Fund (FWF) through grant number F65 and the Esprit Programme [ESP 156], and by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 716117).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' For the purpose of Open Access, the authors have applied a CC BY public copy- right licence to any Author Accepted Manuscript (AAM) version arising from this submission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Dirichlet forms and domination of associated semigroups In this section we introduce notation and review some basic definitions and results about Dirichlet forms and domination of the associated semigroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Unless stated otherwise, all functions are real-valued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Throughout (X, A, m) is a σ-finite measure 4 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' KELLER, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' LENZ, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' SCHMIDT, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' SCHWARZ, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' WIRTH space and Q denotes a nonnegative quadratic form with domain D(Q) ⊆ L2(X, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' We tacitly identify Q and the bilinear form it induces by polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' In particular, we have the convention Q(f) = Q(f, f) for f ∈ D(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' The form norm ∥·∥Q is the norm on D(Q) defined by ∥f∥2 Q = Q(f) + ∥f∥2, where ∥·∥ is the norm on L2(X, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' If Q′ is another quadratic form we write Q ⊑ Q′ if D(Q) ⊆ D(Q′) and Q(f) ≥ Q′(f) for all f ∈ D(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' The induced order relation ⊑ on all quadratic forms is called the natural order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' We say that a quadratic form is positive if Q(f, g) ≥ 0 for all nonnegative f, g ∈ D(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' It is called local if fg = 0 implies Q(f, g) = 0 for all f, g ∈ D(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Moreover, Q is called monotone if |f| ≤ |g| implies Q(f) ≤ Q(g) whenever f, g ∈ D(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' In this case, Q(f) only depends on the absolute value of f and not on its sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' We discuss these properties for forms whose domains are continuous functions in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' (Regular) Dirichlet forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' A densely defined closed quadratic form Q on L2(X, m) is called Dirichlet form if f ∈ D(Q) implies f+∧1 ∈ D(Q) and Q(f+∧1) ≤ Q(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' The second Beurling-Deny criterion [RS78, Theorem XIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='51] asserts that Q is a Dirichlet form if and only if the semigroup (e−tL) generated by the positive self-adjoint operator L associated with Q is Markovian, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=', 0 ≤ f ≤ 1 implies 0 ≤ e−tLf ≤ 1 for all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' If Q is a Dirichlet form, then D(Q) ∩ L∞(X, m) is an algebra with respect to pointwise multiplication and Q(fg)1/2 ≤ ∥g∥∞Q(f)1/2 + ∥f∥∞Q(g)1/2 for all f, g ∈ D(Q) ∩ L∞(X, m), see [FOT11, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' A Dirichlet form Q is called regular if the following are satisfied: X is a locally compact separable metric space and m is a Radon measure of full support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' D(Q)∩Cc(X) is uniformly dense in Cc(X) and in D(Q) with respect to ∥·∥Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' In this case, the Q-capacity (or simply capacity if Q is fixed) of an open set O ⊆ X is defined by cap(O) = inf{∥f∥2 Q | f ∈ D(Q) with f ≥ 1 m-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' on O}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Here we use the convention cap(O) = ∞ if there does not exist f ∈ D(Q) with f ≥ 1 on O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' For an arbitrary set A ⊆ X, the capacity is defined by cap(A) = inf{cap(O) | O open with A ⊆ O}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' The capacity is inner regular, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=', for any Borel set A ⊆ X it satisfies cap(A) = sup{cap(K) | K compact with K ⊆ A}, see [FOT11, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Moreover, by [FOT11, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='7], the capacity for compact K ⊆ X can alternatively be described as cap(K) = inf{∥f∥2 Q | f ∈ D(Q) ∩ Cc(X) with f ≥ 1 on K}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' INTERMEDIATE DIRICHLET FORMS 5 A subset A of X is called polar if Cap(A) = 0 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' A property is said to hold quasi everywhere, abbreviated q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=', if it holds on the complement of a polar set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' A measurable function f : X → [−∞, ∞] is said to be quasi continuous if for every ε > 0 there is an open set O with Cap(O) < ε such that f|X\\O is finite-valued and continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' If Q is a regular Dirichlet form, then every f in D(Q) has a unique (up to equality quasi everywhere) quasi continuous representative ˜f, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' [CF12, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Domination of Dirichlet forms and semigroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' If U, V are sublattices of L2(X, m), we say that U is an order ideal in V if f ∈ U, g ∈ V and |g| ≤ |f| implies g ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' If U, V are subalgebras of L∞(X, m) we say U is an algebraic ideal in V if f ∈ U and g ∈ V implies fg ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' We will frequently use the following characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' The equivalence of (i) and (ii) is Ouhabaz’ domination criterion [Ouh96, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='7], whereas the equivalence with (iii) is taken from [Sch20a, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='1 (Characterization of Domination).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Let Q, Q′ be Dirichlet forms with associated self-adjoint operators L, L′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' The following assertions are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' (i) For all nonnegative f ∈ L2(X, m) and all t ≥ 0 we have e−tLf ≤ e−tL′f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' (ii) D(Q) ⊆ D(Q′), D(Q) is an order ideal in D(Q′) and Q(f, g) ≥ Q′(f, g) for all non-negative f, g ∈ D(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' (iii) D(Q) ⊆ D(Q′), D(Q) ∩ L∞(X, m) is an algebraic ideal in D(Q′) ∩ L∞(X, m) and Q(f, g) ≥ Q′(f, g) for all non-negative f, g ∈ D(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' If Q and Q′ satisfy one of the conditions of this proposition, we say that Q′ dominates Q and write Q ⪯ Q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Similarly, in this situation we write (e−tL) ⪯ (e−tL′) and say that the semigroup (e−tL′) dominates the semigroup (e−tL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Domination also induces an order relation on the set of all Dirichlet forms on L2(X, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Note that in general Q ⪯ Q′ does not imply Q ⊑ Q′ nor the other way round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' The maximal dominating form and an abstract characterization of sandwiched semigroups For every Dirichlet form Q there is a maximal Dirichlet form Q(M) (with respect to the natural order) that dominates the given Dirichlet form Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' In this section we describe the construction of this maximal form and give an abstract characterization 6 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' KELLER, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' LENZ, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' SCHMIDT, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' SCHWARZ, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' WIRTH of all Dirichlet forms Q′ that satisfy Q ⪯ Q′ ⪯ Q(M) provided that Q(M) is an extension of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' We denote by (Tt) the semigroup generated by Q and by (T (M) t ) the semigroup generated by Q(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' According to the discussion in Subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='2, any self-adjoint C0-semigroup (St) with (Tt) ⪯ (St) ⪯ (T (M) t ) corresponds to a Dirichlet form Q′ with Q ⪯ Q′ ⪯ Q(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Hence, our result can be seen as an abstract characterization of all semigroups sandwiched between (Tt) and (T (M) t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' The active main part and the killing part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' We will next recall the def- inition of the active main part and the killing part of a Dirichlet form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' For two concrete examples see Examples 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='6 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Let Q be a Dirichlet form on L2(X, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' For ϕ ∈ D(Q) with 0 ≤ ϕ ≤ 1 we define the domain of the quadratic form ˜Qϕ on L2(X, m) by D( ˜Qϕ) = {f ∈ L2(X, m) ∩ L∞(X, m) | fϕ, f 2ϕ ∈ D(Q)}, on which it acts by ˜Qϕ(f) = Q(ϕf) − Q(ϕf 2, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Since D(Q) ∩ L∞(X, m) is an algebra, we have D(Q) ∩ L∞(X, m) ⊆ D( ˜Qϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' The form ˜Qϕ is closable on L2(X, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Indeed, [Sch20a, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='1] shows that ˜Qϕ is lower semicontinuous on its domain with respect to local convergence in measure and hence it is lower semicontinuous on its domain with respect to L2-convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' We denote its closure by Qϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' The next proposition summarizes further important properties of Qϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Let ϕ, ψ ∈ D(Q) with 0 ≤ ϕ ≤ ψ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' (a) Qϕ is a Dirichlet form and its domain satisfies D(Qϕ) ∩ L∞(X, m) = D( ˜Qϕ) = {f ∈ L2(X, m) ∩ L∞(X, m) | fϕ ∈ D(Q)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' (b) D(Q) ⊆ D(Qϕ) and Qϕ(f) ≤ Q(f), f ∈ D(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' (c) D(Qψ) ⊆ D(Qϕ) and Qϕ(f) ≤ Qψ(f), f ∈ D(Qψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' This follows from [Sch20a, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' The proofs given there treat an extension of Qϕ to all measurable m-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' defined functions that is lower semicontin- uous with respect to local convergence in measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Restricting this form with larger domain to L2(X, m) yields all the claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' □ INTERMEDIATE DIRICHLET FORMS 7 Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Part (a) of this proposition is important because it yields a formula for Qϕ for bounded functions in its domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Namely, for f, g ∈ D(Qϕ) ∩ L∞(X, m) we have f, g ∈ D( ˜Qϕ) and hence Qϕ(f, g) = ˜Qϕ(f, g) = Q(ϕf, ϕg) − Q(ϕfg, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' For the last equality, we used the definition of ˜Qϕ and polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='3 (Active main part).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' The active main part Q(M) of Q is defined as follows: Its domain D(Q(M)) consists of all f ∈ L2(X, m) that satisfy f ∈ D(Qϕ) for all ϕ ∈ D(Q) with 0 ≤ ϕ ≤ 1 such that {ϕ ∈ D(Q) | 0 ≤ ϕ ≤ 1} → [0, ∞), ϕ �→ Qϕ(f) is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' On it Q(M) acts by Q(M)(f) = sup{Qϕ(f) | ϕ ∈ D(Q) with 0 ≤ ϕ ≤ 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Since ϕ �→ Qϕ(f) is monotone increasing, the form Q(M) is indeed a Dirichlet form, see [Sch20a, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' It turns out that Q(M) is the maximal Dirichlet form with respect to the natural order that dominates Q, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=', Q ⪯ Q(M) and for all Dirichlet forms Q′ with Q ⪯ Q′ we have Q′ ⊑ Q(M), see [Sch20a, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' However, Q(M) need not be an extension of Q and hence we introduce the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='4 (Killing part).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' The difference Q(k) = Q − Q(M) with domain D(Q(k)) = D(Q) is called the killing part of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' The killing part is a local and positive quadratic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Both properties are a consequence of Q(k) being monotone, see [Sch20a, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='11] for monotonicity and [Sch20a, Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='1] for how monotonicity implies the other properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' In particular, the value of Q(k)(f) only depends on |f| and not on the sign of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' We illustrate these objects with an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' It shows that the active main part is an abstract way of constructing operators with Neumann boundary conditions from the quadratic forms leading to Dirichlet boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='5 (Dirichlet and Neumann Laplacian on domains).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Let Ω ⊆ Rn be open (or more generally let Ω be a Riemannian manifold) and let V ∈ L1 loc(Ω) be nonnegative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' We consider the Dirichlet form E(N) V with domain D(E(N) V ) = {f ∈ H1(Ω) | V 1/2f ∈ L2(Ω)}, on which it acts by E(N) V (f) = � Ω |∇f|2dx + � Ω |f|2V dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' The associated operator is the self-adjoint realization of the Schrödinger operator H = −∆ + V with (abstract) Neumann boundary conditions, which we denote by H(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Moreover, we let E(D) V be the restriction of E(N) V to D(E(D) V ) = {f ∈ H1 0(Ω) | 8 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' KELLER, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' LENZ, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' SCHMIDT, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' SCHWARZ, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' WIRTH V 1/2f ∈ L2(Ω)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' This is a regular Dirichlet form and the associated operator is the self-adjoint realization of the Schrödinger operator H = −∆ + V with (abstract) Dirichlet boundary conditions, which we denote by H(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' The active main part of E(D) V is given by E(N) 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Hence, the self-adjoint operator associated to the active main part is −∆(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' For f ∈ D(E(D) V ), the killing part of E(D) V is given by (E(D) V )(k)(f) = � Ω |f|2V dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' In particular, if V = 0, this discussion shows that a Dirichlet form Q satisfies E(D) 0 ⪯ Q ⪯ (E(D) 0 )(M) if and only if the associated semigroup (St) satisfies (et∆(D)) ⪯ (St) ⪯ (et∆(N)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Hence, forms sandwiched between E(D) 0 and (E(D) 0 )(M) = E(N) 0 correspond to semigroups sandwiched between the Dirichlet and the Neumann semigroup of the Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' This is precisely the situation studied in [AW03].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Here we only sketch the main ideas of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' For the details we refer to [Sch20a, Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' We only consider bounded functions, the general case can be treated through approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Let f ∈ H1(Ω)∩L∞(Ω) and let ϕ ∈ C∞ c (Ω) with 0 ≤ ϕ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' A direct computation using the product rule for ∇ shows f ∈ D((E(D) V )ϕ) and (E(D) V )ϕ(f) = � Ω ϕ2|∇f|2dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Letting ϕ ր 1 and taking into account that C∞ c (Ω) is dense in D(E(D) V ) yields f ∈ D((E(D) V )(M)) and the formula for the action of (E(D) V )(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Similarly, if f ∈ D((E(D) V )(M))∩L∞(Ω), by the definition of (E(D) V )ϕ and the active main part, we have ϕf ∈ D(E(D) V ) = H1 0(Ω) ∩ L2(Ω, V · dx) for every ϕ ∈ C∞ c (Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' This yields ∇f ∈ ⃗L2 loc(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' With this at hand, an application of the product rule for ∇ as above shows (E(D) V )ϕ(f) = � Ω ϕ2|∇f|2dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Since (E(D) V )ϕ(f) ≤ (E(D) V )(M)(f) and ϕ is arbitrary, we conclude ∇f ∈ ⃗L2(Ω) so that f ∈ H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' The statement on the killing part is an immediate consequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' □ Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='6 (Fractional Laplacians).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' As above we let Ω ⊆ Rn be open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' For a background on fractional Sobolev spaces we refer to [DNPV12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' For 0 < s < 1, we denote by Qs,(N) the Dirichlet form with domain D(Qs,(N)) = W s(Ω) on which it acts by Qs,(N)(f) = 1 2 � Ω×Ω |f(x) − f(y)|2 |x − y|n+2s dx dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' The restriction of this form to W s 0 (Ω) is denoted by Qs,(D), it is a regular Dirichlet form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Note that at least if Ω is bounded and has C∞-boundary, the spaces W s 0 (Ω) and W s(Ω) coincide for 0 < s ≤ 1 2 by [LM72, Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='1], which makes the problem of finding the Dirichlet forms sandwiched between Qs,(D) and Qs,(N) trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' INTERMEDIATE DIRICHLET FORMS 9 It is well-known that the associated self-adjoint operators H(N) s and H(D) s are restrictions of the restricted fractional Laplacian Hs given by Hsf(x) = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' � Ω f(x) − f(y) |x − y|n+2s dy = lim ε→0+ � Ω\\Bε(x) f(x) − f(y) |x − y|n+2s dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Hence, they can be viewed as realizations of Hs with abstract Neumann and Dirich- let boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Note that we ignore a constant so that our fractional Laplacian is only a constant multiple of the ’usual’ restricted fractional Laplacian, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' [DNPV12, Section 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Similar as in the previous example the active main part of Qs,(D) is Qs,(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Here we only show the statement on the active main part of Qs,(D), the rest is well-known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Since Qs,(N) and (Qs,(D))(M) are Dirichlet forms, it suffices to prove D(Qs,(N)) ∩ L∞(Ω) = D((Qs,(D))(M)) ∩ L∞(Ω) and that Qs,(N) and (Qs,(D))(M) agree on these sets (use that bounded functions are dense in the domains of Dirichlet forms, see [FOT11, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' We first proof that Qs,(N) is a restriction of (Qs,(D))(M) (on L∞(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Let f ∈ W s(Ω) ∩ L∞(Ω) and let ϕ ∈ W s 0 (Ω) with 0 ≤ ϕ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Then fϕ ∈ W s 0 (Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' We infer Qs,(D)(ϕf) − Qs,(D)(ϕf 2, ϕ) = 1 2 � Ω×Ω (ϕ(x)f(x) − ϕ(y)f(y))2 |x − y|n+2s dx dy − 1 2 � Ω×Ω (ϕ(x)f(x)2 − ϕ(y)f(y)2)(ϕ(x) − ϕ(y)) |x − y|n+2s dx dy = 1 2 � Ω×Ω ϕ(x)ϕ(y)|f(x) − f(y)|2 |x − y|n+2s dx dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Taking the supremum over such ϕ yields f ∈ D((Qs,(D))(M)) and (Qs,(D))(M)(f) = Qs,(N)(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' It remains to prove D((Qs,(D))(M)) ∩ L∞(Ω) ⊆ W s(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Let f ∈ D((Qs,(D))(M)) ∩ L∞(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' For ϕ ∈ W s 0(Ω) with 0 ≤ ϕ ≤ 1, we have by definition of the main part ϕf, ϕf 2 ∈ W s 0 (Ω) and (Qs,(D))(M)(f) ≥ Qs,(D)(ϕf) − Qs,(D)(ϕf 2, ϕ) = 1 2 � Ω×Ω ϕ(x)ϕ(y)|f(x) − f(y)|2 |x − y|n+2s dx dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' For the last equality we used the same computation as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Since ϕ was arbitrary, this shows f ∈ W s(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' These examples show that it is a good intuition to think of a regular Dirichlet form Q with Q(k) = 0 as being a form with ‘Dirichlet type’ boundary con- ditions and Q(M) being the ’same’ form with ‘Neumann type’ boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' 10 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' KELLER, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' LENZ, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' SCHMIDT, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' SCHWARZ, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' WIRTH 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' An abstract characterization of sandwiched semigroups and forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' The main abstract result of this paper is the following characterization of Dirichlet forms sandwiched between a Dirichlet form without killing and its active main part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Let Q, Q′ be Dirichlet forms on L2(X, m) with Q(k) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' The fol- lowing assertions are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' (i) Q ⪯ Q′ ⪯ Q(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' (ii) (a) D(Q′) ⊆ D(Q(M)) and D(Q′) is an order ideal in D(Q(M)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' (b) Q′ − Q(M) is a positive and local form on D(Q′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' (c) Q′ is an extension of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' (i) =⇒ (ii): (a) This is a consequence of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' (b) The positivity of Q′ − Q(M) follows directly from Q′ ⪯ Q(M), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Proposi- tion 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' In order to see that Q′ − Q(M) is local, we let f, g ∈ D(Q′) with fg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Without loss of generality we may assume f, g ≥ 0, for otherwise we can decom- pose f, g into positive and negative parts and use f±g± = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Since Q′ and Q(M) are Dirichlet forms, we can further assume that f, g are bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' As we already established positivity, it remains to prove Q′(f, g) − Q(M)(f, g) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Let ϕ ∈ D(Q) with 0 ≤ ϕ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' According to Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='1 we have fϕ, gϕ ∈ D(Q), so that by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='1 f, g ∈ D(Qϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Using fg = 0 and Q ⪯ Q′ we obtain Q′(f, g) − Qϕ(f, g) = Q′(f, g) − Q(ϕf, ϕg) + Q(ϕfg, ϕ) = Q′(f, g) − Q′(ϕf, ϕg) = Q′((1 − ϕ)f, g) + Q′(ϕf, (1 − ϕ)g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' The functions η = (1 − ϕ)f and ζ = g are nonnegative and satisfy ηζ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' The Dirichlet form property of Q′ implies Q′(η + ζ) = Q′(|η + ζ|) = Q′(|η − ζ|) ≤ Q′(η − ζ), from which we deduce Q′(η, ζ) ≤ 0 by bilinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' The same argument applies to η = ϕf and ζ = (1 − ϕ)g so that we obtain Q′(f, g) − Qϕ(f, g) = Q′((1 − ϕ)f, g) + Q′(ϕf, (1 − ϕ)g) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' By the definition of Q(M) we can choose ϕ such that Qϕ(f, g) is arbitrarily close to Q(M)(f, g) and hence obtain locality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' (c) The domination Q ⪯ Q′ ⪯ Q(M) and Q(k) = 0 yield for all nonnegative f, g ∈ D(Q) the inequality Q(f, g) = Q(M)(f, g) ≤ Q′(f, g) ≤ Q(f, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' By splitting functions into positive and negative parts this shows Q = Q′ on D(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' (ii) =⇒ (i): Q′ ⪯ Q(M) follows directly from (a) and (b) and the characterization of domination Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' INTERMEDIATE DIRICHLET FORMS 11 Q ⪯ Q′: Since D(Q′) is contained in D(Q(M)) and D(Q) is an order ideal in D(Q(M)) we obtain that D(Q) is also an order ideal in D(Q′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Since Q′ is an extension of Q, this already implies domination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' □ We can rephrase this theorem slightly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Let Q be a Dirichlet form with Q(k) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' We say that a pair (F, q) consisting of a vector lattice F ⊆ D(Q(M)) that is an order ideal in D(Q(M)) and a quadratic form q with D(q) = F is an abstract admissible pair for Q, if it satisfies the following properties: D(Q) ⊆ D(q) and q(f) = 0 for f ∈ D(Q), q is local and positive, the form QF,q = Q(M)|F + q is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Let Q be Dirichlet forms with Q(k) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' The following assertions are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' (i) Q′ is a Dirichlet form with Q ⪯ Q′ ⪯ Q(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' (ii) There exists an abstract admissible pair (F, q) such that Q′ = QF,q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' (i) =⇒ (ii): This is a reformulation of the previous theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' (ii) =⇒ (i): Using the previous theorem it suffices to show that QF,q is a Dirichlet form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Since closedness and density of D(QF,q) = F are part of the definition of abstract admissible pairs, it suffices to prove the Markov property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' By assumption F is an order ideal in D(Q(M)) and for f ∈ F we have f+ ∧ 1 ∈ D(Q(M)) and |f+ ∧ 1| ≤ |f|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' This shows f+ ∧ 1 ∈ F whenever f ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Moreover, as already discussed after introducing the killing part, q being local and positive yields that q is monotone, see [Sch20a, Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' These observations and Q(M) being Markovian imply QF,q(f+ ∧ 1) = Q(M)(f+ ∧ 1) + q(f+ ∧ 1) ≤ Q(M)(f) + q(f) = QF,q(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' This corollary shows that in order to determine all sandwiched forms between Q and Q(M) we need to characterize all abstract admissible pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' This is possible when Q(M) is a regular Dirichlet form on a metric space K containing X as a dense open subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' In the next section we will prove that in this case: (a) Positive and local forms correspond to measures if their domain contains suf- ficiently many continuous functions, see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' If these forms satisfy q(f) = 0 for f ∈ D(Q), the corresponding measure is supported on the boundary K \\ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' (b) Closed order ideals in D(Q(M)) correspond to functions vanishing outside an open set (under some additional density assumption for continuous functions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' This then allows us to identify abstract admissible pairs with pairs of open subsets of the boundary and certain measures on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' 12 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' KELLER, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' LENZ, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' SCHMIDT, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' SCHWARZ, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' WIRTH 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Domination for parts of regular Dirichlet forms Let Q be a regular Dirichlet form on L2(X, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Let O ⊆ X be an open set and let µ be a Radon measure on the Borel σ-algebra of O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' We define the quadratic form Qc O,µ by letting D(Qc O,µ) = D(Q) ∩ Cc(O) and Qc O,µ(f) = Q(f) + � O f 2dµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Here Cc(O) is tacitly identified with {ϕ ∈ Cc(X) | supp ϕ ⊆ O}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' The following assertions are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' (i) µ charges no sets of Q-capacity zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' (ii) The quadratic form Qc O,µ is closable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' In this case, the closure QO,µ of Qc O,µ is given by D(QO,µ) = {f ∈ D(Q) | ˜f = 0 q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' on X \\ O and � O ˜f 2dµ < ∞}, QO,µ(f) = Q(f) + � O ˜f 2dµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' (i) =⇒ (ii): This follows as in [Sto92, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' (ii) =⇒ (i): By the inner regularity of the capacity and the inner regularity of the Radon measure µ it suffices to show for compact sets K ⊆ O that cap(K) = 0 implies µ(K) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Let now K ⊆ O be compact with cap(K) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Since Q is regular, there exists a sequence (ϕn) in D(Q) ∩ Cc(X) such that ∥ϕn∥Q → 0, 0 ≤ ϕn ≤ 1 and ϕn ≥ 1 on K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Let G be open and relatively compact with K ⊆ G ⊆ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Using regularity of Q again yields the existence of a function ψ ∈ D(Q) ∩ Cc(X) with 0 ≤ ψ ≤ 1, ψ = 1 on K and supp ψ ⊆ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' We now consider fn := ψ · ϕn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Since Q is a Dirichlet form, it satisfies fn ∈ D(Q) ∩ Cc(O) = D(Qc O,µ) and Q(fn)1/2 ≤ Q(ψ)1/2 + Q(ϕn)1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' The inequality 0 ≤ fn ≤ 1 and supp fn ⊆ G imply ∥fn∥ → 0 as n → ∞ and � O |fn|2dµ ≤ µ(G), n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' In particular, these estimates show that (fn) is bounded with respect to the form norm ∥·∥QO,µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Let QO,µ be the closure of Qc O,µ, which exists by (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' The Banach–Saks theorem implies that for some subsequence (fnk) the sequence of Césaro means gN := 1 N N � k=1 fnk converges to some g ∈ D(QO,µ) with respect to ∥·∥QO,µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' The form norm of QO,µ is larger than ∥·∥ and hence we obtain gn → g with respect to ∥·∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' But since INTERMEDIATE DIRICHLET FORMS 13 ∥fn∥ → 0, we conclude g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' By the choice of (fn) we also have gN ∈ D(Q)∩Cc(O), 0 ≤ gN ≤ 1 and gN ≥ 1 on K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Putting everything together we obtain µ(K) ≤ � O |gN|2dµ ≤ QO,µ(gN) → 0 as N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' This yields the desired µ(K) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' The proof of the formula for QO,µ follows as in [SV96, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' □ Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' A pair (O, µ) satisfying one of the conditions of the previous the- orem is called an admissible pair for the form Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' In this case, we write QO,µ for the closure of the form Qc O,µ above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Let (Oi, µi), i = 1, 2, be admissible pairs for Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' The following assertions are equivalent: (i) QO1,µ1 ⪯ QO2,µ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' (ii) cap(O1 \\ O2) = 0 and µ2(A) ≤ µ1(A) for every Borel set A ⊆ O1 ∩ O2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' (ii) =⇒ (i): This follows immediately from (ii) and the formula for QOi,µi given in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' (i) =⇒ (ii): Since D(QO1,µ1) is a lattice and an order ideal in D(QO2,µ2), we have D(QO1,µ1) ⊆ D(QO2,µ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Hence, every f ∈ D(QO1,µ1) satisfies ˜f = 0 q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' on X \\ O2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Now, suppose cap(O1 \\ O2) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' By [FOT11, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='1] there exists a compact set K ⊆ O1 \\ O2 with cap(K) > 0 and by [FOT11, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='5] there exists f ∈ D(Q) with 0 ≤ f ≤ 1 and ˜f = 1 q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' on K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' By the regularity of Q there exists ϕ ∈ D(Q) ∩ Cc(O1) with ϕ ≥ 1 on K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' We obtain ϕf ∈ D(QO1,µ1) as ϕ ˜f = 0 q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' on X \\ O1 and � O1 |ϕ ˜f|2dµ1 ≤ � O1 |ϕ|2dµ1 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Furthermore, ϕ ˜f ≥ 1 q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' on K ⊆ X \\ O2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' This and cap(K) > 0 are a contradiction to the fact that functions in D(QO1,µ1) vanish q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' on X \\ O2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' It remains to prove the inequality for the measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Domination implies Q(ϕ) + � O2 |ϕ|2dµ2 ≤ Q(ϕ) + � O1 |ϕ|2dµ1 for all nonnegative ϕ ∈ D(Q) ∩ Cc(O1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' For any compact set K ⊆ O1 ∩ O2 and any open neighborhood G of K with G ⊆ O1 ∩ O2 there exists ψ ∈ Cc(X) ∩ D(Q) with supp ψ ⊆ G, 0 ≤ ψ ≤ 1 and ψ ≥ 1 on K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Plugging this into the last inequality yields µ2(K) ≤ � O2 |ψ|2dµ2 ≤ � O1 |ψ|2dµ1 ≤ µ1(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Thus we obtain µ2(K) ≤ µ1(K) from the outer regularity of the Radon measure µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' By inner regularity of Radon measures this implies the statement for all Borel sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' □ 14 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' KELLER, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' LENZ, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' SCHMIDT, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' SCHWARZ, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' WIRTH 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' A boundary for regular Dirichlet forms and a characterization of sandwiched semigroups 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' A boundary for regular Dirichlet forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Let Q be a regular Dirichlet form on L2(X, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' In this subsection we introduce a locally compact separable metric space K that contains X as an open subset and extend m to a Radon measure ˆm on K such that Q(M) can be considered to be a regular form on L2(K, ˆm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' The spaces Cc(X) and L2(X, m) are separable because X is locally compact sep- arable metric space and m is a Radon measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' The map L2(X, m) → D(Q(M)), f �→ (L(M) + 1)−1f is continuous with respect to the form norm ∥·∥Q(M) (here L(M) denotes the positive self-adjoint operator associated with Q(M)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' It has dense image D(L(M)) in D(Q(M)) with respect to ∥·∥Q(M), showing that (D(Q(M)), ∥·∥Q(M)) is also separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Moreover, [Sch20a, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='3] asserts that for a regular Dirichlet form Q the space D(Q(M))∩ Cb(X) is dense in D(Q(M)) with respect to ∥·∥Q(M) (this is an abstract version of the Meyers-Serrin theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Combining these observations yields the existence of a subalgebra C of D(Q(M))∩ Cb(X) with the following three properties: C is countably generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' C is ∥·∥Q(M)-dense in D(Q(M)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' C ∩ Cc(X) is uniformly dense in Cc(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Let A be the uniform closure of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Its complexification AC = {f + ig | f, g ∈ A} is a commutative C∗-algebra that satisfies C0(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' C) ⊆ AC ⊆ Cb(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' By Gelfand theory there exists a unique (up to homeomorphism) locally compact, separable Hausdorff space K with the following properties: X is a dense and open subset of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Every f ∈ AC can be extended to a function ˆf ∈ C0(K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' C) and C0(K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' C) = { ˆf | f ∈ AC}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' As C is countably generated, the space K is metrizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Hence, K is Polish, that is, separable and completely metrizable, since every locally compact, separable, second countable space is Polish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Since X is dense in K, the continuous extension of a function from A to K is unique and we will therefore not distinguish between ele- ments of A and their extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' For real-valued functions this interpretation leads to A = C0(K)(= C0(K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' R)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' The measure m on X can be extended to a Borel measure ˆm on K by setting ˆm(A) = m(A ∩ X), A ∈ B(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' The measure ˆm is again a Radon measure of full support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' By this definition the space L2(K, ˆm) can be naturally identified with L2(X, m) via the unitary map R: L2(K, ˆm) → L2(X, m), f �→ f|X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' INTERMEDIATE DIRICHLET FORMS 15 Our discussion shows R−1(A∩L2(X, m)) = C0(K)∩L2(K, ˆm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Since R also preserves the order relation, any Dirichlet form on L2(X, m) can be viewed as a Dirichlet form on L2(K, ˆm) under this transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' In particular, the form Q(M) is a regular Dirichlet form on L2(K, ˆm), see [Sch20a, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' The following remark sketches the uniqueness of the space K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' We leave details (especially the involved definitions, which can be found in [FOT11, Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='4]) to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='1 (Uniqueness of K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' The space K depends on the choice of the algebra C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' However, given two algebras C, C′ with the required properties and the corresponding spaces K, K′, there exists a unitary order isomorphism U : L2(K, ˆm) → L2(K′, ˆm′) such that 0 ≤ fn ≤ 1, fn ր 1 implies Ufn ր 1 and U intertwines Q(M) and Q(M) (when considererd as a form on the corresponding space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' This implies that both forms are equivalent in the sense of [FOT11, Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Since they are also regular, [FOT11, Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='2] yields that K and K′ are quasi-homeomorphic (and establishes further properties of a corresponding quasi-homeomorphism).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' In view of the previous remark we make the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' The set ∂X = K \\ X is called the boundary of X relative to the form Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='3 (Dirichlet and Neumann Laplacian – continued).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' We use the situation and notation of Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='5 and assume that the potential vanishes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=', V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' As discussed in Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='5 we have (E(D) 0 )(M) = E(N) 0 so that D((E(D) 0 )(M)) = H1(Ω) and the standard Sobolev norm on H1(Ω) coincides with the form norm of (E(D) 0 )(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' If Ω ⊆ Rn has continuous boundary (for a precise definition see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' [EE18, Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='1]), the space {f|Ω | f ∈ C∞ c (Rn)} is dense in H1(Ω) with respect to the standard Sobolev norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Hence, in this case we can choose the algebra C to be a subset of {f|Ω | f ∈ C∞ c (Rn)} ⊆ Cc(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Since by Stone-Weierstraß {f|Ω | f ∈ C∞ c (Rn)} is dense in C0(Ω), we can further assume that C is dense in C0(Ω), so that the algebra A, the uniform closure of C, equals C0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Our construction then yields K = Ω (up to homeomorphism) and that the boundary of Ω relative to E(D) 0 coincides with the metric boundary ∂Ω = Ω \\ Ω in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='4 (Fractional Laplacian – continued).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' We use the situation and notation of Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' As discussed above we have D((Qs,(D))(M)) = W s(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Moreover, if Ω has Lipschitz boundary, then {f|Ω | f ∈ C∞ c (Rn)} is dense in W s(Ω) with respect to the form norm of Qs,(N), which coincides with the ususal norm on W s(Ω), see [DNPV12, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' With this at hand the same argument as in the previous example yields that we can choose K = Ω such that the boundary of Ω relative to Qs,(D) coincides with the metric boundary ∂Ω = Ω \\ Ω in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' 16 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' KELLER, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' LENZ, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' SCHMIDT, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' SCHWARZ, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' WIRTH 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' A characterization of sandwiched semigroups for regular Dirichlet forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' In this subsection we prove the main result of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Let Q be a regular Dirichlet form on L2(X, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' We apply the theory developed in Section 3 to the form Q(M) when considered as a regular Dirichlet form on L2(K, ˆm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' We start with a simple observation that follows from the previous discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Let Q be a regular Dirichlet form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Then (K, 0) and (X, 0) are admissible pairs for the regular Dirichlet form Q(M) on L2(K, ˆm) and we have Q(M) = (Q(M))K, 0 and Q = (Q(M))X, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' In particular, D(Q) = {f ∈ D(Q(M)) | ˜f = 0 q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' on ∂X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' The following is the main result of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Let Q be a regular Dirichlet form with Q(k) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' For a Dirichlet form Q′ the following assertions are equivalent: (i) There exists an admissible pair (O, µ) for Q(M) with X ⊆ O and µ(X) = 0 such that Q′ = (Q(M))O,µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' (ii) (a) Q ⪯ Q′ ⪯ Q(M) (b) D(Q′) ∩ Cc(K) is dense in D(Q′) with respect to ∥·∥Q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' (i) =⇒ (ii): (a) follows from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='3 and the identities discussed in Propostion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' The density of D((Q(M))O,µ) ∩ Cc(O) in D((Q(M))O,µ) with respect to ∥·∥(Q(M))O,µ is part of the definition of the form (Q(M))O,µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' (ii) =⇒ (i): Let D be the uniform closure of the algebra D(Q′) ∩ Cc(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Since D(Q′) ∩ Cc(K) is an algebraic ideal in D(Q(M)) ∩ Cc(K) (here we use domination and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='3), D is a uniformly closed ideal in D(Q(M)) ∩ Cc(K) ∥·∥∞ = C0(K) (here we use the regularity of Q(M)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Moreover, by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='8 we have D(Q) ⊆ D(Q′) so that D(Q)∩Cc(X) ⊆ D(Q′)∩Cc(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Since Q is regular on L2(X, m), this yields C0(X) ⊆ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' By the characterization of closed ideals in C0(K) there exists an open set X ⊆ O ⊆ K such that D = {f ∈ C0(K) | f = 0 on K \\ O}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Altogether this discussion shows that D(Q′) ∩ Cc(O) is ∥·∥Q′ dense in D(Q′) and uniformly dense in Cc(O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Next, we show D(Q′) ∩ Cc(O) = D(Q(M)) ∩ Cc(O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Let ϕ ∈ D(Q(M)) ∩ Cc(O) and let K = supp ϕ ⊆ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Since Q′ is a Dirichlet form and D(Q′) ∩ Cc(O) is uniformly dense in Cc(O), there exists ψ ∈ D(Q′) ∩ Cc(O) with ψ = 1 on K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' We obtain ϕ = ψϕ ∈ D(Q′) ∩ Cc(O) since D(Q′) ∩ Cc(O) is an algebraic ideal in D(Q(M)) ∩ Cc(O) (here we use domination and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' According to Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='8 the domination (a) implies that the form q = Q′−Q(M) with domain D(q) = D(Q(M)) ∩ Cc(O) is positive, local and satisfies q(f) = 0 for INTERMEDIATE DIRICHLET FORMS 17 all f ∈ D(Q) ∩ Cc(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' By Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='4 there exists a Radon measure µ on O such that q(f) = � O |f|2dµ, f ∈ D(Q(M)) ∩ Cc(O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Since D(Q) ∩ Cc(X) is uniformly dense in Cc(X), the property q(f) = 0 for f ∈ D(Q) ∩ Cc(X) implies µ(X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' For f ∈ D(Q(M)) ∩ Cc(O), we have by definition of q that Q′(f) = Q(M)(f) + � O |f|2dµ = (Q(M))c O,µ(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Since Q′ is closed and D(Q(M)) ∩ Cc(O) is ∥·∥Q′-dense in D(Q′), this implies that (O, µ) is an admissible pair for Q(M) and Q′ = (Q(M))O,µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' □ We can reformulate this theorem as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Let Q be a regular Dirichlet form with Q(k) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' For a Dirichlet form Q′, the following assertions are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' (i) There exists an open subset ∂µX ⊆ ∂X and a Radon measure µ on ∂µX that does not charge sets of Q(M)-capacity zero such that D(Q′) = {f ∈ D(Q(M)) | ˜f = 0 q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' on ∂X \\ ∂µX and � ∂µX | ˜f|2dµ < ∞} and Q′(f) = Q(M)(f) + � ∂µX | ˜f|2dµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' (ii) (a) Q ⪯ Q′ ⪯ Q(M) (b) D(Q′) ∩ Cc(K) is dense in D(Q′) with respect to ∥·∥Q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' As an application of this result and our examples we obtain one of the main results of [AW03] under slightly less restrictive assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Again we use the situation of Schrödinger operators on Ω of Exam- ple 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='5 with V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Assume further that Ω ⊆ Rn has continuous boundary and let Q be a Dirichlet form on L2(Ω) with associated Markovian semigroup (St).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Let ∂Ω = Ω \\ Ω be the metric boundary of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' The disscusion in Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='5 and Ex- ample 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='3 combined with the previous corollary yield that the following assertions are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' (i) There exists an open subset ∂µΩ ⊆ ∂Ω and a Radon measure µ on ∂µΩ that does not charge sets of E(N) 0 capacity zero such that D(Q) = {f ∈ H1(Ω) | ˜f = 0 q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' on ∂Ω \\ ∂µΩ and � ∂µΩ | ˜f|2dµ < ∞} and Q(f) = � Ω |∇f|2dx + � ∂µΩ | ˜f|2dµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' (ii) (a) (et∆(D)) ⪯ (St) ⪯ (et∆(N)) 18 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' KELLER, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' LENZ, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' SCHMIDT, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' SCHWARZ, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' WIRTH (b) D(Q) ∩ Cc(Ω) is dense in D(Q) with respect to ∥·∥Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' This is precisely the statement of [AW03, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='1] under the slightly less restric- tive assumption of Ω having continuous boundary instead of Lipschitz boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' We use the situation of fractional Laplacians of Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Assume further that Ω ⊆ Rn has Lipschitz boundary and let Q be a Dirichlet form on L2(Ω) with associated Markovian semigroup (St).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Let ∂Ω = Ω \\ Ω be the metric boundary of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' The disscusion in Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='6 and Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='4 combined with the previous corollary yield that the following assertions are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' (i) There exists an open subset ∂µΩ ⊆ ∂Ω and a Radon measure µ on ∂µΩ that does not charge sets of Qs,(N)-capacity zero such that D(Q) = {f ∈ W s(Ω) | ˜f = 0 q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' on ∂Ω \\ ∂µΩ and � ∂µΩ | ˜f|2dµ < ∞} and Q(f) = 1 2 � Ω×Ω |f(x) − f(y)|2 |x − y|n+2s dx dy + � ∂µΩ | ˜f|2dµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' (ii) (a) (e−tH(D) s ) ⪯ (St) ⪯ (e−tH(N) s ) (b) D(Q) ∩ Cc(Ω) is dense in D(Q) with respect to ∥·∥Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' The implication (i) =⇒ (ii) was also proved for Dirichlet forms associated with a re- lated, but different fractional Laplacian by Claus and Warma [CW20, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' As mentioned in the introducion we wanted to provide a version of the results of [AW03] for general Dirichlet forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' In the abstract framework we were as general as possible but held back in generality for regular Dirichlet forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' In the following remarks we collect what else can be deduced from our general framework (at the cost of brevity and technical simplicity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' (a) Another result of [AW03] is the descritption of the operators corresponding to semigroups (et∆(D)) ⪯ (St) ⪯ (et∆(N)) as Laplacians with Robin type boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Something similar is possible here after equipping the abstract boundary ∂X with so-called harmonic measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' This allows for the definition of densities of normal derivatives and leads to abstract Robin boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' In the Euclidean setting with Ω having Lipschitz boundary the harmonic measures are mutually absolutely continuous with respect to the surface measure on ∂Ω and the abstract normal derivatives are given by the usual normal derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' (b) In Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='6 we used that D(Q′) ∩ Cc(K) is dense in D(Q′) because we constructed the set O as the complement of the zero set of the closed ideal D(Q′) ∩ Cc(K) ∥·∥∞ in C0(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' One can drop the density assumption and replace this argument by the characterization of closed ideals in regular Dirichlet spaces given in [Sto93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' INTERMEDIATE DIRICHLET FORMS 19 In this case, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='6 remains true without assertion (ii)(b) but with O open replaced by O quasi-open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' (c) We always assumed that the killing part vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' If Q(k) ̸= 0, then there are two possible choices of reference for the maximal form: (1) One can characterize all Dirichlet forms Q′ with Q ⪯ Q′ ⪯ Q(M) via abstract admissible pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Since in this case Q(M) is not an extension of Q, the form q of the abstract admissible pair corresponding to Q′ does not vanish on D(Q) but is bounded above by Q(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' In the regular setting this implies that the measure µ from the admissible pair corresponding to Q′ is not necessarily supported only on ∂X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' It satisfies µ ≤ k on X, where k is the measure corresponding to the local and positive form Q(k) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' (2) Instead of comparing Q′ with Q(M) one can characterize Q ⪯ Q′ ⪯ Qref, where Qref is the active reflected Dirichlet form of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' It arises by adding a suitable extension of Q(k) to Q(M), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' [Sch20a, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' In this case, our main theorems still hold true but the proofs become substantially longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Bilinear forms on Cc(X) Let X be a locally compact metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' In this section we provide a character- ization of positive and local forms defined on Cc(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' First we show that densely defined positive forms on Cc(X) can be extended to the whole of Cc(X) if their domain is a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' In a second step we prove a representation theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Certainly both results are well-known to experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Since we could not find a proper reference, we include the proofs for the convenience of the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' In the following lemma we write C(K) for the subspace {f ∈ Cc(X) | supp f ⊆ K}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Let q be a densely defined (with respect to the uniform norm) quadratic form on D(q) ⊆ Cc(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Suppose q is positive and D(q) is a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' (a) For any compact K ⊆ X the restriction of q to D(q) ∩ C(K) is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' (b) q can be uniquely extended to a positive quadratic form on Cc(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' We first show that for any compact set K ⊆ X the restriction of q to D(q) ∩ C(K) is continuous with respect to the supremum norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Let f, g ∈ D(q) ∩ C(K) be nonnegative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Let θK ∈ D(q) be such that θK ≥ 0 and θK ≥ 1 on K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Such a functions exists because D(q) is a dense lattice in Cc(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Without loss of generality we assume q(∥f∥∞θK, g) − q(∥g∥∞θK, f) ≤ 0, for otherwise we could interchange f and g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Then, using the positivity of q, we get 0 ≤ q(∥f∥∞θK − f, ∥g∥∞θK + g) = −q(f, g) + ∥f∥∞∥g∥∞q(θK, θK) + q(∥f∥∞θK, g) − q(∥g∥∞θK, f) ≤ −q(f, g) + ∥f∥∞∥g∥∞q(θK, θK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' 20 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' KELLER, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' LENZ, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' SCHMIDT, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' SCHWARZ, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' WIRTH This implies 0 ≤ q(f, g) ≤ ∥f∥∞∥g∥q(θK, θK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' For arbitrary f, g ∈ D(q) ∩ C(K) we have f +, f −, g+, g− ∈ D(q) ∩ C(K) because D(q) is a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' We obtain |q(f, g)| ≤ q(f +, g+) + q(f +, g−) + q(f −, g+) + q(f −, g−) ≤ 4∥f∥∞∥g∥∞q(θK, θK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Using this continuity in order to prove that q can be uniquely extended to a posi- tive quadratic form on Cc(X) it suffices to show the following: For every nonnegative ϕ ∈ Cc(X) there exists a compact K ⊆ X with supp ϕ ⊆ K such that ϕ can be approximated by nonnegative functions in D(q) ∩ C(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' To this end, we choose a nonnegative θ ∈ D(q) with θ ≥ ∥ϕ∥∞ on supp ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Such a function exists because D(q) is a dense lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Let K = supp θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Since q is densely defined, there exists ( ˜ϕn) in D(q) with ˜ϕn → ϕ uniformly, as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Since D(q) is a lattice, the sequence ϕn = ( ˜ϕn)+ ∧ θ belongs to D(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' It is nonnegative and supp ϕn ⊆ supp θ = K for all n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Moreover, using that 0 ≤ ϕ ≤ θ, we obtain ϕn → ϕ uniformly, as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' □ The following theorem provides a characterization of monotone quadratic forms on Cc(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Let q: Cc(X) → [0, ∞) be a quadratic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' The following asser- tions are equivalent: (i) q is positive and local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' (ii) For all f, g ∈ Cc(X) the inequality fg ≥ 0 implies q(f, g) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' (iii) For all f, f ′, g, g′ ∈ Cc(X) the inequality fg ≥ f ′g′ implies q(f, g) ≥ q(f ′, g′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' (iv) q is monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' (v) There exists a Radon measure µ on X such that q(u) = � X f 2dµ, f ∈ Cc(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' In this case, the measure µ is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Clearly, (ii) implies (i), (iii) implies (ii) and (v) implies all other assertions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' (i) =⇒ (iv): Let f, g ∈ Cc(X) with |g| ≤ |f|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' The positivity of q yields q(|f|) = q(|g|, |f|) + q(|f| − |g|, |f|) ≥ q(|g|, |f|) = q(|g|) + q(|g|, |f| − |g|) ≥ q(|g|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' It is left to show q(f) = q(|f|) for every f ∈ Cc(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Since f +, f − ∈ Cc(X) and f +f − = 0, the locality of q implies q(f +, f −) = 0 and hence q(f) = q(f +) − 2q(f +, f −) + q(f −) = q(f +) + 2q(f +, f −) + q(f −) = q(|f|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' INTERMEDIATE DIRICHLET FORMS 21 (iv) =⇒ (ii): Let f, g ∈ Cc(X) with fg ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Then |f + g| ≥ |f − g| so that by monotonicty q(f) + q(g) − 2q(f, g) = q(f − g) ≤ q(f + g) ≤ q(f) + q(g) + 2q(f, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' This shows (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' We already proved the equivalence of (i),(ii) and (iv) and that these assertions are implied by (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Next we prove that they imply (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Let f, f ′, g, g′ ∈ Cc(X) with fg ≥ f ′g′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' If f = f ′, the inequality q(f, g) ≥ q(f ′, g′) directly follows from (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' With the help of an approximation we reduce the case f ̸= f ′ to this one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' We start with the following observation: Locality of q implies that for ϕ, χ ∈ Cc(X) the value q(ϕ, χ) is independent of χ as long as χ = 1 on supp ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' In this case, we write I(ϕ) := q(ϕ, χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Let ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' By compacteness of the supports we can choose finitely many rela- tively compact open sets Gj, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' , N, that cover the union of the supports of f, f ′, g, g′, and choose ξj ∈ Gj, such that sup x∈Gj |f(x) − f(ξj)| < ε and sup x∈Gj |f ′(x) − f ′(ξj)| < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' We let χj ∈ Cc(X), j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=', N, be a subordinate partition of unity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' 0 ≤ χj ≤ 1, suppχj ⊆ Gj and N � j=1 χj = 1 on N � j=1 Gj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Such a partition of unity exists because metric spaces are normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' We define ˜f = N � j=1 f(ξj)χj and ˜f ′ = N � j=1 f ′(ξj)χj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Then, using � j χj = 1 on the supports of f, g, we obtain |q( ˜f, g) − q(f, g)| ≤ N � j=1 |q(χj(f − f(ξj)), g)| ≤ εq( N � j=1 χj, |g|) = εI(|g|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' For the second inequality we used |q(ϕ, ψ)| ≤ q(|ϕ|, |ψ|), which directly follows from the positivity of q, and the fact that |χj(f − f(ξj))| ≤ εχj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Similarly, we have 22 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' KELLER, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' LENZ, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' SCHMIDT, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' SCHWARZ, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' WIRTH |q( ˜f ′, g′) − q(f ′, g′)| ≤ εI(|g′|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Moreover, q( ˜f, g) − q( ˜f ′, g′) = N � j=1 (q(f(ξj)χj, g) − q(f ′(ξj)χj, g′)) = N � j=1 q(χj, f(ξj)g − f ′(ξj)g′) ≥ N � j=1 q(χj, fg − f ′g′) − εq( N � j=1 χj, |g| + |g|′) ≥ −εI(|g| + |g′|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' For the first inequality we used (ii) and the estimate χj(f(ξj)g − f ′(ξj)g′) ≥ χj(fg − f ′g′ − ε(|g| + |g′|)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' The last inequality follows from χj(fg − f ′g′) ≥ 0 and � j χj = 1 on the support of |g| + |g′|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Since ε > 0 was arbitrary, these estimates show (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' (iii) =⇒ (v): As above we define I : Cc(X) → R by letting I(ϕ) = q(χ, ϕ) for some χ ∈ Cc(X) with χ = 1 on the support of ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' It follows from (iii) that this is well-defined and positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Moreover, I is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' By the Riesz-Markov-Kakutani representation theorem there exists a unique Radon measure µ such that I(ϕ) = � X ϕdµ for all ϕ ∈ Cc(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Let now f, g ∈ Cc(X) an let χ ∈ Cc(X) such that χ = 1 on the supports of f and g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Since fg = χ(fg), property (iii) yields q(f, g) = q(χ, fg) = I(fg) = � X fgdµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Thus, µ is the desired measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' □ Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' The statement of the theorem is not only valid for quadratic forms on continuous functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' The equivalence of (ii) and (iv) was observed in [Sch20a, Appendix B] for quadratic forms on sublattices of L0(Y, m), where Y is an arbitrary set and m is a measure on Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Indeed, the above proof yields the equivalence of (i),(ii) and (iv) in this situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' The equivalence with (iii) requires the existence of suitable partitions of unity in the domain of q and the equivalence with (v) requires that the domain of q is an algebra and a representation theorem for positive functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Let q be a densely defined positive and local quadratic form on Cc(X) such that D(q) is a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Then there exists a unique Radon measure µ on X such that q(f) = � X f 2dµ, f ∈ D(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' INTERMEDIATE DIRICHLET FORMS 23 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' As noted in the previous remark the form q is also monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' By Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='1 it can be uniquely extended to a positive quadratic form on Cc(X) and by the continuity of restrictions to compact sets this extension is also monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Hence, the statement follows from the previous theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' □ References [ACD21] Sahiba Arora, Ralph Chill, and Jean-Daniel Djida.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Domination of semigroups generated by regular forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' [Akh18] Khalid Akhlil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Locality and domination of semigroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Results Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=', 73(2):Art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' 59, 11, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' [AW03] Wolfgang Arendt and Mahamadi Warma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Dirichlet and Neumann boundary conditions: What is in between?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Evol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Equ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=', 3(1):119–135, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Dedicated to Philippe Bénilan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' [CF12] Zhen-Qing Chen and Masatoshi Fukushima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Symmetric Markov processes, time change, and boundary theory, volume 35 of London Mathematical Society Monographs Series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Princeton University Press, Princeton, NJ, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' [Cla21] Burkhard Claus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Non-linear Dirichlet forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' PhD thesis, TU Dresden, Dresden, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' [CW12] Ralph Chill and Mahamadi Warma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Dirichlet and Neumann boundary conditions for the p-Laplace operator: what is in between?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Proceedings of the Royal Society of Ediburgh: Section A Mathematics, 142(5):975–1002, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' [CW20] Burkhard Claus and Mahamadi Warma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Realization of the fractional Laplacian with nonlocal exterior conditions via form methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Evol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Equ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=', 20(4):1597–1631, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' [DNPV12] Eleonora Di Nezza, Giampiero Palatucci, and Enrico Valdinoci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Hitchhiker’s guide to the fractional Sobolev spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=', 136(5):521–573, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' [EE18] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Edmunds and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Evans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Spectral theory and differential operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Oxford Mathematical Monographs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Oxford University Press, Oxford, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' [FOT11] Masatoshi Fukushima, Yoichi Oshima, and Masayoshi Takeda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Dirichlet forms and sym- metric Markov processes, volume 19 of De Gruyter Studies in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Walter de Gruyter & Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=', Berlin, extended edition, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' [KLSS19] Matthias Keller, Daniel Lenz, Marcel Schmidt, and Michael Schwarz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Boundary repre- sentation of Dirichlet forms on discrete spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Pures Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' (9), 126:109–143, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' [LM72] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Lions and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Magenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Non-homogeneous boundary value problems and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Die Grundlehren der mathematischen Wissenschaften, Band 181.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Springer-Verlag, New York-Heidelberg, 1972.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Translated from the French by P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Kenneth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' [Ouh96] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Ouhabaz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Invariance of closed convex sets and domination criteria for semigroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Potential Analysis, 5(6):611–625, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' [Pos14] Andrea Posilicano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Markovian extensions of symmetric second order elliptic differential operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Nachr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=', 287(16):1848–1885, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' [RS78] Michael Reed and Barry Simon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Methods of modern mathematical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Analysis of operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' [Sch17] Marcel Schmidt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Energy forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' PhD thesis, Mar 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' [Sch20a] Marcel Schmidt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' A note on reflected Dirichlet forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Potential Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=', 52(2):245–279, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' [Sch20b] Michael Schwarz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Nodal Domains and Boundary Representation for DirichletForms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' PhD thesis, Jan 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' [Sto92] Peter Stollmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Smooth perturbations of regular Dirichlet forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=', 116(3):747–752, 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' [Sto93] Peter Stollmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Closed ideals in Dirichlet spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Potential Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=', 2(3):263–268, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' 24 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' KELLER, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' LENZ, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' SCHMIDT, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' SCHWARZ, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' WIRTH [SV96] Peter Stollmann and Jürgen Voigt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Perturbation of Dirichlet forms by measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Potential Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=', 5(2):109–138, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='Keller, Institut für Mathematik, Universität Potsdam, Campus Golm, Haus 9, Karl-Liebknecht-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' 24-25, 14476 Potsdam OT Golm, Germany Email address: matthias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='keller@uni-potsdam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='de D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='Lenz, Institut für Mathematik, Friedrich-Schiller-Universität Jena, 07737 Jena, Germany Email address: daniel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='lenz@uni-jena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='de M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Schmidt, Mathematisches Institut, Universität Leipzig, Augustusplatz 10, 04109 Leipzig, Germany Email address: marcel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='schmidt@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='uni-leipzig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='de M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Schwarz, dotSource GmbH, Goethestr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' 1, 07743 Jena, Germany Email address: m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='schwarz@dotSource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='de M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content=' Wirth, Institute of Science and Technology Austria (ISTA), Am Campus 1, 3400 Klosterneuburg, Austria Email address: melchior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='wirth@ist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} +page_content='at' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAzT4oBgHgl3EQfGvvx/content/2301.01035v1.pdf'} diff --git a/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf b/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..b36b58ed382ab33d5a1ac291a4d8f73ee64de665 --- /dev/null +++ b/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:cab12ad8bf2a20dde026abc6624020f55b33b8607dfa3da569542690d7c76511 +size 6153325 diff --git a/F9AzT4oBgHgl3EQfxP6U/content/tmp_files/2301.01735v1.pdf.txt b/F9AzT4oBgHgl3EQfxP6U/content/tmp_files/2301.01735v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1b365c686a369baab8172b9175622b524a5879e8 --- /dev/null +++ b/F9AzT4oBgHgl3EQfxP6U/content/tmp_files/2301.01735v1.pdf.txt @@ -0,0 +1,552 @@ +arXiv:2301.01735v1 [math.MG] 4 Jan 2023 +NORMED SPACES USING INTRINSICALLY LIPSCHITZ SECTIONS +AND EXTENSION THEOREM FOR THE INTRINSICALLY HÖLDER +SECTIONS +DANIELA DI DONATO +Abstract. The purpose of this article is twofold: first of all, we want to define two norms +using the space of intrinsically Lipschitz sections. On the other hand, we want to generalize +an Extension Theorem proved by the author in the context of the intrinsically Hölder sections +with target a topological space Y. Here our target will be Y × Rs with s ≥ 1 instead of Y. +Contents +1. +Introduction +1 +2. +Intrinsically Hölder sections +3 +2.1. +Intrinsically Hölder sections: when Y is bounded +3 +2.2. +Equivalent definition for intrinsic Hölder sections +4 +3. +Normed space for the intrinsically Lipschitz sections +5 +3.1. +Normed space: Version 1 +5 +3.2. +Normed space: Version 2 +6 +4. +Level sets and extensions +7 +References +9 +1. Introduction +In this paper, we focus our attention on a new point of view for the intrinsically Lipschitz +graphs in the Franchi-Serapioni-SerraCassano sense [FSSC01, FSSC03b, FSSC03a] (see also +[SC16, FS16]) in metric spaces. +They introduced and analized this notion in order to establish a good notion of rectifi- +ability in a particular metric spaces called subRiemannian Carnot groups [ABB19, BLU07, +CDPT07]. +In the usual way, Federer [Fed69] says that a subset of Rn is countably d-rectifiable if it is +possible to cover it with a countable union of suitable graphs. More precisely, he considers +graphs of Lipschitz maps f : Rd → Rn−d. However, Ambrosio and Kirchheim [AK00] (see +also [Mag04]) proved that this definition does not work in Carnot groups and so many +mathematicians give other notions of rectifiability. The reader can see [AM22a, AM22b, +Bat21, DS91, DS93, CP06, Pau04, NY18]. As we said, another possible solution is given +by Franchi-Serapioni-SerraCassano with the so-called "Intrinsic Lipschitz maps" in Carnot +Date: 5th January 2023. +1 + +groups. The idea is similar to Euclidean case: firstly, they introduce suitable cones called +intrinsic cones which are not equivalent with the Euclidean ones; and, then, they say that a +map ϕ is intrinsic Lipschitz if it is possible to have for any point p belongs to the graph of ϕ +an empty intersection between a suitable intrinsic cone with vertex p and the graph of this +map. +Recently, Le Donne and the author generalize this concept in metric spaces (see [DDLD22]). +A basic difference is the following: in Franchi, Serapioni and Serra Cassano approach, they +consider intrinsically Lipschitz maps. On the other hand, in our approach we consider the +graphs and this a bit change is so important because: +• The setting are more general: the class of the metric spaces is larger than the class +of Carnot groups. +• The proofs are elegantly short and simple. +• We use basic mathematical tools in the proofs. +In a natural way, in [DD22c] the author introduce the notion of intrinsically Hölder sections +which extend the Lipschitz ones. Here, our setting is the following. We have a metric space +X, a topological space Y , and a quotient map π : X → Y , meaning continuous, open, and +surjective. The standard example for us is when X is a metric Lie group G (meaning that the +Lie group G is equipped with a left-invariant distance that induces the manifold topology), +for example a subRiemannian Carnot group, and Y is the space of left cosets G/H, where +H < G is a closed subgroup and π : G → G/H is the projection modulo H, g �→ gH. +Definition 1.1 (Intrinsic Hölder section). Let (X, d) be a metric space and let Y be a +topological space. We say that a map ϕ : Y → X is a section of a quotient map π : X → Y +if +π ◦ ϕ = idY . +Moreover, we say that ϕ is an intrinsically (L, α)-Hölder section with constant L > 0 and +α ∈ (0, 1) if in addition +(1) +d(ϕ(y1), ϕ(y2)) ≤ Ld(ϕ(y1), π−1(y2))α + d(ϕ(y1), π−1(y2)), +for all y1, y2 ∈ Y. +Equivalently, we are requesting that +d(x1, x2) ≤ Ld(x1, π−1(π(x2)))α + d(x1, π−1(π(x2))), +for all x1, x2 ∈ ϕ(Y ). +When α = 1, a section ϕ is intrinsic Lipschitz in the sense of [DDLD22]; and, if in addition, +π is a Lipschitz quotient or submetry [BJL+99, VN88], the results trivialize, since in this +case being intrinsically Lipschitz is equivalent to biLipschitz embedding, see Proposition +2.4 in [DDLD22]. In a natural way, following the seminal papers [AGS14a, LV09, Sav22] +(see also [AGS15, AGS14b, FSS22, Stu06, Vil09]), the author introduced and studied the +link between the intrinsic sections/intrinsic Lipschitz sections and the intrinsic Hopf-Lax +semigroups [DD22b, DD22e]. +The purpose of this article is twofold: first of all, we want to define two norms using +the notion of Lipschitz sections. Second, we want to generalize an Extension Theorem with +target Y which is a topological space; in this paper, our target will be Y × Rs instead of Y. +More precisely, in Section 3, the main results are Theorem 3.1 and 3.4. Here, we define +two possible normed spaces using the following ingredients: +2 + +• we know that there is a large class of intrinsically Hölder sections and so Lipschitz +sections that is a vector space over R or C (see Theorem 2.7); +• we can define two different norms noting the following simple fact: in the usual +case, we have that d(x, y) = d(y, x) for any point x, y ∈ X; on the other hand, in +our intrinsic context, in general, we have that d(f(x), π−1(y)) ̸= d(f(y), π−1(x)), for +every x, y ∈ X. +• we obtain the homogeneity of our norms defined in (5) and in (9) thanks to linearity +of π and, in particular, to Lemma 3.2. +Finally, in Section 4 the main result is Theorem 4.1 which generalizes Extension Theorem +for the intrinsically Hölder sections proved in [DD22c, Theorem 1.3]. The main difference is +that, in this project, the target space is a topological space Y × Rs with s ≥ 1 instead of Y. +As in Vittone’s case, we build each component fi for i = 1, . . . , s separately and then join +them without any additional assumptions. However, the final step when the target space is +only Y does not provide a Lipschitz map f = (f1, . . . , fs). +2. Intrinsically Hölder sections +2.1. Intrinsically Hölder sections: when Y is bounded. Definition 1.1 it is very natural +if we think that what we are studying graphs of appropriate maps. However, in the following +proposition, we introduce an equivalent condition of (1) when Y is a bounded set. +Proposition 2.1. Let π : X → Y be a quotient map between a metric space X and a +topological and bounded space Y and let α ∈ (0, 1). The following are equivalent: +(1) there is L > 0 such that +d(ϕ(y1), ϕ(y2)) ≤ Ld(ϕ(y1), π−1(y2))α + d(ϕ(y1), π−1(y2)), +for all y1, y2 ∈ Y. +(2) there is K ≥ 1 such that +(2) +d(ϕ(y1), ϕ(y2)) ≤ Kd(ϕ(y1), π−1(y2))α, +for all y1, y2 ∈ Y. +We further rephrase the definition as saying that ϕ(Y ), which we call the graph of ϕ, +avoids some particular sets (which depend on α, L and ϕ itself): +Proposition 2.2. Let π : X → Y be a quotient map between a metric space and a topological +space, ϕ : Y → X be a section of π, α ∈ (0, 1) and L > 0. Then ϕ is intrinsically (L, α)- +Hölder if and only if +ϕ(Y ) ∩ Rx,L = ∅, +for all x ∈ ϕ(Y ), +where +Rx,L := +� +x′ ∈ X | Ld(x′, π−1(π(x)))α + d(x′, π−1(π(x))) < d(x′, x) +� +. +Proposition 2.2 is a triviality, still its purpose is to stress the analogy with the intrinsically +Lipschitz sections theory introduced in [DDLD22] when α = 1. In particular, the sets Rx,L +are the intrinsic cones in the sense of Franchi, Serapioni and Serra Cassano when X is a +subRiemannian Carnot group and α = 1. The reader can see [DD22d] for a good notion of +intrinsic cones in metric groups. +3 + +2.2. Equivalent definition for intrinsic Hölder sections. +Definition 2.3 (Intrinsically Hölder with respect to a section). Given sections ϕ, ψ : Y → X +of π. We say that ϕ is intrinsically (L, α)-Hölder with respect to ψ at point ˆx, with L > +0, α ∈ (0, 1) and ˆx ∈ X, if +(1) ˆx ∈ ψ(Y ) ∩ ϕ(Y ); +(2) ϕ(Y ) ∩ Cψ +ˆx,L = ∅, +where +Cψ +ˆx,L := {x ∈ X : d(x, ψ(π(x))) > Ld(ˆx, ψ(π(x)))α + d(ˆx, ψ(π(x)))}. +Remark 2.4. Definition 2.3 can be rephrased as follows. A section ϕ is intrinsically (L, α)- +Hölder with respect to ψ at point ˆx if and only if there is ˆy ∈ Y such that ˆx = ϕ(ˆy) = ψ(ˆy) +and +(3) +d(x, ψ(π(x))) ≤ Ld(ˆx, ψ(π(x)))α + d(ˆx, ψ(π(x))), +∀x ∈ ϕ(Y ), +which equivalently means +(4) +d(ϕ(y), ψ(y)) ≤ Ld(ψ(ˆy), ψ(y))α + d(ψ(ˆy), ψ(y)), +∀y ∈ Y. +Notice that Definition 2.3 does not induce an equivalence relation because of lack of +symmetry in the right-hand side of (4). However, following Cheeger theory [Che99] (see also +[Kei04, KM16]), in [DD22c, Theorem 4.2] we give an equivalent property of Hölder section. +Being intrinsically Lipschitz section is equivalent to the last definition as proved in [DD22c, +Proposition 1.5] +Proposition 2.5. Let X be a metric space, Y a topological and bounded space, π : X → Y +a quotient map, L ≥ 1 and α, β, γ ∈ (0, 1). Assume that every point x ∈ X is contained in +the image of an intrinsic (L, α)-Hölder section ψx for π. Then for every section ϕ : Y → X +of π the following are equivalent: +(1) for all x ∈ ϕ(Y ) the section ϕ is intrinsically (L1, β)-Hölder with respect to ψx at x; +(2) the section ϕ is intrinsically (L2, γ)-Hölder. +We conclude this section recall an important concept which we will be used later. +Definition 2.6 (Intrinsic Hölder set with respect to ψ). Let α ∈ (0, 1] and ψ : Y → X a +section of π. We define the set of all intrinsically Hölder sections of π with respect to ψ at +point ˆx as +Hψ,ˆx,α := {ϕ : Y → X a section of π : ϕ is intrinsically (˜L, α)-Hölder w.r.t. ψ at point ˆx +for some ˜L > 0}. +In particular, in [DD22c] we have the following statement regarding the set Hψ,ˆx,α. +Theorem 2.7 (Theorem 3.5 [DD22c]). Let π : X → Y is a linear and quotient map from a +normed space X to a topological space Y. Assume also that ψ : Y → X is a section of π and +{λˆx : λ ∈ R+} ⊂ X with ˆx ∈ ψ(Y ). +Then, for any α ∈ (0, 1], the set � +λ∈R+ Hλψ,λˆx,α ∪ {0} is a vector space over R or C. +Notice that it is no possible to obtain the statement for Hψ,ˆx,α since the simply observation +that if ψ(ˆy) = ˆx then ψ(ˆy) + ψ(ˆy) ̸= ˆx. +4 + +3. Normed space for the intrinsically Lipschitz sections +3.1. Normed space: +Version 1. In this section, we consider the case of intrinsically +Lipschitz sections, i.e., α = 1. +Let π : Rκ → Y be a quotient map with Y ⊂ Rκ. +Assume also that K ⊂ Y is a +compact set and ψ|K : K → R is an intrinsically L-Lipschitz section of π with L ≥ 1 and +ˆx = ψ(¯y) ∈ ψ(Y ). We will use the following notation +ILSλψ|K,λˆx := Hλψ|K,λˆx,1. +Here, we show that +(L, ∥.∥) := +� � +λ∈R+ +ILSλψ|K,λˆx ∪ {0}, ∥.∥ +� +is a normed space for a suitable norm ∥.∥= ∥.∥ILSλψ,λˆx: L → R+ defined as for any ϕ ∈ L, +(5) +∥ϕ∥ILSλψ,λˆx:= ∥ϕ∥∞+[ϕ]λψ,λˆx, +where ∥ϕ∥∞:= supy∈K|ϕ(y)| and +[ϕ]λψ,λˆx := sup +y∈K +d(λϕ(y), (1/λπ)−1(π(ˆx))). +Then, we are able to give the main result of this section. +Theorem 3.1. Let π : Rκ → Y be a linear and quotient map with Y ⊂ Rκ. Assume also +that ψ : K → Rκ is an intrinsically L-Lipschitz section of π with K ⊂ Y compact, L ≥ 1 +and ˆx ∈ X. Then, the set L endowed with ∥·∥ILSψ,ˆx is a normed space. +We need the following lemma. +Lemma 3.2 (Lemma 4.7 [DD22a]). Let X be a normed space, Y be a topological space and +π : X → Y be a linear and quotient map. Then +(6) +|λ|d(x1, π−1(y2)) = d(λx1, (1/λπ)−1(y2)), +∀x1 ∈ Rκ, y2 ∈ Y, λ ∈ R − {0}. +Remark 3.3. An easy corollary of Lemma 3.2 when Y ⊂ R and X = Rκ is that +lim +h→0+ +d(hϕ(t + h), (1/hπ)−1(t))) +h += 0, +lim +h→0+ +d(hϕ(t), (1/hπ)−1(t + h))) +h += 0, +for t ∈ Y. Indeed, for h > 0 +d(hϕ(t + h), (1/hπ)−1(t)) +h += d(ϕ(t + h), π−1(t)) ≤ d(ϕ(t + h), ϕ(t)), +and so take to the limit for h → 0, we obtain the first limit thank to the continuity of ϕ. In +a similar way, it is possible to see the second limit. +At this point, we give the proof of Theorem 3.1. +5 + +Proof of Theorem 3.1. The fact ∥ϕ∥≡ 0 if and only if ϕ ≡ 0 follows because ∥.∥∞ is a norm. +On the other hand, since π is linear map and thanks to Lemma 3.2, we have +sup +y∈K +d(δϕ(y), (1/δπ)−1(π(ˆx))) = sup +y∈K +|δ|d(ϕ(y), π−1(π(ˆx))) +for any δ ∈ R − {0} and so +(7) +∥δϕ∥∞+[δϕ]ψ,ˆx = |δ|(∥ϕ∥∞+[ϕ]ψ,ˆx), +for any ϕ ∈ L. +Finally, the triangle inequality follows using again the linearity of π. Indeed, if ϕ, η ∈ +L − {0} and, in particular, ϕ, η ∈ ILS(λ1+λ2)ψ|K,(λ1+λ2)ˆx then for xϕ, xη ∈ Rκ such that +[ϕ](λ1+λ2)ψ,(λ1+λ2)ˆx = sup +y∈K +d((λ1 + λ2)ϕ(y), (1/(λ1 + λ2)π)−1(π(ˆx))) = d((λ1 + λ2)ϕ(y), xϕ) +[η](λ1+λ2)ψ,(λ1+λ2)ˆx = sup +y∈K +d((λ1 + λ2)η(y), (1/(λ1 + λ2)π)−1(π(ˆx))) = d((λ1 + λ2)η(y), xη) +one gets +[ϕ + η](λ1+λ2)ψ,(λ1+λ2)ˆx = sup +y∈K +d((λ1 + λ2)(ϕ(y) + η(y)), (2/(λ1 + λ2)π)−1(π(ˆx))) +≤ ∥(λ1 + λ2)ϕ(y) + (λ1 + λ2)η(y) − (xϕ + xη)∥ +≤ ∥(λ1 + λ2)ϕ(y) − xϕ∥+∥(λ1 + λ2)η(y) − xη∥, +[ϕ](λ1+λ2)ψ,(λ1+λ2)ˆx + [η](λ1+λ2)ψ,(λ1+λ2)ˆx, +where in the first equality, by linearity of π, we used the fact +π((λ1 + λ2)(ϕ(y) + η(y))) = π((λ1 + λ2)ϕ(y)) + π((λ1 + λ2)η(y)) += (λ1 + λ2)(π(ϕ(y)) + (π(η(y))) += 2(λ1 + λ2)y. +Hence, +[ϕ + η](λ1+λ2)ψ,(λ1+λ2)ˆx ≤ [ϕ](λ1+λ2)ψ,(λ1+λ2)ˆx + [η](λ1+λ2)ψ,(λ1+λ2)ˆx, +and this complete the proof of the statement. +□ +3.2. Normed space: Version 2. In the usual case, we have that d(x, y) = d(y, x) for any +point x, y ∈ X; on the other hand, in our intrinsic context, in general, we have that +d(f(x), π−1(y)) ̸= d(f(y), π−1(x)), +for every x, y ∈ X. In particular, it holds +(8) +d(f(y), π−1(x)) − d(f(z), π−1(x)) ≤ d(f(y), f(z)), +∀x, y, z ∈ Y +d(f(x), π−1(y)) − d(f(x), π−1(z)) ≰ d(f(y), f(z)), +for some x, y, z ∈ Y. +In fact, for any fixed x, y, z ∈ Y, if we choose a ∈ π−1(x) such that +d(f(z), π−1(x)) = d(f(z), a), +6 + +we deduce that +d(f(y), π−1(x)) − d(f(z), π−1(x)) ≤ d(f(y), a) − d(f(z), π−1(x)) +≤ d(f(y), f(z)) + d(f(z), a) − d(f(z), π−1(x)) += d(f(y), f(z)), +i.e., the first inequality of (8) holds. On the other hand, for the second inequality in (8), +we give the following example. let X ⊂ R2 the set given by the three lines with vertex +(0, 8), (8, 8); (1, 4), (8, 6) and (0, 3), (8, 7) and the subset Y of R2 defined as the line with +vertex (0, 0) and (8, 0). On X we consider the usual distance on R2. Then, if we consider a +continuous section f : Y → X of the projection on the first component π : X → Y with +f(x) = f((1, 0)) = (1, 4), f(y) = f((7, 0)) = (8, 7) and f(z) = f((6, 0)) = (8, 6), it is easy to +see that +d(f(x), π−1(y)) − d(f(x), π−1(z)) = +� +5 +4, +d(f(y), f(z)) = 1, +and so +d(f(x), π−1(y)) − d(f(x), π−1(z)) ≰ d(f(y), f(z)). +Then, it is not trivial to consider the norm |||.||| defined as +(9) +|||ϕ|||ILSψ,ˆx:= ∥ϕ∥∞+{ϕ}λψ,λˆx, +where ∥ϕ∥∞:= supy∈K|ϕ(y)| and +{ϕ}λψ,λˆx := sup +y∈K +d(λˆx, (1/λπ)−1(y)). +and to prove as above the following statement. +Theorem 3.4. Let π : Rκ → Y be a linear and quotient map with Y a metric space. Assume +also that ψ : K → Rκ is an intrinsically L-Lipschitz section of π with K ⊂ Y compact, L ≥ 1 +and ˆx ∈ X. Then, the set L endowed with |||·|||ILSψ,ˆx as in (9) is a normed space. +Proof. The proof follows in a similar way as in Theorem 3.1. +□ +4. Level sets and extensions +In this section we prove the following theorem. +Theorem 4.1 (Extensions as level sets). Let π : X → Y × Rs be a quotient map between a +metric space X and a topological space Y × Rs. +Assume that X is geodesic and that there exist k ≥ 1, ρ : X × X → R k-biLipschitz +equivalent to the distance of X, and τ = (τ1, . . . , τs) : X → Rs k-Lipschitz and k-biLipschitz +on fibers such that for all τ0 ∈ Rs +(1) the set τ −1 +1 (τ0) is an intrinsically k-Lipschitz graph of a section ϕ1,τ0 : Y × R × +{0}s−1 → X; the set τ −1 +2 (τ0) is an intrinsically k-Lipschitz graph of a section ϕ2,τ0 : +Y × {0} × R × {0}s−2 → X; . . . , the set τ −1 +s (τ0) is an intrinsically k-Lipschitz graph +of a section ϕs,τ0 : Y × {0}s−1 × R → X; +7 + +(2) for all x0 ∈ τ −1 +i +(τ0) the map X → R, x �→ δi,τ0(x) := ρ(x0, ϕi,τ0(π(x))) is k-Lipschitz +on the set {|τi|≤ δi,τ0}. +Let Y ′ × Rs ⊂ Y × Rs a set and L ≥ 1. Then for every intrinsically L-Lipschitz section +ϕ : Y ′ × Rs → π−1(Y ′ × Rs) of π|π−1(Y ′×Rs): π−1(Y ′ × Rs) → Y ′ × Rs, there exists a map +f : X → Rs that is K-Lipschitz and K-biLipschitz on fibers, with K = 2k(Lk + 2), such that +(10) +ϕ(Y ′ × Rs) ⊆ f −1(0). +In particular, each ‘partially defined’ intrinsically Lipschitz graph ϕ(Y ′ × Rs) is a subset of +a ‘globally defined’ intrinsically Lipschitz graph f −1(0). +We need to mention that there have been several earlier partial results on extensions of +Lipschitz graphs, as for example in [FSSC06], [Vit, Theorem 1.5], [Mon14, Proposition 4.8], +in the Heisenberg group with codimension larger than one; [Vit12, Proposition 3.4], for the +case of codimension one in the Heisenberg groups; [FS16, Theorem 4.1], for the case of +codimension one in Carnot groups. Finally, for the general metric spaces the reader can see +[AP20, LN05, Oht09]. +Proof of Theorem 4.1 (4.1.i). It is proved in [DDLD22]. +□ +Proof of Theorem 4.1 (4.1.ii). Step 1. Fix i = 1, . . . , s and, for simplicity, we write τ −1, fx0 +instead of τ −1 +i +and fx0,i. Fix x0 ∈ τ −1(τ0). We consider the map fx0 : X → R defined as +(11) +fx0(x) = + + + +2(τ(x) − τ(x0) − αδτ0(x)) +if |τ(x) − τ(x0)|≤ 2αδτ0(x) +τ(x) − τ(x0) +if τ(x) − τ(x0) > 2αδτ0(x) +3(τ(x) − τ(x0)) +if τ(x) − τ(x0) < −2αδτ0(x) +where α := kL + 1. We prove that fx0 satisfies the following properties: +(i): fx0 is K-Lipschitz; +(ii): fx0(x0) = 0; +(iii): fx0 is biLipschitz on fibers. +where K = max{3k, 2k + 2αk} = 2k + 2αk because α > 1. The property (i) derives using +that τ, δτ0 are both Lipschitz and X is a geodesic space. On the other hand, (ii) is true +noting that δτ0(x0) = ρ(x0, ϕτ0(π(x0))) = 0 because x0 ∈ ϕτ0(Y ). +Finally, for any x, x′ ∈ π−1(y) we have that ρ(x0, ϕτ0(π(x))) = ρ(x0, ϕτ0(π(x′))) and so f +is biLipschitz on fibers because τ is so too. Hence (iii) holds. +At this point, we consider the map f : X → R given by +f(x) := +sup +x0∈ϕ(Y ) +fx0(x), +∀x ∈ X, +and we want to prove that it is the map we are looking for. The Lipschitz properties are +true recall that the function δx0 is constant on the fibers. Consequently, the only non trivial +fact to show is (10). Fix ¯x0 ∈ ϕ(Y ′). By (ii) we have that f¯x0(¯x0) = 0 and so it is sufficient +to prove that fx0(¯x0) ≤ 0 for x0 ∈ ϕ(Y ′). Let x0 ∈ ϕ(Y ′). Then using in addition that τ is +k-Lipschitz, and that ϕ is intrinsically L-Lipschitz, we have +|τ(¯x0) − τ(x0)|≤ kd(¯x0, x0) ≤ Lkd(x0, π−1(π(¯x0))) ≤ Lkd(x0, ϕτ0(π(¯x0))) < αδτ0(¯x0), +8 + +and so +fx0(¯x0) = 2(τ(¯x0) − τ(x0) − αδτ0(¯x0)) < 0, +i.e., (10) holds. +Step 2. We consider f = (f1, . . . , fs) where each fi is given by τ −1 +i +. Roughly speaking, +here the problem is that when we put together (f1, . . . , fs) using τ −1 +1 , . . . , τ −1 +s +then f can not +Lipschitz. But now f is Lipschitz thanks to the construction of Y × Rs and in particular of +ϕ1,τ0 : Y × R × {0}s−1 → X; . . . ; ϕs,τ0 : Y × {0}s−1 × R → X. +□ +References +[ABB19] +Andrei Agrachev, Davide Barilari, and Ugo Boscain. A comprehensive introduction to sub- +Riemannian geometry. Cambridge Studies in Advanced Mathematics, Cambridge Univ. Press, +181:762, 2019. +[AGS14a] +L. Ambrosio, N. Gigli, and G. Savaré. Calculus and heat flow in metric measure spaces and +applications to spaces with Ricci bounds from below. Invent. Math., 195(2):289–391, 2014. +[AGS14b] +L. Ambrosio, N. Gigli, and G. Savaré. Metric measure spaces with Riemannian Ricci curvature +bounded from below. Duke Math. J., 163(7):1405–1490, 2014. +[AGS15] +L. Ambrosio, N. Gigli, and G. Savaré. Bakry-Émery curvature-dimension condition and Rieman- +nian Ricci curvature bounds. Ann. Probab., 43(1):339–404, 2015. +[AK00] +Luigi Ambrosio and Bernd Kirchheim. Rectifiable sets in metric and Banach spaces. Math. Ann., +318(3):527–555, 2000. +[AM22a] +G. Antonelli and A. Merlo. On rectifiable measures in Carnot groups: existence of density. +Accepted in Journal of Geometric Analysis, 2022. +[AM22b] +G. Antonelli and A. Merlo. On rectifiable measures in Carnot groups: Marstrand–Mattila recti- +fiability criterion. Accepted in Journal of Functional Analysis, 2022. +[AP20] +Luigi Ambrosio and Daniele Puglisi. Linear extension operators between spaces of Lipschitz maps +and optimal transport. Journal für die reine und angewandte Mathematik (Crelles Journal), +2020(764):1–21, 2020. +[Bat21] +D. Bate. Characterising rectifiable metric spaces using tangent spaces. Accepted, Invent. math., +2021. +[BJL+99] +S. Bates, W. B. Johnson, J. Lindenstrauss, D. Preiss, and G. Schechtman. Affine approximation +of Lipschitz functions and nonlinear quotients. Geom. Funct. Anal., 9(6):1092–1127, 1999. +[BLU07] +A. Bonfiglioli, E. Lanconelli, and F. Uguzzoni. Stratified Lie groups and potential theory for their +sub-Laplacians. Springer Monographs in Mathematics. Springer, Berlin, 2007. +[CDPT07] Luca Capogna, Donatella Danielli, Scott D. Pauls, and Jeremy T. Tyson. An introduction to +the Heisenberg group and the sub-Riemannian isoperimetric problem, volume 259 of Progress in +Mathematics. Birkhäuser Verlag, Basel, 2007. +[Che99] +J. Cheeger. Differentiability of Lipschitz functions on metric measure spaces. Geom. Funct. Anal., +9(8):428–517, 1999. +[CP06] +D.R. Cole and S.D. Pauls. C1 hypersurfaces of the Heisenberg group are N-rectifiable. Houston +J. Math., 32(3):713–724, 2006. +[DD22a] +Daniela Di Donato. Intrinsic Cheeger energy for the intrinsically Lipschitz constants. preprint, +2022. +[DD22b] +Daniela Di Donato. The intrinsic Hopf-Lax semigroup vs. the intrinsic slope. preprint, 2022. +[DD22c] +Daniela Di Donato. Intrinsically Hölder sections in metric spaces. preprint, 2022. +[DD22d] +Daniela Di Donato. Intrinsically Lipschitz graphs on semidirect products of groups. preprint, +2022. +[DD22e] +Daniela Di Donato. Non-symmetric intrinsic Hopf-Lax semigroup vs. intrinsic Lagrangian. pre- +print, 2022. +9 + +[DDLD22] Daniela Di Donato and Enrico Le Donne. Intrinsically Lipschitz sections and applications to +metric groups. preprint, 2022. +[DS91] +G. David and S. Semmes. Singular integrals and rectifiable sets in Rn: Beyond lipschitz graphs, +astérisque. (193):171–279, 1991. +[DS93] +G. David and S. Semmes. Analysis of and on uniformly rectifiable sets. Mathematical Surveys +and Monographs, 38, 1993. +[Fed69] +Herbert Federer. Geometric measure theory. Die Grundlehren der mathematischen Wis- +senschaften, Band 153. Springer-Verlag New York Inc., New York, 1969. +[FS16] +Bruno Franchi and Raul Paolo Serapioni. Intrinsic Lipschitz graphs within Carnot groups. J. +Geom. Anal., 26(3):1946–1994, 2016. +[FSS22] +M. Fornasier, G. Savaré, and G.E. Sodini. Density of subalgebras of lipschitz functions in metric +sobolev spaces and applications to wasserstein sobolev spaces. preprint, 2022. +[FSSC01] +B. Franchi, R. Serapioni, and F. Serra Cassano. Rectifiability and perimeter in the Heisenberg +group. Math. Ann., 321(3):479–531, 2001. +[FSSC03a] B. Franchi, R. Serapioni, and F. Serra Cassano. Regular hypersurfaces, intrinsic perimeter and +implicit function theorem in Carnot groups. Comm. Anal. Geom., 11(5):909–944, 2003. +[FSSC03b] Bruno Franchi, Raul Serapioni, and Francesco Serra Cassano. On the structure of finite perimeter +sets in step 2 Carnot groups. The Journal of Geometric Analysis, 13(3):421–466, 2003. +[FSSC06] +Bruno Franchi, Raul Serapioni, and Francesco Serra Cassano. Intrinsic Lipschitz graphs in Heis- +enberg groups. J. Nonlinear Convex Anal., 7(3):423–441, 2006. +[Kei04] +S. Keith. A differentiable structure for metric measure spaces. Adv. Math. 183, pages 271–315, +2004. +[KM16] +B. Kleiner and J.M. Mackay. Differentiable structures on metric measure spaces: a primer. Ann. +Sc. Norm. Super. Pisa Cl. Sci. (5) Vol. XVI, pages 41–64, 2016. +[LN05] +J.R. Lee and A. Naor. Extending lipschitz functions via random metric partitions. Invent. Math., +160(1):59–95, 2005. +[LV09] +John Lott and Cédric Villani. Ricci curvature for metric-measure spaces via optimal transport. +Ann. of Math. (2), 169(3):903–991, 2009. +[Mag04] +V. Magnani. Unrectifiability and rigidity in stratified groups. Arch. Math., 83(6):568–576, 2004. +[Mon14] +Roberto Monti. Lipschitz approximation of H-perimeter minimizing boundaries. Calc. Var. Par- +tial Differential Equations, 50(1-2):171–198, 2014. +[NY18] +Assaf Naor and Robert Young. Vertical perimeter versus horizontal perimeter. Ann. of Math., +(2) 188, (1):171–279, 2018. +[Oht09] +S.I. Ohta. Extending Lipschitz and Hölder maps between metric spaces. Positivity 13, 2:407–425, +2009. +[Pau04] +S.D. Pauls. A notion of rectifiability modeled on Carnot groups. Indiana Univ. Math. J., 53(1):49– +81, 2004. +[Sav22] +G. Savaré. Sobolev spaces in extended metric-measure spaces. In New Trends on Analysis and +Geometry in Metric Spaces, Springer, pages 117–276, 2022. +[SC16] +Francesco Serra Cassano. Some topics of geometric measure theory in Carnot groups. In Geo- +metry, analysis and dynamics on sub-Riemannian manifolds. Vol. 1, EMS Ser. Lect. Math., +pages 1–121. Eur. Math. Soc., Zürich, 2016. +[Stu06] +K.T. Sturm. On the geometry of metric measure spaces. I, II. Acta Math., 196(1):65– 131 and +133–177, 2006. +[Vil09] +Cédric Villani. Optimal transport, Old and new, volume 338 of Grundlehren der Mathemat- +ischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Ber- +lin, 2009. +[Vit] +Davide Vittone. Lipschitz graphs and currents in Heisenberg groups. Forum of Mathematics, +Sigma, 10, E6, 2022. +[Vit12] +Davide Vittone. Lipschitz surfaces, perimeter and trace theorems for BV functions in Carnot- +Carathéodory spaces. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 11(4):939–998, 2012. +10 + +[VN88] +Berestovskii Valerii Nikolaevich. Homogeneous manifolds with intrinsic metric. Sib Math J, +I(29):887–897, 1988. +Daniela Di Donato: Dipartimento di Ingegneria Industriale e Scienze Matematiche, Via +Brecce Bianche, 12 60131 Ancona, Universitá Politecnica delle Marche. +Email address: d.didonato@staff.univpm.it +11 + diff --git a/F9AzT4oBgHgl3EQfxP6U/content/tmp_files/load_file.txt b/F9AzT4oBgHgl3EQfxP6U/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fd0d6229e67da5107481f45d632cdcf86d31cfee --- /dev/null +++ b/F9AzT4oBgHgl3EQfxP6U/content/tmp_files/load_file.txt @@ -0,0 +1,572 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf,len=571 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='01735v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='MG] 4 Jan 2023 NORMED SPACES USING INTRINSICALLY LIPSCHITZ SECTIONS AND EXTENSION THEOREM FOR THE INTRINSICALLY HÖLDER SECTIONS DANIELA DI DONATO Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' The purpose of this article is twofold: first of all, we want to define two norms using the space of intrinsically Lipschitz sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' On the other hand, we want to generalize an Extension Theorem proved by the author in the context of the intrinsically Hölder sections with target a topological space Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Here our target will be Y × Rs with s ≥ 1 instead of Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Introduction 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Intrinsically Hölder sections 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Intrinsically Hölder sections: when Y is bounded 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Equivalent definition for intrinsic Hölder sections 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Normed space for the intrinsically Lipschitz sections 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Normed space: Version 1 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Normed space: Version 2 6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Level sets and extensions 7 References 9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Introduction In this paper, we focus our attention on a new point of view for the intrinsically Lipschitz graphs in the Franchi-Serapioni-SerraCassano sense [FSSC01, FSSC03b, FSSC03a] (see also [SC16, FS16]) in metric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' They introduced and analized this notion in order to establish a good notion of rectifi- ability in a particular metric spaces called subRiemannian Carnot groups [ABB19, BLU07, CDPT07].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' In the usual way, Federer [Fed69] says that a subset of Rn is countably d-rectifiable if it is possible to cover it with a countable union of suitable graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' More precisely, he considers graphs of Lipschitz maps f : Rd → Rn−d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' However, Ambrosio and Kirchheim [AK00] (see also [Mag04]) proved that this definition does not work in Carnot groups and so many mathematicians give other notions of rectifiability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' The reader can see [AM22a, AM22b, Bat21, DS91, DS93, CP06, Pau04, NY18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' As we said, another possible solution is given by Franchi-Serapioni-SerraCassano with the so-called "Intrinsic Lipschitz maps" in Carnot Date: 5th January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' 1 groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' The idea is similar to Euclidean case: firstly, they introduce suitable cones called intrinsic cones which are not equivalent with the Euclidean ones;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' and, then, they say that a map ϕ is intrinsic Lipschitz if it is possible to have for any point p belongs to the graph of ϕ an empty intersection between a suitable intrinsic cone with vertex p and the graph of this map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Recently, Le Donne and the author generalize this concept in metric spaces (see [DDLD22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' A basic difference is the following: in Franchi, Serapioni and Serra Cassano approach, they consider intrinsically Lipschitz maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' On the other hand, in our approach we consider the graphs and this a bit change is so important because: The setting are more general: the class of the metric spaces is larger than the class of Carnot groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' The proofs are elegantly short and simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' We use basic mathematical tools in the proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' In a natural way, in [DD22c] the author introduce the notion of intrinsically Hölder sections which extend the Lipschitz ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Here, our setting is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' We have a metric space X, a topological space Y , and a quotient map π : X → Y , meaning continuous, open, and surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' The standard example for us is when X is a metric Lie group G (meaning that the Lie group G is equipped with a left-invariant distance that induces the manifold topology), for example a subRiemannian Carnot group, and Y is the space of left cosets G/H, where H < G is a closed subgroup and π : G → G/H is the projection modulo H, g �→ gH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='1 (Intrinsic Hölder section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Let (X, d) be a metric space and let Y be a topological space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' We say that a map ϕ : Y → X is a section of a quotient map π : X → Y if π ◦ ϕ = idY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Moreover, we say that ϕ is an intrinsically (L, α)-Hölder section with constant L > 0 and α ∈ (0, 1) if in addition (1) d(ϕ(y1), ϕ(y2)) ≤ Ld(ϕ(y1), π−1(y2))α + d(ϕ(y1), π−1(y2)), for all y1, y2 ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Equivalently, we are requesting that d(x1, x2) ≤ Ld(x1, π−1(π(x2)))α + d(x1, π−1(π(x2))), for all x1, x2 ∈ ϕ(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' When α = 1, a section ϕ is intrinsic Lipschitz in the sense of [DDLD22];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' and, if in addition, π is a Lipschitz quotient or submetry [BJL+99, VN88], the results trivialize, since in this case being intrinsically Lipschitz is equivalent to biLipschitz embedding, see Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='4 in [DDLD22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' In a natural way, following the seminal papers [AGS14a, LV09, Sav22] (see also [AGS15, AGS14b, FSS22, Stu06, Vil09]), the author introduced and studied the link between the intrinsic sections/intrinsic Lipschitz sections and the intrinsic Hopf-Lax semigroups [DD22b, DD22e].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' The purpose of this article is twofold: first of all, we want to define two norms using the notion of Lipschitz sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Second, we want to generalize an Extension Theorem with target Y which is a topological space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' in this paper, our target will be Y × Rs instead of Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' More precisely, in Section 3, the main results are Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Here, we define two possible normed spaces using the following ingredients: 2 we know that there is a large class of intrinsically Hölder sections and so Lipschitz sections that is a vector space over R or C (see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='7);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' we can define two different norms noting the following simple fact: in the usual case, we have that d(x, y) = d(y, x) for any point x, y ∈ X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' on the other hand, in our intrinsic context, in general, we have that d(f(x), π−1(y)) ̸= d(f(y), π−1(x)), for every x, y ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' we obtain the homogeneity of our norms defined in (5) and in (9) thanks to linearity of π and, in particular, to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Finally, in Section 4 the main result is Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='1 which generalizes Extension Theorem for the intrinsically Hölder sections proved in [DD22c, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' The main difference is that, in this project, the target space is a topological space Y × Rs with s ≥ 1 instead of Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' As in Vittone’s case, we build each component fi for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' , s separately and then join them without any additional assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' However, the final step when the target space is only Y does not provide a Lipschitz map f = (f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' , fs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Intrinsically Hölder sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Intrinsically Hölder sections: when Y is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='1 it is very natural if we think that what we are studying graphs of appropriate maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' However, in the following proposition, we introduce an equivalent condition of (1) when Y is a bounded set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Let π : X → Y be a quotient map between a metric space X and a topological and bounded space Y and let α ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' The following are equivalent: (1) there is L > 0 such that d(ϕ(y1), ϕ(y2)) ≤ Ld(ϕ(y1), π−1(y2))α + d(ϕ(y1), π−1(y2)), for all y1, y2 ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' (2) there is K ≥ 1 such that (2) d(ϕ(y1), ϕ(y2)) ≤ Kd(ϕ(y1), π−1(y2))α, for all y1, y2 ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' We further rephrase the definition as saying that ϕ(Y ), which we call the graph of ϕ, avoids some particular sets (which depend on α, L and ϕ itself): Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Let π : X → Y be a quotient map between a metric space and a topological space, ϕ : Y → X be a section of π, α ∈ (0, 1) and L > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Then ϕ is intrinsically (L, α)- Hölder if and only if ϕ(Y ) ∩ Rx,L = ∅, for all x ∈ ϕ(Y ), where Rx,L := � x′ ∈ X | Ld(x′, π−1(π(x)))α + d(x′, π−1(π(x))) < d(x′, x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='2 is a triviality, still its purpose is to stress the analogy with the intrinsically Lipschitz sections theory introduced in [DDLD22] when α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' In particular, the sets Rx,L are the intrinsic cones in the sense of Franchi, Serapioni and Serra Cassano when X is a subRiemannian Carnot group and α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' The reader can see [DD22d] for a good notion of intrinsic cones in metric groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Equivalent definition for intrinsic Hölder sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='3 (Intrinsically Hölder with respect to a section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Given sections ϕ, ψ : Y → X of π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' We say that ϕ is intrinsically (L, α)-Hölder with respect to ψ at point ˆx, with L > 0, α ∈ (0, 1) and ˆx ∈ X, if (1) ˆx ∈ ψ(Y ) ∩ ϕ(Y );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' (2) ϕ(Y ) ∩ Cψ ˆx,L = ∅, where Cψ ˆx,L := {x ∈ X : d(x, ψ(π(x))) > Ld(ˆx, ψ(π(x)))α + d(ˆx, ψ(π(x)))}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='3 can be rephrased as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' A section ϕ is intrinsically (L, α)- Hölder with respect to ψ at point ˆx if and only if there is ˆy ∈ Y such that ˆx = ϕ(ˆy) = ψ(ˆy) and (3) d(x, ψ(π(x))) ≤ Ld(ˆx, ψ(π(x)))α + d(ˆx, ψ(π(x))), ∀x ∈ ϕ(Y ), which equivalently means (4) d(ϕ(y), ψ(y)) ≤ Ld(ψ(ˆy), ψ(y))α + d(ψ(ˆy), ψ(y)), ∀y ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Notice that Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='3 does not induce an equivalence relation because of lack of symmetry in the right-hand side of (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' However, following Cheeger theory [Che99] (see also [Kei04, KM16]), in [DD22c, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='2] we give an equivalent property of Hölder section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Being intrinsically Lipschitz section is equivalent to the last definition as proved in [DD22c, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='5] Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Let X be a metric space, Y a topological and bounded space, π : X → Y a quotient map, L ≥ 1 and α, β, γ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Assume that every point x ∈ X is contained in the image of an intrinsic (L, α)-Hölder section ψx for π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Then for every section ϕ : Y → X of π the following are equivalent: (1) for all x ∈ ϕ(Y ) the section ϕ is intrinsically (L1, β)-Hölder with respect to ψx at x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' (2) the section ϕ is intrinsically (L2, γ)-Hölder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' We conclude this section recall an important concept which we will be used later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='6 (Intrinsic Hölder set with respect to ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Let α ∈ (0, 1] and ψ : Y → X a section of π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' We define the set of all intrinsically Hölder sections of π with respect to ψ at point ˆx as Hψ,ˆx,α := {ϕ : Y → X a section of π : ϕ is intrinsically (˜L, α)-Hölder w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' ψ at point ˆx for some ˜L > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' In particular, in [DD22c] we have the following statement regarding the set Hψ,ˆx,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='7 (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='5 [DD22c]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Let π : X → Y is a linear and quotient map from a normed space X to a topological space Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Assume also that ψ : Y → X is a section of π and {λˆx : λ ∈ R+} ⊂ X with ˆx ∈ ψ(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Then, for any α ∈ (0, 1], the set � λ∈R+ Hλψ,λˆx,α ∪ {0} is a vector space over R or C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Notice that it is no possible to obtain the statement for Hψ,ˆx,α since the simply observation that if ψ(ˆy) = ˆx then ψ(ˆy) + ψ(ˆy) ̸= ˆx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Normed space for the intrinsically Lipschitz sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Normed space: Version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' In this section, we consider the case of intrinsically Lipschitz sections, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=', α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Let π : Rκ → Y be a quotient map with Y ⊂ Rκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Assume also that K ⊂ Y is a compact set and ψ|K : K → R is an intrinsically L-Lipschitz section of π with L ≥ 1 and ˆx = ψ(¯y) ∈ ψ(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' We will use the following notation ILSλψ|K,λˆx := Hλψ|K,λˆx,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Here, we show that (L, ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='∥) := � � λ∈R+ ILSλψ|K,λˆx ∪ {0}, ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='∥ � is a normed space for a suitable norm ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='∥= ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='∥ILSλψ,λˆx: L → R+ defined as for any ϕ ∈ L, (5) ∥ϕ∥ILSλψ,λˆx:= ∥ϕ∥∞+[ϕ]λψ,λˆx, where ∥ϕ∥∞:= supy∈K|ϕ(y)| and [ϕ]λψ,λˆx := sup y∈K d(λϕ(y), (1/λπ)−1(π(ˆx))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Then, we are able to give the main result of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Let π : Rκ → Y be a linear and quotient map with Y ⊂ Rκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Assume also that ψ : K → Rκ is an intrinsically L-Lipschitz section of π with K ⊂ Y compact, L ≥ 1 and ˆx ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Then, the set L endowed with ∥·∥ILSψ,ˆx is a normed space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' We need the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='2 (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='7 [DD22a]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Let X be a normed space, Y be a topological space and π : X → Y be a linear and quotient map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Then (6) |λ|d(x1, π−1(y2)) = d(λx1, (1/λπ)−1(y2)), ∀x1 ∈ Rκ, y2 ∈ Y, λ ∈ R − {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' An easy corollary of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='2 when Y ⊂ R and X = Rκ is that lim h→0+ d(hϕ(t + h), (1/hπ)−1(t))) h = 0, lim h→0+ d(hϕ(t), (1/hπ)−1(t + h))) h = 0, for t ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Indeed, for h > 0 d(hϕ(t + h), (1/hπ)−1(t)) h = d(ϕ(t + h), π−1(t)) ≤ d(ϕ(t + h), ϕ(t)), and so take to the limit for h → 0, we obtain the first limit thank to the continuity of ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' In a similar way, it is possible to see the second limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' At this point, we give the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' 5 Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' The fact ∥ϕ∥≡ 0 if and only if ϕ ≡ 0 follows because ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='∥∞ is a norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' On the other hand, since π is linear map and thanks to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='2, we have sup y∈K d(δϕ(y), (1/δπ)−1(π(ˆx))) = sup y∈K |δ|d(ϕ(y), π−1(π(ˆx))) for any δ ∈ R − {0} and so (7) ∥δϕ∥∞+[δϕ]ψ,ˆx = |δ|(∥ϕ∥∞+[ϕ]ψ,ˆx), for any ϕ ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Finally, the triangle inequality follows using again the linearity of π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Indeed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' if ϕ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' η ∈ L − {0} and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' in particular,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' ϕ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' η ∈ ILS(λ1+λ2)ψ|K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='(λ1+λ2)ˆx then for xϕ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' xη ∈ Rκ such that [ϕ](λ1+λ2)ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='(λ1+λ2)ˆx = sup y∈K d((λ1 + λ2)ϕ(y),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' (1/(λ1 + λ2)π)−1(π(ˆx))) = d((λ1 + λ2)ϕ(y),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' xϕ) [η](λ1+λ2)ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='(λ1+λ2)ˆx = sup y∈K d((λ1 + λ2)η(y),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' (1/(λ1 + λ2)π)−1(π(ˆx))) = d((λ1 + λ2)η(y),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' xη) one gets [ϕ + η](λ1+λ2)ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='(λ1+λ2)ˆx = sup y∈K d((λ1 + λ2)(ϕ(y) + η(y)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' (2/(λ1 + λ2)π)−1(π(ˆx))) ≤ ∥(λ1 + λ2)ϕ(y) + (λ1 + λ2)η(y) − (xϕ + xη)∥ ≤ ∥(λ1 + λ2)ϕ(y) − xϕ∥+∥(λ1 + λ2)η(y) − xη∥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' [ϕ](λ1+λ2)ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='(λ1+λ2)ˆx + [η](λ1+λ2)ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='(λ1+λ2)ˆx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' where in the first equality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' by linearity of π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' we used the fact π((λ1 + λ2)(ϕ(y) + η(y))) = π((λ1 + λ2)ϕ(y)) + π((λ1 + λ2)η(y)) = (λ1 + λ2)(π(ϕ(y)) + (π(η(y))) = 2(λ1 + λ2)y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Hence, [ϕ + η](λ1+λ2)ψ,(λ1+λ2)ˆx ≤ [ϕ](λ1+λ2)ψ,(λ1+λ2)ˆx + [η](λ1+λ2)ψ,(λ1+λ2)ˆx, and this complete the proof of the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Normed space: Version 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' In the usual case, we have that d(x, y) = d(y, x) for any point x, y ∈ X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' on the other hand, in our intrinsic context, in general, we have that d(f(x), π−1(y)) ̸= d(f(y), π−1(x)), for every x, y ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' In particular, it holds (8) d(f(y), π−1(x)) − d(f(z), π−1(x)) ≤ d(f(y), f(z)), ∀x, y, z ∈ Y d(f(x), π−1(y)) − d(f(x), π−1(z)) ≰ d(f(y), f(z)), for some x, y, z ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' In fact, for any fixed x, y, z ∈ Y, if we choose a ∈ π−1(x) such that d(f(z), π−1(x)) = d(f(z), a), 6 we deduce that d(f(y), π−1(x)) − d(f(z), π−1(x)) ≤ d(f(y), a) − d(f(z), π−1(x)) ≤ d(f(y), f(z)) + d(f(z), a) − d(f(z), π−1(x)) = d(f(y), f(z)), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=', the first inequality of (8) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' On the other hand, for the second inequality in (8), we give the following example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' let X ⊂ R2 the set given by the three lines with vertex (0, 8), (8, 8);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' (1, 4), (8, 6) and (0, 3), (8, 7) and the subset Y of R2 defined as the line with vertex (0, 0) and (8, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' On X we consider the usual distance on R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Then, if we consider a continuous section f : Y → X of the projection on the first component π : X → Y with f(x) = f((1, 0)) = (1, 4), f(y) = f((7, 0)) = (8, 7) and f(z) = f((6, 0)) = (8, 6), it is easy to see that d(f(x), π−1(y)) − d(f(x), π−1(z)) = � 5 4, d(f(y), f(z)) = 1, and so d(f(x), π−1(y)) − d(f(x), π−1(z)) ≰ d(f(y), f(z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Then, it is not trivial to consider the norm |||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='||| defined as (9) |||ϕ|||ILSψ,ˆx:= ∥ϕ∥∞+{ϕ}λψ,λˆx, where ∥ϕ∥∞:= supy∈K|ϕ(y)| and {ϕ}λψ,λˆx := sup y∈K d(λˆx, (1/λπ)−1(y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' and to prove as above the following statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Let π : Rκ → Y be a linear and quotient map with Y a metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Assume also that ψ : K → Rκ is an intrinsically L-Lipschitz section of π with K ⊂ Y compact, L ≥ 1 and ˆx ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Then, the set L endowed with |||·|||ILSψ,ˆx as in (9) is a normed space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' The proof follows in a similar way as in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Level sets and extensions In this section we prove the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='1 (Extensions as level sets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Let π : X → Y × Rs be a quotient map between a metric space X and a topological space Y × Rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Assume that X is geodesic and that there exist k ≥ 1, ρ : X × X → R k-biLipschitz equivalent to the distance of X, and τ = (τ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' , τs) : X → Rs k-Lipschitz and k-biLipschitz on fibers such that for all τ0 ∈ Rs (1) the set τ −1 1 (τ0) is an intrinsically k-Lipschitz graph of a section ϕ1,τ0 : Y × R × {0}s−1 → X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' the set τ −1 2 (τ0) is an intrinsically k-Lipschitz graph of a section ϕ2,τ0 : Y × {0} × R × {0}s−2 → X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' , the set τ −1 s (τ0) is an intrinsically k-Lipschitz graph of a section ϕs,τ0 : Y × {0}s−1 × R → X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' 7 (2) for all x0 ∈ τ −1 i (τ0) the map X → R, x �→ δi,τ0(x) := ρ(x0, ϕi,τ0(π(x))) is k-Lipschitz on the set {|τi|≤ δi,τ0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Let Y ′ × Rs ⊂ Y × Rs a set and L ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Then for every intrinsically L-Lipschitz section ϕ : Y ′ × Rs → π−1(Y ′ × Rs) of π|π−1(Y ′×Rs): π−1(Y ′ × Rs) → Y ′ × Rs, there exists a map f : X → Rs that is K-Lipschitz and K-biLipschitz on fibers, with K = 2k(Lk + 2), such that (10) ϕ(Y ′ × Rs) ⊆ f −1(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' In particular, each ‘partially defined’ intrinsically Lipschitz graph ϕ(Y ′ × Rs) is a subset of a ‘globally defined’ intrinsically Lipschitz graph f −1(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' We need to mention that there have been several earlier partial results on extensions of Lipschitz graphs, as for example in [FSSC06], [Vit, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='5], [Mon14, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='8], in the Heisenberg group with codimension larger than one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' [Vit12, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='4], for the case of codimension one in the Heisenberg groups;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' [FS16, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='1], for the case of codimension one in Carnot groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Finally, for the general metric spaces the reader can see [AP20, LN05, Oht09].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='1 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' It is proved in [DDLD22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' □ Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='1 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Fix i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' , s and, for simplicity, we write τ −1, fx0 instead of τ −1 i and fx0,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Fix x0 ∈ τ −1(τ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' We consider the map fx0 : X → R defined as (11) fx0(x) = \uf8f1 \uf8f2 \uf8f3 2(τ(x) − τ(x0) − αδτ0(x)) if |τ(x) − τ(x0)|≤ 2αδτ0(x) τ(x) − τ(x0) if τ(x) − τ(x0) > 2αδτ0(x) 3(τ(x) − τ(x0)) if τ(x) − τ(x0) < −2αδτ0(x) where α := kL + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' We prove that fx0 satisfies the following properties: (i): fx0 is K-Lipschitz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' (ii): fx0(x0) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' (iii): fx0 is biLipschitz on fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' where K = max{3k, 2k + 2αk} = 2k + 2αk because α > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' The property (i) derives using that τ, δτ0 are both Lipschitz and X is a geodesic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' On the other hand, (ii) is true noting that δτ0(x0) = ρ(x0, ϕτ0(π(x0))) = 0 because x0 ∈ ϕτ0(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Finally, for any x, x′ ∈ π−1(y) we have that ρ(x0, ϕτ0(π(x))) = ρ(x0, ϕτ0(π(x′))) and so f is biLipschitz on fibers because τ is so too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Hence (iii) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' At this point, we consider the map f : X → R given by f(x) := sup x0∈ϕ(Y ) fx0(x), ∀x ∈ X, and we want to prove that it is the map we are looking for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' The Lipschitz properties are true recall that the function δx0 is constant on the fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Consequently, the only non trivial fact to show is (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Fix ¯x0 ∈ ϕ(Y ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' By (ii) we have that f¯x0(¯x0) = 0 and so it is sufficient to prove that fx0(¯x0) ≤ 0 for x0 ∈ ϕ(Y ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Let x0 ∈ ϕ(Y ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Then using in addition that τ is k-Lipschitz, and that ϕ is intrinsically L-Lipschitz, we have |τ(¯x0) − τ(x0)|≤ kd(¯x0, x0) ≤ Lkd(x0, π−1(π(¯x0))) ≤ Lkd(x0, ϕτ0(π(¯x0))) < αδτ0(¯x0), 8 and so fx0(¯x0) = 2(τ(¯x0) − τ(x0) − αδτ0(¯x0)) < 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=', (10) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' We consider f = (f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' , fs) where each fi is given by τ −1 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Roughly speaking, here the problem is that when we put together (f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' , fs) using τ −1 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' , τ −1 s then f can not Lipschitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' But now f is Lipschitz thanks to the construction of Y × Rs and in particular of ϕ1,τ0 : Y × R × {0}s−1 → X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' ϕs,τ0 : Y × {0}s−1 × R → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' □ References [ABB19] Andrei Agrachev, Davide Barilari, and Ugo Boscain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' A comprehensive introduction to sub- Riemannian geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Cambridge Studies in Advanced Mathematics, Cambridge Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Press, 181:762, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' [AGS14a] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Ambrosio, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Gigli, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Savaré.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Invent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=', 195(2):289–391, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' [AGS14b] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Ambrosio, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Gigli, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Savaré.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Metric measure spaces with Riemannian Ricci curvature bounded from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Duke Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=', 163(7):1405–1490, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' [AGS15] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Ambrosio, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Gigli, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Savaré.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Bakry-Émery curvature-dimension condition and Rieman- nian Ricci curvature bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=', 43(1):339–404, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' [AK00] Luigi Ambrosio and Bernd Kirchheim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Rectifiable sets in metric and Banach spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=', 318(3):527–555, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' [AM22a] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Antonelli and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Merlo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' On rectifiable measures in Carnot groups: existence of density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Accepted in Journal of Geometric Analysis, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' [AM22b] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Antonelli and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Merlo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' On rectifiable measures in Carnot groups: Marstrand–Mattila recti- fiability criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Accepted in Journal of Functional Analysis, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' [AP20] Luigi Ambrosio and Daniele Puglisi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Linear extension operators between spaces of Lipschitz maps and optimal transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Journal für die reine und angewandte Mathematik (Crelles Journal), 2020(764):1–21, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' [Bat21] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Bate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Characterising rectifiable metric spaces using tangent spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Accepted, Invent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=', 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' [BJL+99] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Bates, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Johnson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Lindenstrauss, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Preiss, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Schechtman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Affine approximation of Lipschitz functions and nonlinear quotients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=', 9(6):1092–1127, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' [BLU07] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Bonfiglioli, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Lanconelli, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Uguzzoni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Stratified Lie groups and potential theory for their sub-Laplacians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Springer Monographs in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Springer, Berlin, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' [CDPT07] Luca Capogna, Donatella Danielli, Scott D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Pauls, and Jeremy T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Tyson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' An introduction to the Heisenberg group and the sub-Riemannian isoperimetric problem, volume 259 of Progress in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Birkhäuser Verlag, Basel, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' [Che99] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Cheeger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Differentiability of Lipschitz functions on metric measure spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=', 9(8):428–517, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' [CP06] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Cole and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Pauls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' C1 hypersurfaces of the Heisenberg group are N-rectifiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Houston J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=', 32(3):713–724, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' [DD22a] Daniela Di Donato.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Intrinsic Cheeger energy for the intrinsically Lipschitz constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' preprint, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' [DD22b] Daniela Di Donato.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' The intrinsic Hopf-Lax semigroup vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' the intrinsic slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' preprint, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' [DD22c] Daniela Di Donato.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Intrinsically Hölder sections in metric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' preprint, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' [DD22d] Daniela Di Donato.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Intrinsically Lipschitz graphs on semidirect products of groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' preprint, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' [DD22e] Daniela Di Donato.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Non-symmetric intrinsic Hopf-Lax semigroup vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' intrinsic Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' pre- print, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' 9 [DDLD22] Daniela Di Donato and Enrico Le Donne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Intrinsically Lipschitz sections and applications to metric groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' preprint, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' [DS91] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' David and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Semmes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Singular integrals and rectifiable sets in Rn: Beyond lipschitz graphs, astérisque.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' (193):171–279, 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' [DS93] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' David and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Semmes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Analysis of and on uniformly rectifiable sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Mathematical Surveys and Monographs, 38, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' [Fed69] Herbert Federer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Geometric measure theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Die Grundlehren der mathematischen Wis- senschaften, Band 153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Springer-Verlag New York Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=', New York, 1969.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' [FS16] Bruno Franchi and Raul Paolo Serapioni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Intrinsic Lipschitz graphs within Carnot groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=', 26(3):1946–1994, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' [FSS22] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Fornasier, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Savaré, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Sodini.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Density of subalgebras of lipschitz functions in metric sobolev spaces and applications to wasserstein sobolev spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' preprint, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' [FSSC01] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Franchi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Serapioni, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Serra Cassano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Rectifiability and perimeter in the Heisenberg group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=', 321(3):479–531, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' [FSSC03a] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Franchi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Serapioni, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Serra Cassano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Regular hypersurfaces, intrinsic perimeter and implicit function theorem in Carnot groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=', 11(5):909–944, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' [FSSC03b] Bruno Franchi, Raul Serapioni, and Francesco Serra Cassano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' On the structure of finite perimeter sets in step 2 Carnot groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' The Journal of Geometric Analysis, 13(3):421–466, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' [FSSC06] Bruno Franchi, Raul Serapioni, and Francesco Serra Cassano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Intrinsic Lipschitz graphs in Heis- enberg groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Nonlinear Convex Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=', 7(3):423–441, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' [Kei04] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Keith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' A differentiable structure for metric measure spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' 183, pages 271–315, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' [KM16] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Kleiner and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Mackay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Differentiable structures on metric measure spaces: a primer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Super.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Pisa Cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' (5) Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' XVI, pages 41–64, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' [LN05] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Lee and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Naor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Extending lipschitz functions via random metric partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Invent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=', 160(1):59–95, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' [LV09] John Lott and Cédric Villani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Ricci curvature for metric-measure spaces via optimal transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' (2), 169(3):903–991, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' [Mag04] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Magnani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Unrectifiability and rigidity in stratified groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=', 83(6):568–576, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' [Mon14] Roberto Monti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Lipschitz approximation of H-perimeter minimizing boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Calc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Par- tial Differential Equations, 50(1-2):171–198, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' [NY18] Assaf Naor and Robert Young.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Vertical perimeter versus horizontal perimeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=', (2) 188, (1):171–279, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' [Oht09] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Ohta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Extending Lipschitz and Hölder maps between metric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Positivity 13, 2:407–425, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' [Pau04] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Pauls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' A notion of rectifiability modeled on Carnot groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Indiana Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=', 53(1):49– 81, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' [Sav22] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Savaré.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Sobolev spaces in extended metric-measure spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' In New Trends on Analysis and Geometry in Metric Spaces, Springer, pages 117–276, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' [SC16] Francesco Serra Cassano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Some topics of geometric measure theory in Carnot groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' In Geo- metry, analysis and dynamics on sub-Riemannian manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' 1, EMS Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Lect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=', pages 1–121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=', Zürich, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' [Stu06] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Sturm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' On the geometry of metric measure spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' I, II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Acta Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=', 196(1):65– 131 and 133–177, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' [Vil09] Cédric Villani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Optimal transport, Old and new, volume 338 of Grundlehren der Mathemat- ischen Wissenschaften [Fundamental Principles of Mathematical Sciences].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Springer-Verlag, Ber- lin, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' [Vit] Davide Vittone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Lipschitz graphs and currents in Heisenberg groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Forum of Mathematics, Sigma, 10, E6, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' [Vit12] Davide Vittone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Lipschitz surfaces, perimeter and trace theorems for BV functions in Carnot- Carathéodory spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Super.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Pisa Cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' (5), 11(4):939–998, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' 10 [VN88] Berestovskii Valerii Nikolaevich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Homogeneous manifolds with intrinsic metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Sib Math J, I(29):887–897, 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Daniela Di Donato: Dipartimento di Ingegneria Industriale e Scienze Matematiche, Via Brecce Bianche, 12 60131 Ancona, Universitá Politecnica delle Marche.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content=' Email address: d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='didonato@staff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='univpm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} +page_content='it 11' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfxP6U/content/2301.01735v1.pdf'} diff --git a/F9E1T4oBgHgl3EQfXASj/content/tmp_files/2301.03122v1.pdf.txt b/F9E1T4oBgHgl3EQfXASj/content/tmp_files/2301.03122v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..12dcbd7f3d8da8bbb02f2c84049f799747999965 --- /dev/null +++ b/F9E1T4oBgHgl3EQfXASj/content/tmp_files/2301.03122v1.pdf.txt @@ -0,0 +1,2008 @@ +Under consideration for publication in J. Fluid Mech. +1 +Nusselt number scaling in horizontal convection: +boundary conditions and dimensionality +Navid C. Constantinou1†, Cesar B. Rocha2, Stefan G. Llewellyn Smith3, 4, and +William R. Young4 +1Research School of Earth Sciences & ARC Centre of Excellence for Climate Extremes, Australian National +University, Canberra, ACT 2601, Australia +2Department of Marine Sciences, University of Connecticut, Groton, CT 06340, USA +3Department of Mechanical and Aerospace Engineering, University of California San Diego, La Jolla, CA +92093-0411, USA +4Scripps Institution of Oceanography, University of California San Diego, La Jolla, CA 92093-0213, USA +(Received xx; revised xx; accepted xx) +We conduct a numerical study of horizontal convection (HC) at Prandtl number Pr = 1, +with both with no-slip and free-slip boundary conditions. We obtain 2D and 3D solutions and +determine the relation between the Rayleigh number Ra and the Nusselt number Nu. In 2D we +vary Ra between 0 and 6.4 × 1013. In the range 106 ⪅ Ra ⪅ 1010 the Nu–Ra relation is, apart +from minor departures, in agreement with Rossby’s scaling Nu ∼ Ra1/5. With Ra greater than +about 1011 we find a 2D regime with Nu ∼ 𝑅𝑎1/4 over three decades, up to the highest 2D Ra. +In 3D, with maximum Ra = 3.2 × 1011, we only find Rossby scaling regimes. These results +apply to both viscous boundary conditions. The Nu ∼ 𝑅𝑎1/4 regime has a double boundary +layer (BL): there is a thin BL with thickness ∼ Ra−1/4 inside a thicker BL with thickness +∼ Ra−1/5. The Ra−1/4 BL thickness, which determines Nu, coincides with the Kolmogorov +and Batchelor scales of HC. +Our numerical and theoretical results indicate that 3D HC is qualitatively and quantitatively +similar to 2D HC. At the same Ra, the 3D Nu exceeds the non-turbulent 2D Nu by only +20%, i.e., there is very little 3D enhancement of heat transport. Boundary conditions are +more important than dimensionality: the non-turbulent 2D free-slip solutions have larger Nu +than 3D no-slip solutions. The mechanical energy power integral of HC implies that mean +square vorticity of 3D HC is nearly equal to that of 2D HC at the same Ra. Thus vorticity +amplification by vortex stretching does not operate in 3D HC. +1. Introduction +Horizontal convection (HC) is convection driven by imposing non-uniform heating and +cooling along a single horizontal surface, such as the top of a rectangular enclosure; there is +no flux of heat through the other boundaries (Hughes & Griffiths 2008). Oceanography is an +important motivation for consideration of HC (Sandström 1908; Rossby 1965; Coman et al. +2006; Kuhlbrodt 2008), and in that connection the dependence of horizontal heat transport on +the strength of buoyancy forcing applied at the ocean surface is a prime question. Buoyancy +forcing is quantified via the horizontal-convective Rayleigh number, Ra, and horizontal heat +transport by a suitably defined Nusselt number Nu (Rocha et al. 2020b). +HC is an interesting counterpoint to Rayleigh-Bénard convection because HC buoyancy +transport in the interior of the domain cannot be easily interpreted as the vertical motion +of thermal plumes. Instead, heat enters the fluid where the non-uniform heated surface is +† Email address for correspondence: navid.constantinou@anu.edu.au +arXiv:2301.03122v1 [physics.flu-dyn] 8 Jan 2023 + +2 +N. C. Constantinou, C. B. Rocha, S. G. L. Smith, and W. R. Young +Figure 1: Snapshots of 2D free-slip HC at Ra = 6.4 × 107 in panels (a) and (b) and +Ra = 6.4 × 109 in (c) and (d); Pr = 1. Contours are streamlines. At the top surface +−1 ⩽ 𝑏/𝑏★ ⩽ +1; the narrow range of the buoyancy color scale in (b) and (d) makes the +small interior buoyancy variations visible. The sinusoidal surface buoyancy in (2.4) defines +the buoyancy scale 𝑏★; ℎ is the layer depth. +hotter than average and exits where it is colder. This horizontal transport is associated with a +prominent boundary layer (BL) adjacent to the non-uniform surface. +The oldest result for the high-Ra variation of horizontal-convective Nu is the scaling law of +Rossby (1965) +Nu ∼ Pr0 Ra1/5 , +(1.1) +where Pr is the Prandtl number. (Parameters Ra, Pr, and Nu are defined in section 2.) Figure 1 +shows two numerical solutions in the Rossby-scaling regime (1.1). +Rossby (1965) was motivated by experiments, mainly with Pr ∼ 103 and 104, and their +reasoning leading to (1.1) (reviewed in section 4) assumes visco-diffusive balances in the BL. + +(a) Ra=6.4 × 107 +vorticity, (azu - axw) Vh/b* +1.0 +3 +2 +1 + 0.5 +0 +-1 +-2 +0.0 +(b) +buoyancy, b/b* +1.0 +-0.60 +-0.62 +-0.64 +%0.5 +-0.66 +0.68 +0.70 +0.0 +x/h +(c) Ra=6.4 × 109 +vorticity, (azu - axw) Vh/b* +1.0 +3 +2 +1 +%0.5 +0 +-1 +-2 +0.0 +(d) +buoyancy, b/b* +1.0 +-0.74 +%0.5 +-0.75 +-0.76 +0.0 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +x/hHorizontal convection Nu scaling +3 +At moderately large Ra, the exponent 1/5 is supported by both laboratory work and numerical +solutions (Rossby 1998; Mullarney et al. 2004; Siggers et al. 2004; Wang & Huang 2005; +Chiu-Webster et al. 2008; Sheard & King 2011; Ilicak & Vallis 2012). Perhaps because of +Rossby’s assumption that the boundary layer is visco-diffusive, the scaling (1.1) is sometimes +characterized as applying to steady laminar HC, e.g., the flow in figure 1(a) and (b). But (1.1) +has much wider applicability: some studies find that the one-fifth law (1.1) applies in unsteady +and three-dimensional (3D) HC regimes. For example, Gayen et al. (2014), with Pr = 5, find +the exponent 1/5 in a steady 2D flow regime with Ra ⩽ 1010, and also in a turbulent 3D +regime with Ra ⩾ 1012. In the transition, 1010 ⩽ Ra ⩽ 1012, the constant multiplying Ra1/5 +increases from 0.37 to 0.51. Gayen et al. (2014) make the important point that the exponent +1/5 also arises with a non-Rossby balance between inertia and buoyancy in the surface BL. In +section 3 we provide further examples of (1.1) in unsteady three-dimensional regimes with +Pr = 1. The tenacity of 1/5 as the primary exponent is striking, and rationalizes (1.1) in cases +far removed from Rossby’s visco-diffusive scenario. +Increasing Ra, with Pr fixed, results in qualitative changes in the structure of HC: compare +figure 2 with figure 1. It is unlikely that a single Nu–Ra power-law can persist across the six +decades of Ra spanned by these illustrations. Tsai et al. (2020) have investigated 2D HC with +Pr = 6.14 and no-slip boundary conditions. At moderately large Ra, say 108 < Ra < 1010, +Tsai et al. (2020) find Rossby’s scaling law (1.1). But in the range 1010 < Ra < 1014, and +provided that the imposed surface buoyancy varies linearly with the horizontal coordinate 𝑥, +Tsai et al. (2020) report the scaling +Nu ∼ Ra1/4 . +(1.2) +In section 3, using the sinusoidal surface buoyancy in (2.4), we also find the scaling (1.2) in a +suite of 2D solutions at Pr = 1. We show that in 2D, with both no-slip and free-slip boundary +conditions, (1.2) applies from about Ra = 1011 to at least 6.4 × 1013. In 3D, however, we +achieve maximum Ra = 3.2 × 1011 and there is no numerical evidence for the one-fourth +scaling (1.2). +The vortices evident in figure 2 indicate that the interior buoyancy fluctuations are so small +that the interior dynamics is essentially that of a vortex gas characteristic of freely evolving 2D +turbulence (Benzi et al. 1987, 1988; Carnevale et al. 1991; Dritschel et al. 2008; McWilliams +1984, 1990). +Tsai et al. (2020) note that the exponent one-fourth in (1.2) corresponds to one of the +exponents proposed by Shishkina et al. (2016) in their phase diagram of the (Ra, Pr) parameter +plane. We review the proposal of Shishkina et al. (2016) in sections 4 and 5, and conclude +that this correspondence is accidental: diagnosis of the solutions in figure 2 does not confirm +essential features of the one-fourth Ra–Nu scaling regimes in the phase diagram of Shishkina +et al. (2016). In section 6 we suggest instead that scaling (1.2) follows if the thinnest BL has a +thickness scaling as the Kolmogorov length. In section 7 we argue that our numerical results +indicate that HC does not exhibit defining features of turbulence. +2. Formulation of the horizontal convection problem +We consider a Boussinesq fluid with density 𝜌 = 𝜌0(1 − 𝑔−1𝑏), where 𝜌0 is a constant +reference density and 𝑏 is the “buoyancy”. If, for example, the fluid is stratified by temperature +variations then 𝑏 = 𝑔𝛼(𝑇 − 𝑇0), where 𝑇0 is a reference temperature and 𝛼 is the thermal + +4 +N. C. Constantinou, C. B. Rocha, S. G. L. Smith, and W. R. Young +Figure 2: Snapshots of 2D free-slip solutions at Ra = 6.4 × 1011 in panels (a) and (b) and +Ra = 6.4 × 1013 in (c) and (d); Pr = 1. Contours are streamlines. This illustration uses true +aspect ratio so that axisymmetric vortices look circular. The buoyancy color scale is narrow +in order to reveal the small interior buoyancy variations, which are localized within the +cores of axisymmetric vortices. Across the six-decade range of Ra in figures 1 and 2 +vorticity scales with +√︁ +𝑏★/ℎ. +expansion coefficient. The Boussinesq equations of motion are +𝒖𝑡 + 𝒖 · ∇𝒖 + ∇𝑝 = 𝑏 ˆ𝒛 + 𝜈∇2𝒖 , +(2.1) +𝑏𝑡 + 𝒖 · ∇𝑏 = 𝜅∇2𝑏 , +(2.2) +∇ · 𝒖 = 0 . +(2.3) +The kinematic viscosity is 𝜈 and the thermal diffusivity is 𝜅. + +(a) Ra=6.4 × 1011 +vorticity, (azu - axw) h/b* +1.0 +3 +2 +L + 0.5 +0 +-1 +0.0 +(b) +buoyancy, b/b* +1.0 +-0.735 +00.5 +-0.745 +-0.755 +0.0 +xih +vorticity, (azu- axw) Vh/b* +(c) Ra=6.4 × 1013 +1.0 + 0.5 +0 +-1 +0.0 +(d) +buoyancy,b/b +1.0 +0.846 +-0.848 + 0.5 +-0.850 +-0.852 +0.0 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +x/hHorizontal convection Nu scaling +5 +2.1. Horizontal convective boundary conditions and control parameters +We suppose the fluid occupies a domain with depth ℎ, length ℓ𝑥, width ℓ𝑦; we assume +periodicity in the 𝑥- and 𝑦-directions. At the bottom surface (𝑧 = 0) and top surface (𝑧 = ℎ) +the primary boundary conditions on the velocity, 𝒖 = (𝑢, 𝑣, 𝑤), is that 𝑤 = 0; the viscous +boundary condition is either no slip (NS hereafter), 𝑢 = 𝑣 = 0, or free slip (FS hereafter), +𝑢𝑧 = 𝑣𝑧 = 0. At the bottom 𝑧 = 0 the buoyancy boundary condition is no flux, 𝜅𝑏𝑧 = 0. At +the top, 𝑧 = ℎ, the boundary condition is 𝑏 = 𝑏s(𝑥), where the top surface buoyancy 𝑏s is a +prescribed function of 𝑥. As a surface buoyancy field we use +𝑏s(𝑥) = 𝑏★ cos 𝑘𝑥 , +(2.4) +where 𝑘 = 2𝜋/ℓ𝑥. +As an idealization of conditions at the sea surface, FS is better than NS. But the main +reason for considering different viscous boundary conditions is to test scaling arguments. +We find only minor quantitative differences in the Nu–Ra scaling between the two boundary +conditions. Thus scaling arguments that rely on special properties of NS, such as analogies +with the Blasius BL, should be reconsidered: in the numerical solutions described below +the main features of the Nu–Ra scaling relation are independent of the viscous boundary +condition. +The problem is characterized by four non-dimensional parameters: the Rayleigh and Prandtl +numbers +Ra +def= ℓ3 +𝑥𝑏★ +𝜈𝜅 , +and +Pr +def= 𝜈 +𝜅 , +(2.5) +and the aspect ratios 𝐴𝑥 +def= ℓ𝑥/ℎ and 𝐴𝑦 +def= ℓ𝑦/ℎ. With periodic boundary conditions in 𝑦 +(no side walls), 2D HC is the special case 𝐴𝑦 = 0. +2.2. Mechanical energy dissipation +We use an overbar to denote an average over 𝑥, 𝑦, and 𝑡, taken at any fixed 𝑧 and angle brackets +⟨ ⟩ to denote a total volume average over 𝑥, 𝑦, 𝑧, and 𝑡. Using this notation, we recall some +results from Paparella & Young (2002) that are used below. +Horizontally averaging the buoyancy equation (2.2) we obtain the zero-flux constraint +𝑤𝑏 − 𝜅 ¯𝑏𝑧 = 0 . +(2.6) +Taking ⟨𝒖 · (2.1)⟩, we obtain the kinetic energy power integral +𝜀 = ⟨𝑤𝑏⟩ , +(2.7) +where 𝜀 +def= 𝜈⟨|∇𝒖|2⟩ is the rate of dissipation of kinetic energy and ⟨𝑤𝑏⟩ is rate of conversion +between potential and kinetic energy. +Vertically integrating (2.6) from 𝑧 = 0 to ℎ, and using the fact that 𝑏s = 0, we obtain another +expression for ⟨𝑤𝑏⟩; substituting this into (2.7) we find +𝜀 = −𝜅 ¯𝑏(0) +ℎ +. +(2.8) +In (2.8), ¯𝑏(0) is the (𝑥, 𝑦, 𝑡)-average of the buoyancy at the bottom 𝑧 = 0. +2.3. The Nusselt number of horizontal convection +Following Rocha et al. (2020b), we use the dissipation of buoyancy variance, +𝜒 +def= 𝜅⟨|∇𝑏|2⟩ , +(2.9) + +6 +N. C. Constantinou, C. B. Rocha, S. G. L. Smith, and W. R. Young +to define the Nusselt number as +Nu +def= 𝜒�𝜒diff . +(2.10) +Above, 𝜒diff +def= 𝜅⟨|∇𝑏diff|2⟩ is the buoyancy dissipation of the diffusive solution, i.e., +𝜅∇2𝑏diff = 0 with 𝑏diff satisfying the same boundary conditions as 𝑏. +Application of variational methods to HC (Siggers et al. 2004; Winters & Young 2009; +Rocha et al. 2020a) results in bounds on 𝜒 taking the form Nu ≲ Ra1/3. The exponent 1/3 is +safely larger than the exponents 1/5 and 1/4 reported in numerical studies of HC, including +this work. +Rocha et al. (2020b) show that there is also a “surface Nusselt number” +Nus +def= 𝑏s 𝜅𝑏𝑧(ℎ) +� +𝑏s 𝜅𝑏diff 𝑧(ℎ) . +(2.11) +Above, 𝜅𝑏𝑧(ℎ) is the buoyancy flux through the top surface 𝑧 = ℎ. With sufficient temporal +averaging Nu = Nus; in physical terms the interior entropy production, 𝜒, is balanced by +entropy flux through the surface 𝑧 = ℎ; Nus is the non-dimensional entropy flux though the +surface. In numerical solutions described below, in which the temporal average is computed +over a finite time interval, Nu ≈ Nus is a check on the estimated Nusselt number. +3. A numerical study of horizontal convection with Pr = 1 +In this section we present the results of a numerical study directed at characterizing the +variation of the Nusselt number Nu in (2.10) as a function of Ra. Computations are performed +using Dedalus, a spectral framework for solving partial differential equations (Burns et al. 2020, +www.dedalus-project.org). We use Fourier bases in the horizontal, periodic directions +and a Chebyshev basis in the vertical; the equations are time stepped using a fourth-order +implicit-explicit Runge–Kutta scheme. +We limit attention to Pr = 1 and the sinusoidal surface buoyancy forcing 𝑏s(𝑥) in (2.4). We +discuss both NS and FS boundary conditions and consider 2D solutions with aspect ratios +ℓ𝑥/ℎ = 4 , +ℓ𝑦/ℎ = 0 , +(3.1) +and 3D solutions with +ℓ𝑥/ℎ = 4 , +ℓ𝑦/ℎ = 1 . +(3.2) +Thus we have four solution suites: 2DFS, 3DFS, 2DNS, and 3DNS. The resulting estimates +of Nusselt number are summarized in table 1 and figure 3. +3.1. The low-Ra regime +Analysis of the low-Ra regime in appendix A shows that with ℓ𝑥/ℎ = 4 the first variation of +the Nusselt number away from unity is +Nu𝐹𝑆 = 1 + +� +Ra +21 567.5 +�2 ++ ord�Ra4� , +(3.3) +and +Nu𝑁 𝑆 = 1 + +� +Ra +87 789.8 +�2 ++ ord�Ra4� . +(3.4) +The low-Ra regime means that the Ra2 term in (3.3) and (3.4) is less than one, i.e., that the +convective buoyancy transport is a weak enhancement of the diffusive transport. For FS low Ra +means that Ra is somewhat less than about 104 and for NS low Ra means that Ra is somewhat + +Horizontal convection Nu scaling +7 +Free-slip Nu +No-slip Nu +Highest resolution +Ra +2D ◦ +3D • +2D □ +3D ■ +𝑛𝑥, 𝑛𝑧 +1.28e03 +1.00 +1.00† +1.00 +1.00† +128, 32 +3.20e03 +1.02 +1.02† +1.00 +1.00† +128, 32 +4.48e03 +1.04 +1.04† +1.00 +1.00† +128, 32 +6.40e03 +1.08 +1.08† +1.01 +1.01† +128, 32 +1.28e04 +1.22 +1.22† +1.02 +1.02† +128, 32 +1.92e04 +1.38 +1.38† +1.04 +1.04† +128, 32 +3.20e04 +1.64 +1.64† +1.11 +1.11† +128, 32 +6.40e04 +2.07 +2.07† +1.28 +1.28† +128, 32 +1.28e05 +2.55 +2.55† +1.59 +1.59† +128, 32 +2.56e05 +3.04 +3.04† +1.96 +1.96† +128, 32 +6.40e05 +3.71 +3.71† +2.47 +2.47† +128, 32 +1.60e06 +4.50 +4.50† +3.01 +3.01† +256, 64 +3.20e06 +5.14 +5.14† +3.45 +3.45† +256, 64 +6.40e06 +5.80 +5.80† +3.93 +3.93† +256, 64 +1.60e07 +6.77 +6.77† +4.70 +4.70† +256, 64 +3.20e07 +7.61 +7.61† +5.38 +5.44♯ +256, 64 +6.40e07 +8.65 +8.68∗ +6.17 +6.59♯ +256, 64 +1.60e08 +10.48 +10.57∗ +7.41 +8.46♯ +256, 64 +3.20e08 +12.17∗ +12.19∗ +8.51 +10.11♯ +256, 64 +6.40e08 +14.01∗ +14.09∗ +9.69 +11.97∗ +256, 64 +1.60e09 +16.99∗ +17.07∗ +11.77∗ +14.75∗ +256, 64 +3.20e09 +19.60∗ +20.32∗ +13.53∗ +17.20∗ +512, 128 +6.40e09 +22.41∗ +24.64∗ +15.35∗ +19.86∗ +512, 128 +1.60e10 +27.32∗ +31.38∗ +18.83∗ +23.67∗ +512, 128 +3.20e10 +31.48∗ +37.28∗ +21.88∗ +27.61∗ +512, 128 +6.40e10 +36.08∗ +43.94∗ +25.50∗ +30.68∗ +512, 128 +1.28e11 +41.85∗ +49.74∗ +29.87∗ +36.67∗ +1024, 256 +1.60e11 +43.80∗ +32.02∗ +1024, 256 +3.20e11 +50.71∗ +59.91∗ +37.45∗ +44.01∗ +1024, 256 +6.40e11 +58.36∗ +44.23∗ +1024, 256 +1.60e12 +71.43∗ +55.65∗ +1024, 256 +3.20e12 +86.26∗ +66.97∗ +2048, 512 +6.40e12 101.32∗ +78.86∗ +4096, 1024 +1.60e13 128.44∗ +97.66∗ +4096, 1024 +6.40e13 178.55∗ +133.16∗ +4096, 1024 +Table 1: Nu–Ra data for HC DNS. All runs have 𝑃𝑟 = 1 and ℓ𝑥/ℎ = 4. 3D runs have +ℓ𝑦/ℎ = 1 and 𝑛𝑦 = 𝑛𝑧. The surface buoyancy is the sinusoid in (2.4). Unsteady solutions +are indicated by a superscript ∗ on Nu; strictly 2D solutions (no 𝑦-dependence and 𝑣 = 0) of +3D computations are marked by a superscript †. The four NS runs with superscript ♯ are 3D +but steady. +less than about 4 × 104. These analytic results are compared with numerical solutions in the +insert of figure 3. In table 1 all values of Nu are rounded to 2 decimal places, e.g., the 2DFS +solution at Ra = 6.4 has Nu − 1 = 8.8 × 10−8 and at Ra = 640, Nu − 1 = 8.8 × 10−4. (These +boring runs are not reported in table 1.) + +8 +N. C. Constantinou, C. B. Rocha, S. G. L. Smith, and W. R. Young +10 +0 +10 +1 +10 +2 +10 +3 +10 +4 +10 +5 +10 +6 +10 +7 +10 +8 +10 +9 +10 +10 +10 +11 +10 +12 +10 +13 +10 +14 +Ra +1 +2 +5 +10 +20 +40 +100 +150 +250 +Nu +Ra +1/5 +Ra +1/4 +3D +2D +free-slip +no-slip +0 +1 +2 +3 +4 +Ra +× +10 +−4 +1.0 +1.1 +Nu +Figure 3: Variation of Nusselt number Nu with Rayleigh number Ra using the data from +table 1. The inset compares the low-Ra numerical results with the low-Ra analytic +results (3.3) and (3.4). Note that some solid markers fall on top of open markers, indicating +that the 3D solutions evolve to become 2D, or that the three dimensionality is weak. The +four vertical grey line segments mark Ra’s of the solutions in figures 1 and 2. +3.2. Nu–Ra scaling regimes: one-fifth and one-fourth +Between Ra ∼ 104 and 105 we do not see a simple relation between Nu and Ra. But once Ra +is greater than about 6.4 × 105 we find Rossby’s scaling, +Nu ∼ 𝐾1/5Ra1/5 , +(3.5) +in all four solution suites. See section 4 for a discussion of Rossby’s scaling argument, and +for more recent arguments that also predict the exponent 1/5 in (3.5) (Gayen et al. 2014; +Shishkina et al. 2016). Starting at around Ra ∼ 1011 in the 2DNS suite and 1012 in the 2DFS +suite there is a transition from the one-fifth regime (3.5) to the one-fourth regime, +Nu ∼ 𝐾1/4 Ra1/4 . +(3.6) +The one-fourth regime with NS has been documented previously by Tsai et al. (2020). +In figure 4 we show the data from figure 3, replotted using the compensated Nusselt number +Ra−1/5Nu in panel (a) and Ra−1/4Nu in panel (b). Table 2 summarizes the exponents determined +by least-squares fitting the Ra–Nu data over selected ranges. Least-squares exponents are +broadly in agreement with the scaling regimes determined by visual inspection of figure 4 and +other compensated plots. We use least-squares because it is objective and reproducible. Least +squares also assesses the sensitivity of estimated exponents to the points at the beginning and +end of a putative scaling range. + +Horizontal convection Nu scaling +9 +10 +4 +10 +5 +10 +6 +10 +7 +10 +8 +10 +9 +10 +10 +10 +11 +10 +12 +10 +13 +10 +14 +Ra +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +Ra +−1/5 +Nu +0.17 +0.215 +0.25 +0.3 +3D +2D +(a) +10 +4 +10 +5 +10 +6 +10 +7 +10 +8 +10 +9 +10 +10 +10 +11 +10 +12 +10 +13 +10 +14 +Ra +0.04 +0.06 +0.08 +0.10 +0.12 +0.14 +Ra +−1/4 +Nu +0.065 +0.050 +(b) +free-slip +no-slip +Figure 4: Variation of “compensated Nusselt numbers” (a) Ra−1/5Nu and (b) Ra−1/4Nu +with Rayleigh number Ra. The four vertical grey line segments mark Ra’s of the solutions +in figures 1 and 2. +3.3. Discussion of the no-slip solutions +We begin with easiest case, which is the 2DNS solution suite. The one-fifth scaling (3.5) is +found across the four-decade range in row 1 of table 2; this is the plateau at 𝐾1/5 = 0.17 in +figure 4(a). Least-squares estimates of exponent and prefactor, 𝐾1/5, are robust to changes in +the range, e.g., row 2 of table 2. The 2DNS suite transitions to the one-fourth scaling (3.6) at +around Ra = 1011 and forms the plateau at 𝐾1/4 = 0.05 in figure 4(b); see rows 4 through 6 of +table 2. +The 3DNS solution suite is more complicated. With moderate Ra (rows 7 through 9 of +table 2) the 3DNS solutions coincide with their 2DNS partners and the scaling is again (3.5) +with 𝐾1/5 = 0.17. At a critical Ra, roughly 3.2 × 107, the 3DNS suite becomes unstable to 3D +perturbations. With further increases in Ra the 3DNS solutions have larger Nu than their 2DNS +colleagues: the four steady 3DNS solutions in the interval 3.20 × 107 ⩽ Ra ⩽ 3.20 × 108, +are here. One might hope that development of 3D flow, albeit steady 3D flow, signals the +beginning of a new scaling regime, with an exponent greater than one-fifth. But alas, this is +the transition discovered by Gayen et al. (2014): at about Ra = 1.60 × 109 the 3DNS solutions +enter a new one-fifth regime: see rows 10 through 12 of table 2 and the 3DNS plateau at +0.215 in figure 4(a). With maximum Ra = 3.20 × 1011, we did not find convincing evidence +of the one-fourth scaling (3.6) in the 3DNS solution suite. +The NS computations of Gayen et al. (2014) used Pr = 5 and a piecewise constant surface + +10 +N. C. Constantinou, C. B. Rocha, S. G. L. Smith, and W. R. Young +row +suite +range +points least squares Nu +1 +2DNS +6.40 × 105 ⩽ Ra ⩽ 6.40 × 109 +13 +0.177 Ra0.198 +2 +2DNS +1.60 × 106 ⩽ Ra ⩽ 3.20 × 109 +11 +0.178 Ra0.197 +4 +2DNS 6.40 × 1010 ⩽ Ra ⩽ 6.40 × 1013 +10 +0.062 Ra0.242 +5 +2DNS 6.40 × 1010 ⩽ Ra ⩽ 1.60 × 1013 +9 +0.056 Ra0.246 +6 +2DNS 1.28 × 1011 ⩽ Ra ⩽ 1.60 × 1013 +8 +0.055 Ra0.246 +7 +3DNS +6.40 × 105 ⩽ Ra ⩽ 3.20 × 107 +6 +0.173 Ra0.199 +8 +3DNS +6.40 × 105 ⩽ Ra ⩽ 6.40 × 106 +4 +0.167 Ra0.202 +9 +3DNS +3.20 × 106 ⩽ Ra ⩽ 3.20 × 107 +4 +0.179 Ra0.197 +10 +3DNS +1.60 × 109 ⩽ Ra ⩽ 3.20 × 1011 +8 +0.195 Ra0.204 +11 +3DNS +1.60 × 109 ⩽ Ra ⩽ 1.60 × 1010 +4 +0.191 Ra0.205 +12 +3DNS 3.20 × 1010 ⩽ Ra ⩽ 3.20 × 1011 +4 +0.180 Ra0.208 +20 +2DFS +6.40 × 105 ⩽ Ra ⩽ 1.60 × 1011 +18 +0.253 Ra0.199 +21 +2DFS +6.40 × 105 ⩽ Ra ⩽ 3.20 × 108 +9 +0.312 Ra0.186 +22 +2DFS +6.40 × 108 ⩽ Ra ⩽ 1.60 × 1011 +9 +0.215 Ra0.206 +23 +2DFS 6.40 × 1011 ⩽ Ra ⩽ 6.40 × 1013 +6 +0.074 Ra0.245 +24 +2DFS 6.40 × 1011 ⩽ Ra ⩽ 3.20 × 1012 +3 +0.082 Ra0.241 +25 +2DFS 6.40 × 1012 ⩽ Ra ⩽ 6.40 × 1013 +3 +0.073 Ra0.245 +26 +3DFS +6.40 × 105 ⩽ Ra ⩽ 3.20 × 108 +9 +0.308 Ra0.187 +27 +3DFS 6.40 × 1010 ⩽ Ra ⩽ 3.20 × 1011 +3 +0.358 Ra0.193 +Table 2: Summary of least-squares fits to various scaling regimes. Where possible, we +assess the sensitivity of the exponent by varying the range. +buoyancy. Instead of (2.10), Gayen et al. (2014) defined Nu based on the buoyancy flux +through the destabilized portion of the non-uniformly heated surface. Despite these differences, +Gayen et al. (2014) document analogous 3DNS behavior within the one-fifth scaling regime: +the constant 𝐾1/5 takes different values on either side of a smooth transition. +3.4. Discussion of the free-slip solutions +Turning to the 2DFS solutions, the most generous identification of the one-fifth regime in (3.5) +is the five-decade range in row 20 of table 2. These 18 points correspond to the plateau +𝐾1/5 = 0.25 in figure 4(a). We are concerned, however, by 9 points in the first half of this +range, i.e., row 21 of table 2. These 9 points undulate around the 𝐾1/5 = 0.25 plateau with an +amplitude of about ±0.01 and the least-squares exponent 0.186 is uncomfortably different +from 1/5. These wayward points, at only moderately large Ra, correspond to solutions that are +either steady, or weakly time dependent. Thus insufficient time-averaging in the estimate of Nu +is not an issue. Moreover in this range the 2D and 3D solutions coincide. We conducted several +tests by changing the spatial resolution and found no significant variation in the numerical +estimate of Nu. If one views the exponent 0.186 as close to 1/5 then the wayward points +are the lower end of a five-decade 2DFS scaling regime: the undulation is a pre-asymptotic +imperfection in the first half of this regime. A more cautious interpretation is that the 2DFS + +Horizontal convection Nu scaling +11 +one-fifth regime begins only at about Ra = 6.40 × 108 and consists of the 9 points in row 22 +of table 2. The 2DFS suite transitions to the one-fourth scaling (3.6) at around Ra = 1012 and +forms the plateau at 𝐾1/4 = 0.065 in figure 4(b); see rows 23 through 25 of table 2. +The 3DFS solutions depart significantly from their 2DFS colleagues first at about Ra = +3.20 × 109. There is no evidence for a one-fourth scaling in the 3DFS suite. Instead, the three +highest Ra 3DFS solutions (row 27 of table 2) indicate a second one-fifth regime e.g. the 0.3 +plateau in figure 4(a). We speculate that the 3DFS suite is recapitulating the phenomenology +seen in the 3DNS suite: two one-fifth scaling regimes separated by a smooth transition. We +caution, however, that this speculation is based on three solutions in row 27 spanning less +than one decade variation in Ra. +4. Review of Nu ∼ Ra1/5 scaling arguments +Rossby (1965) proposed a visco-diffusive balance in the boundary layer adjacent to the +non-uniformly heated surface and so arrived at the one-fifth scaling in (3.5). Rossby identified +the length scale +𝛿1/5 +def= Ra−1/5ℎ +(4.1) +as the thickness of the surface BL. In the following discussion we also need the length +𝛿1/4 +def= Ra−1/4ℎ . +(4.2) +At Ra = 6.4 × 1013, the ratio of these two BL scales is 𝛿1/5/𝛿1/4 ≈ 5. +Central to Rossby’s argument is the assumption that BL buoyancy forces are balanced +by viscosity and that BL inertia is subdominant. At moderately large Ra, the exponent 1/5 +has been supported by subsequent laboratory work and by numerical studies (Rossby 1998; +Mullarney et al. 2004; Siggers et al. 2004; Wang & Huang 2005; Sheard & King 2011; +Ilicak & Vallis 2012). With Pr = ∞, and with both no-slip and free-slip boundary conditions, +Chiu-Webster et al. (2008) provide a compelling confirmation that Nu ∼ Ra1/5 as Ra → ∞. +We emphasize that the scaling (3.5), and the associated BL thickness 𝛿1/5, does not, however, +require that Pr ≫ 1. For example, the experiments of Mullarney et al. (2004) and Wang +& Huang (2005) present evidence of Rossby scaling in unsteady flows. The 2D solutions +shown in figure 1 – including the unsteady solution in panels (c) and (d) – are well within +the 𝐾1/5 = 0.25 regime of figure 4(a). Our unsteady 2DFS solutions exhibit the one-fifth +scaling (3.5) over at least three decades of Ra. +To further complicate the situation, 2D solutions in the one-fourth regime (3.6) still express +the BL scale 𝛿1/5: figure 5 shows a progressively expanded view of the structure of HC near +the upper surface. This 2DNS solution is in the non-Rossby scaling regime (3.6). Nonetheless, +panel (d) of figure 5 indicates that 𝛿1/5 is a useful BL length scale. We conclude that at +sufficiently high Ra there is a double BL: there is a thin-𝛿1/4 layer nestled with a thicker +𝛿1/5-layer. We discuss this double BL further in section 5. +(The 2DNS solution in figure 5(a) exhibits the vortex-gas phenomenology noted previously +in the 2DFS solutions shown in figure 2. At high Ra, no matter the viscous boundary condition, +the interior of 2D HC is characterized as a vortex gas.) +As an alternative to Rossby scaling, Shishkina et al. (2016) proposed a set of scaling +arguments summarized in a phase diagram of the (Ra, Pr)-plane. This diagram shows high-Pr +regions denoted I∗ +ℓ, I∞ and III∞; these three high-Pr regions have Nu ∼ Ra𝜉 with exponent +𝜉 = 1/4 in I∗ +ℓ and III∞ and 1/6 in I∞. This tripartite proposal cannot be reconciled with +the high-Pr results of Rossby (1965) and Chiu-Webster et al. (2008). Instead, in the phase +diagram of Shishkina et al. (2016), the one-fifth scaling (3.5) is found only in the low-Pr +region Iℓ. We discuss the Shishkina et al. (2016) Iℓ regime in more detail below. + +12 +N. C. Constantinou, C. B. Rocha, S. G. L. Smith, and W. R. Young +Figure 5: (a) A snapshot of vorticity in the 2DNS solution at Ra = 6.4 × 1013; this solution +is in the non-Rossby Nu ∼ Ra1/4 scaling regime. Panels (b)-(d) depict the boundary-layer +structure by progressively zooming in to the top surface. Green rectangles in panels (a), (b) +and (c) indicate the regions in panels (b), (c) and (d) respectively. In panels (b), (c) and (d), +both axes are measured in units of 𝛿1/5. The dashed grey line in panel (d) indicates the +distance 2𝛿1/4 below the top surface 𝑧 = ℎ. The contours in all panels are streamlines. +4.1. The spanwise average +To identify the various processes in the BL, we begin taking a spanwise 𝑦-average of the +equations of motion. Denote this spanwise average with a hat so that +𝑏(𝑥, 𝑦, 𝑧, 𝑡) = 1 +ℓ𝑦 +∫ ℓ𝑦 +0 +𝑏(𝑥, 𝑦, 𝑧, 𝑡) d𝑦 +������������������������������������������������ +def= ˆ𝑏(𝑥,𝑧,𝑡) ++ 𝑏′(𝑥, 𝑦, 𝑧, 𝑡) . +(4.3) + +(a) Ra=6.4 × 1013 vorticity, (azu - axw)Vh/b* +1.0 +5.0 +2.5 +0.5 +0.0 +-2.5 +-5.0 +0.0 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +x/h +(b) +0 +20 +z)/h +10 +-40 +0 +-80 +-10 +Ra +-20 +700 +800 +900 +1000 +1100 +Ra1/5x/h +(c) +0 +20 +z)/h +-10 +10 +-10 +20 +Ra +-20 +1000 +1020 +1040 +1060 +1080 +Ra1/5x/h +(d) +0 +30 +Ra 1/5 (h - z) / h +15 +-2 +0 +.4 +-15 +-30 +1075 +1080 +1085 +1090 +Ra 1/5x/ hHorizontal convection Nu scaling +13 +Above, 𝑏′(𝑥, 𝑦, 𝑧, 𝑡) is the three-dimensional departure from the spanwise average. Taking the +spanwise average of the 3D continuity equation (2.3), we obtain a 2D “overturning stream +function” 𝜓(𝑥, 𝑧, 𝑡), such that ( ˆ𝑢 , ˆ𝑤) = (−𝜓𝑧 , 𝜓𝑥). With this notation the 3D velocity is +written as +(𝑢, 𝑣, 𝑤) = (−𝜓𝑧 , 0, 𝜓𝑥) + (𝑢′, 𝑣′, 𝑤′) . +(4.4) +The spanwise-average of the buoyancy equation is +ˆ𝑏𝑡 + 𝜓𝑥 ˆ𝑏𝑧 − 𝜓𝑧 ˆ𝑏𝑥 + 𝜕𝑥 � +𝑢′𝑏′ + 𝜕𝑧 � +𝑤′𝑏′ = 𝜅∇2 ˆ𝑏 , +(4.5) +and the spanwise average of the spanwise vorticity equation is +𝜁𝑡 + 𝜓𝑥𝜁𝑧 − 𝜓𝑧𝜁𝑥 +������������������������������������ +inertia ++ +ˆ𝑏𝑥 +���� +buoyancy +torque ++ (𝜕2 +𝑧 − 𝜕2 +𝑥) � +𝑢′𝑤′ + 𝜕𝑥𝜕𝑧 +�� +𝑢′2 − � +𝑤′2� +�������������������������������������������������������������������������������� +Reynolds stress torque += 𝜈∇2𝜁 +���� +viscosity +, +(4.6) +where 𝜁 +def= −∇2𝜓 is the spanwise-averaged spanwise vorticity. The power integral (2.8) +becomes +𝜀 = 𝜈 +� +𝜁2� ++ 𝜈⟨|∇𝒖′|2⟩ +(4.7) += −𝜅 ¯𝑏(0)/ℎ . +(4.8) +The 2D equations of motion are recovered by suppressing the spanwise averages of quadratic +fluctuations in (4.5), (4.6), and (4.7). +4.2. A review of Nu ∼ Ra1/5 scaling arguments +Following Shishkina et al. (2016), we assume that there is a BL with thickness 𝛿𝑏 in buoyancy +and 𝛿𝑢 in momentum and vorticity. The Reynolds number is +Re +def= 𝑈ℎ +𝜈 , +(4.9) +where +𝑈 +def= +√︁ +⟨|𝒖|2⟩ +(4.10) +is the typical flow velocity. We use the depth ℎ as representative of the domain dimensions, +i.e., (ℓ𝑥, ℓ𝑦) ∼ ℎ; these three length scales are roughly comparable. +Scale analysis of the surface Nusselt number in (2.11), e.g., ˆ𝑏𝑧(ℎ) ∼ 𝑏★/𝛿𝑏, shows that +Nu ∼ ℎ +𝛿𝑏 +. +(4.11) +One reaches the same conclusion via scale analysis of the 𝜒-based Nusselt number in (2.9): +although |∇𝑏|2 ∼ 𝑏2 +★/𝛿2 +𝑏, the 𝜒-BL occupies only a fraction 𝛿𝑏/ℎ of the domain. Thus (4.11) +follows because of the volume average ⟨ ⟩. +Now apply scale analysis to the buoyancy equation (4.5). Using results such as 𝜓𝑧 ˆ𝑏𝑥 ∼ +𝜓𝑥 ˆ𝑏𝑧 ∼ 𝑈𝑏★/ℎ and 𝜅∇2 ˆ𝑏 ∼ 𝜅𝑏★/𝛿2 +𝑏 one has +𝑈 ∼ 𝜅ℎ +𝛿2 +𝑏 +, +or in non-dimensional form +Nu ∼ (RePr)1/2 . +(4.12) +To estimate the viscous dissipation 𝜀 on the right of the power integral (4.7), one assumes +that an order-one fraction of 𝜀 is concentrated in the BL, and this BL occupies a fraction +𝛿𝑢/ℎ of the domain. One can either neglect 𝜈⟨|∇𝒖′|2⟩, or assume that both terms on the right + +14 +N. C. Constantinou, C. B. Rocha, S. G. L. Smith, and W. R. Young +of (4.7) scale in the same way, i.e., as 𝜈𝜁2 ∼ 𝜈(𝑈/𝛿𝑢)2. In either case +𝜀 ∼ 𝜈𝑈2 +𝛿𝑢ℎ . +(4.13) +Scale analysis of the right of (4.8) assumes that the bottom buoyancy, ¯𝑏(0), is an order- +one fraction of the minimum buoyancy, −𝑏★, on the top surface. (The stronger result that +¯𝑏(0) → −𝑏★ as Ra → ∞ is likely true.) Thus +𝜀 ∼ 𝜅𝑏★ +ℎ +. +(4.14) +Combining (4.13) and (4.14) +𝑈2 ∼ 𝜅 +𝜈 𝛿𝑢𝑏★ , +or in non-dimensional form +(RePr)2 ∼ 𝛿𝑢 +ℎ Ra . +(4.15) +Eliminating RePr between (4.12) and (4.15), and then using (4.11) to get rid of ℎ, one finds +Nu5 ∼ 𝛿𝑢 +𝛿𝑏 +Ra . +(4.16) +The final step to obtain the dependence of Nu on Ra and Pr is to express the ratio 𝛿𝑢/𝛿𝑏 on +the right of (4.16) in terms Ra and Pr. There are three arguments in the literature. +The scaling of Rossby (1965). Taking 𝛿𝑢 = 𝛿𝑏 one obtains from (4.16) +Nu ∼ Pr0 Ra1/5 +and +Re ∼ Pr−1Ra2/5 . +(4.17) +Rossby’s 1965 argument did not employ the power integral and its consequence (4.16). +Instead, Rossby assumes ab initio that 𝛿𝑏 = 𝛿𝑢 and balances buoyancy torque with viscosity +in (4.6), leading to 𝑈 ∼ 𝑏★𝛿3 +𝑢/ℓ𝜈. Combining these results with (4.11) and (4.12) one again +finds (4.17). Rossby’s balance between buoyancy torque and viscosity applies to both FS +and NS. In the FS case, the velocity BL results from the vorticity source ˆ𝑏𝑥 in (4.6): this +rationalization of Rossby’s assumption that 𝛿𝑢 = 𝛿𝑏 also applies to NS. +The scaling of Gayen, Griffith & Hughes (2014). In the vorticity equation (4.6), balance +buoyancy torque with either inertia or Reynolds stress torques, leading to 𝑈2 ∼ 𝛿𝑢𝑏★, and +follow Rossby by assuming that 𝛿𝑏 = 𝛿𝑢. Combining these results with (4.11) and (4.12) one +finds +Nu ∼ Pr1/5Ra1/5 +and +Re ∼ Pr−3/5Ra2/5 . +(4.18) +This argument does not use the power integral and it is not consistent with (4.16) unless Pr is +order unity. Using the scaling assumptions above to estimate 𝜀 in (4.7) and (4.8) we find +𝜈 +� +𝜁2� +∼ Ra (𝜅𝜈2/ℎ4) +and +− 𝜅 ¯𝑏(0)/ℎ ∼ Ra (𝜅2𝜈/ℎ4) . +(4.19) +The two terms in (4.19) differ by a factor of Pr: this is a problem if Pr is either very large or +very small. But with Pr of order unity – and here we consider Pr = 1 – there is no problem +closing the mechanical energy budget and thus the scaling of Gayen et al. (2014) is a valid +alternative to that of Rossby. +The scaling of Shishkina, Grossman & Lohse (2016). In the vorticity equation (4.6), +balance inertia with viscosity, leading to 𝑈 ∼ 𝜈ℎ/𝛿2 +𝑢. Eliminating 𝑈 with (4.12) one finds +𝛿𝑢 = Pr1/2 𝛿𝑏, and substituting into the power-integral (4.16) +Nu ∼ Pr1/10 Ra1/5 , +and +Re ∼ Pr−4/5Ra2/5 . +(4.20) +A distinctive feature of this scaling argument is that buoyancy torque ˆ𝑏𝑥 in (4.6) does not +appear in the leading-order BL vorticity balance. This is justified by requiring that Pr ≪ 1, so + +Horizontal convection Nu scaling +15 +that 𝛿𝑢 = Pr1/2 𝛿𝑏 ≪ 𝛿𝑏. In other words, this visco-inertial BL is so thin that both viscosity +and inertia are much greater than the buoyancy torque ˆ𝑏𝑥 ∼ 𝑏★/ℎ. +Despite different physical assumptions, the three arguments summarized above are in +agreement that Nu ∼ Ra1/5: all differences lie in the dependence of Nu on Pr. In this respect +scaling (4.20) – corresponding to region Iℓ in the phase diagram of Shishkina et al. (2016) – +needs clarification. Region Iℓ, with Pr ≪ 1, is referred to by Shishkina et al. (2016) as “Rossby +scaling”. Although the exponent one-fifth is the same as that of Rossby, the dependence on Pr +in (4.20) differs from that of Rossby in (4.17). Moreover Rossby was concerned with 𝑃𝑟 ≫ 1, +while scaling (4.20) ostensibly applies provided that Pr ≪ 1. Thus referring to Iℓ as Rossby +scaling is a misnomer: the phase diagram does not contain a region corresponding to the +original Rossby scaling in (4.17). +5. Nested boundary layers and the Nu ∼ Ra1/4 scaling +Tsai et al. (2020) investigate 2DNS HC with Pr = 6.14. With Ra > 1010, and provided that +the imposed surface buoyancy varies linearly with the horizontal coordinate 𝑥, Tsai et al. +(2020) report the one-fourth scaling (3.6) extending over four decades of Ra. Here, using +the sinusoidal surface buoyancy (2.4), we also find the one-fourth scaling (3.6) in the 2DFS +and 2DNS solution suites. Tsai et al. (2020) speculate that their one-fourth scaling might +correspond to a regime proposed by Shishkina et al. (2016) in their phase diagram of the +(Ra, Pr) parameter plane. In this scheme the (Ra, Pr)-plane is partitioned into seven regions +and the exponent 1/4 is located in regions III∞, IV𝑢, and I∗ +ℓ. But we now show that defining +features of III∞, IV𝑢, and I∗ +ℓ do not agree with the numerical solutions. We conclude that the +exponent one-fourth found here, and likely in the regime identified by Tsai et al. (2020), is +not in agreement with any region of the phase diagram of Shishkina et al. (2016). +5.1. Partitioning of buoyancy dissipation 𝜒 between BL and interior +A main characteristic distinguishing the various regimes by Shishkina et al. (2016) is the +partitioning of kinetic energy dissipation, 𝜀, and buoyancy variance dissipation, 𝜒, between +the BL and the interior of the domain. To quantify the partitioning of 𝜒 we introduce the +function +𝐹𝜒(𝑧) +def= 𝜅 +ℎ +∫ +𝑧 +0 +|∇𝑏|2 d𝑧′�𝜒 , +(5.1) +where the overbar denotes an (𝑥, 𝑦, 𝑡)-average; see figure 6. 𝐹𝜒(𝑧) increases monotonically +from 0 to 1 with 𝑧/ℎ and indicates the fraction of buoyancy-variance dissipation below the +level 𝑧. In the 2DFS case, figure 6(a) shows that 𝜒 is increasingly localized within a BL as +Ra → ∞. Examination of 𝐹𝜒(𝑧) for the 2DNS solutions indicates no significant differences +from the 2DFS results in figure 6(a). With both FS and NS, 𝜒 is increasingly concentrated +within a BL as Ra → ∞. +A main characteristic of regions III∞ and IV𝑢 in the phase diagram of Shishkina et al. +(2016) is that 𝜒 is dominantly in the interior of the domain. Thus figure 6(a) disqualifies +regions III∞ and IV𝑢. The remaining possibility with exponent 1/4 is the Pr ≫ 1 region +I∗ +ℓ, characterized by a momentum BL that is much thicker than the buoyancy diffusion BL. +But region I∗ +ℓ is located at moderate values of Ra in the phase diagram so that 1/4 is the +first exponent encountered if Pr is fixed and Ra is increased from small values. In our 2D +solutions, however, we first find the 1/5 scaling (3.5), which is replaced at higher Ra by 1/4 +in (3.6). This is also the case in the study of Tsai et al. (2020): first 1/5 and then, at higher Ra, +1/4. We conclude that the exponent 1/4 in the 2D solution suites is not related to regions + +16 +N. C. Constantinou, C. B. Rocha, S. G. L. Smith, and W. R. Young +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +free-slip Fχ +−0.30 +−0.25 +−0.20 +−0.15 +−0.10 +−0.05 +0.00 +(z − h)/h +(a) +Ra +6.4 × 106 +6.4 × 107 +6.4 × 108 +6.4 × 109 +6.4 × 1010 +6.4 × 1011 +6.4 × 1012 +6.4 × 1013 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +free-slip Fχ +−6 +−5 +−4 +−3 +−2 +−1 +0 +Ra1/5 (z − h)/h +(b) +107 +108 +109 +1010 +1011 +1012 +1013 +1014 +Ra +1.0 +1.1 +1.2 +1.3 +1.4 +1.5 +1.6 +1.7 +free-slip Ra1/5 δχ/h +(c) +1.2 +1.4 +1.6 +1.8 +2.0 +2.2 +2.4 +no-slip Ra1/5 δχ/h +2D free-slip +2D no-slip +Figure 6: (a) The function 𝐹𝜒(𝑧) defined in (5.1) for the 2DFS solutions. The vertical axis is +distance from the top surface at 𝑧 = ℎ. A BL thickness, 𝛿𝜒, is defined as the distance from +the top at which 𝐹𝜒(𝑧) = 1/2 (dashed vertical line in panel (a)). (b) Same as (a) but with the +vertical axis rescaled with Ra1/5. Horizontal grey dashed lines indicate the distance 2𝛿1/4 +below the top surface 𝑧 = ℎ. (c) The compensated BL thickness, Ra1/5𝛿𝜒/ℎ, as a function +of Ra. The four vertical grey line segments mark Ra’s of the solutions in figures 1 and 2. +III∞, IV𝑢 and I∗ +ℓ of the phase diagram. In section 6 we seek an alternative explanation for the +one-fourth scaling regimes in figure 4(b). +To extract more information from 𝐹𝜒(𝑧), we scale the 𝑧-axis with 𝛿1/5 and re-plot the results +from figure 6(a) in figure 6(b): the curves now fall largely on top of each other, indicating that +the function 𝐹𝜒(𝑧) expresses the BL thickness 𝛿1/5, even if the Nu ∼ Ra1/4. To quantify this, +we define a BL thickness, 𝛿𝜒, by determining the level at which 𝐹𝜒(𝑧) = 1/2: see the dashed +vertical line in figure 6(a). The compensated plot in figure 6(c) then shows that +𝛿𝜒 ≈ 𝐾𝜒𝛿1/5 . +(5.2) +The constant 𝐾𝜒 in (5.2) is about 1.5 for the FS solutions and 1.8 for the NS solutions. +The Ra−1/5 scaling in (5.2) applies in both the one-fifth regime (3.5) and the one-fourth +regime (3.6). Yet the reasoning in section 4, leading to +Nu ∼ ℎ +𝛿𝑏 +, +(5.3) +underpins all scaling arguments and seems inescapable. It must be that in the one-fourth +regime (3.6), the buoyancy BL has a double-layer structure: there is a thin BL, with thickness +𝛿𝑏 = 𝛿1/4, embedded within the thicker 𝛿𝜒-BL in (5.2). +(The scatter of 𝐾𝜒 in figure 6(c) might be considered uncomfortably large. Note, however, + +Horizontal convection Nu scaling +17 +0.00 +0.25 +0.50 +0.75 +1.00 +b1 +−6 +−4 +−2 +0 +Ra1/5 (z − h)/h +(a) +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +Ra1/5 b1z h/b * +(b) +Ra +6.4 × 109 +1.3 × 1011 +6.4 × 1011 +1.6 × 1012 +3.2 × 1012 +6.4 × 1012 +1.6 × 1013 +6.4 × 1013 +0.0 +0.5 +1.0 +1.5 +2.0 +Ra2/5 b1zz h2/b * +(c) +2D free-slip +Figure 7: The structures of (a) 𝑏1(𝑧), (b) 𝑏′ +1(𝑧), and (c) 𝑏′′ +1 (𝑧) for 2DFS solutions at various +Ra. Note that 𝑏1(𝑧) was obtained here just from the final snapshot of each simulation +without any time-averaging. The vertical axis is distance from the surface 𝑧 = ℎ measured in +units of 𝛿1/5. Horizontal grey dashed lines indicate the distance 2𝛿1/4 below the surface. +that Ra is varied by seven decades. This large range encompasses the transition from steady +to strongly time-dependent 2D flows. At Ra = 6.4 × 1013, Ra−1/4 is smaller than Ra−1/5 by a +factor of 5, which is much greater than the ±20% scatter in figure 6(c).) +5.2. The 𝛿1/4-boundary layer +To identify the 𝛿1/4-BL in our solutions, and show consistency with (5.3) in the one-fourth +regime, we notice that with the sinusoidal surface buoyancy in (2.4), the surface Nusselt +number (2.11) is +Nus = 𝑏′ +1(ℎ) +� +𝑏′ +diff1(ℎ) , +(5.4) +where above the prime denotes a 𝑧-derivative and +𝑏1(𝑧) +def= 2 cos 𝑘𝑥 𝑏(𝑥, 𝑦, 𝑧, 𝑡) . +(5.5) +We define the thickness, 𝛿s, of this surface BL in (5.4) by +𝛿s +def= 𝑏★/𝑏′ +1(ℎ) , +(5.6) +where 𝑏★ is the amplitude of the sinusoidal surface buoyancy in (2.4) The numerator in (5.6) +is appropriate since 𝑏★ = 𝑏1(ℎ) = 𝑏diff1(ℎ). +Figure 7 shows 𝑏1(𝑧), and the first two derivatives of this averaged field. The overline +in (5.5) indicates both a horizontal and temporal average. Unfortunately we did not collect a +time series of 𝑏1 and thus figure 7 is based on the horizontal average of single snapshots of +the buoyancy field at the final time. The inner BL, with thickness 𝛿1/4, is not visible in 𝑏1(𝑧) +in figure 7(a). But the higher derivatives of 𝑏1(𝑧) in the panels (b) and (c) reveal the scale 𝛿1/4 +in the solution. In particular, the one-fourth scaling (3.6) results from the increase in 𝑏′ +1(ℎ), +evident in figure 7(b) as Ra increases. The maximum of 𝑏′′ +1 (𝑧) in figure 7(c) appears only in +the one-fourth regime (3.6). +Figure 8(a) shows 𝛿s, diagnosed from (5.6), and compensated by Ra−1/5. In the one-fifth +scaling regime (3.5), with Ra less than about 1011, Ra−1/5𝛿s/ℎ varies between about 2.6 and +2.9. In these cases both 𝛿s and 𝛿𝜒 are ∼ 𝛿1/5. Figure 8(b) shows that at the four or five highest +values of Ra, Ra−1/4𝛿s/ℎ varies between about 10.5 and 11.4. We conclude that in these cases + +18 +N. C. Constantinou, C. B. Rocha, S. G. L. Smith, and W. R. Young +107 +108 +109 +1010 +1011 +1012 +1013 +1014 +Ra +2.0 +2.2 +2.4 +2.6 +2.8 +Ra1/5 δs +(a) +107 +108 +109 +1010 +1011 +1012 +1013 +1014 +Ra +6 +7 +8 +9 +10 +11 +Ra1/4 δs +(b) +Figure 8: The compensated surface BL thickness, 𝛿s in (5.6), for the 2DFS suite. In (a) 𝛿s is +compensated by Ra−1/5 and in (b) by Ra−1/4. The transition between the one-fifth +scaling (3.5) and the one-fourth scaling (3.6) is at about 6.4 × 1011. Vertical grey line +segments mark the Ra values corresponding to the solutions in figures 1 and 2. Vertical grey +line segments mark the Ra values corresponding to the solutions in figures 1 and 2. +𝛿s ∼ 𝛿1/4, but 𝛿𝜒 ∼ 𝛿1/5. Via (5.3), the Nusselt number is determined by the 𝛿1/4 inner BL, +resulting in the one-fourth regime (3.6). +6. A scaling argument for Nu ∼ Ra1/4 +In this section we present a scaling argument applicable to the one-fourth regime of horizontal +convection. Although our numerical solutions revealed the one-fourth scaling regime only +in the 2D cases, we still hold hope that the one-fourth regime might also emerge in 3D at +sufficiently high Ra. With the 3D case in mind, we propose an overarching explanation for the +one-fourth scaling – independent of boundary conditions, dimensionality and 2D vortex-gas +phenomenology. +The one-fourth regime requires an inner buoyancy BL with thickness 𝛿𝑏 ∼ 𝛿1/4. In discussing +this inner BL it is helpful to keep figure 5(d) in mind: the 𝛿1/4-BL is identified by the dashed +grey line. Think of this inner BL as a laminar sub-layer, stirred by the outer flow in the much +thicker 𝛿1/5-BL. The overarching explanation alluded to above is that the thickness of the +laminar sub-layer is related to the Kolmogorov and Batchelor length scales +𝜂K = +� 𝜈3 +𝜀 +�1/4 +and +𝜂B = +� 𝜅2𝜈 +𝜀 +�1/4 +. +(6.1) +These length scales are identified as the smallest scales of fluctuations in momentum and +buoyancy that can survive before the damping by viscosity 𝜈 and diffusion 𝜅 is overwhelming. +By analogy, the HC laminar sub-layer with thickness 𝛿𝑏, is the thinnest BL that can survive in +a horizontal-convective flow that is supplied with kinetic energy at a rate 𝜀. + +Horizontal convection Nu scaling +19 +(In the arguments of Kolmogorov and Batchelor the viscous dissipation rate 𝜀 is also the +energy cascade rate in a 3D inertial range. This interpretation of 𝜀 cannot apply to 2D HC: +there is no vortex stretching in a 2D flow and therefore no forward cascade of energy. We argue +instead that the laminar sub-layer thickness is determined by 𝜀 as the most basic measure +of forcing strength and by the molecular parameters 𝜈 and 𝜅. Thus 𝛿𝑏 ∼ (𝜈 𝑝𝜅𝑞/𝜀)1/4, with +𝑝 + 𝑞 = 3, is dimensionally acceptable; 𝜂K and 𝜂B are the most prominent members of this +family. Saying more would would require varying Pr which is beyond our scope here.) +Following the scaling arguments reviewed section 4, we assume that ¯𝑏(0) ≈ −𝑏★. Then, +once again, the energy power integral (2.8) implies that +𝜀 ∼ 𝜅𝑏★ +ℎ +. +(6.2) +With 𝜀 in (6.2), 𝜂K and 𝜂B in (6.1) can be written as +𝜂K +ℎ = Pr1/2Ra−1/4 +and +𝜂B +ℎ = Pr0Ra−1/4 . +(6.3) +Thus if the laminar sub-layer has thickness 𝛿𝑏 ∼ 𝜂K, or perhaps 𝛿𝑏 ∼ 𝜂B, then Nu ∼ Ra1/4. +7. Discussion: is 3D horizontal convection turbulent? +Paparella & Young (2002) showed that as a consequence of the mechanical energy power +integral (6.2), HC does not satisfy the zeroth law of turbulence. Paparella & Young also noted +that the zeroth law is not a universally accepted as part of a definition of turbulence. For +example, Scotti & White (2011) argued that the zeroth law is irrelevant because “HC can +transport very large quantities of heat and sustain large amounts of diapycnal mixing with +a surprisingly small amount of dissipation”. Similar sentiments, reinforced by arguments +involving exchange between available potential energy and kinetic energy, are expressed by +Gayen et al. (2013, 2014). +(Note that even the very form of zeroth law of turbulence appropriate to HC is also +controversial. Paparella & Young (2002) state that 𝜀 should be non-zero as (𝜈, 𝜅) → (0, 0) +with Pr = 𝜈/𝜅 fixed. In their “clarification” of the zeroth law, Shishkina et al. (2016) prefer a +different limit in which 𝜈 → 0 with 𝜅 fixed. In this case, by inspection of (2.8), the zeroth law +applies in the high-inertial limit in which Ra → ∞ and simultaneously Pr → 0.) +A common response to the question “What is turbulence?” (for example given by Google +or by chatbot ChatGPT), is that “Turbulence is a fluctuation or disturbance in a fluid (such as +air or water) that is characterized by chaotic and irregular movements.” However, there is +more to turbulence than this, else the 2D solutions in figure 2 are turbulent. In addition to +“chaotic and irregular movements” a transition to 3D flow is viewed as essential (Gayen et al. +2014; Tsai et al. 2016, 2020). It is only in 3D that “the high average vorticity which is known +to exist in turbulent motion" can be produced by “extension of vortex filaments in an eddying +fluid” (Taylor 1938). +In this section we use the four solution suites from section 3 to revisit the question of +whether HC is “turbulent”. We show that the 3D solutions, with maximum Ra = 3.2 × 1011, +cannot be considered turbulent. We also present theoretical arguments indicating that 3D +HC cannot become turbulent at any Ra. We set aside the zeroth law of turbulence and focus +instead on accepted characteristics of 3D hydrodynamic turbulence: +(a) chaotic, disordered and irregular fluid motions, irreproducible in detail; +(b) greatly enhanced transport of momentum and heat; +(c) strong vorticity amplification by strain mediated 3D vortex stretching; +(d) a direct cascade of energy in an inertial range; + +20 +N. C. Constantinou, C. B. Rocha, S. G. L. Smith, and W. R. Young +(e) termination of the inertial range at the viscous length scale 𝜂K. +According to Stewart in the educational film “Turbulence” (NCFMF 1968, 1972) these +phenomena “give a defining syndrome, or set of symptoms, for turbulence”. (Stewart listed (a), +(b), and (c); here we have added (d) and (e).) +Chaotic and disordered flow described by symptom (a) applies to both 2D and 3D time- +dependent solutions at moderately high Ra, e.g., beyond 109. Just based on symptom (a) one +would conclude that 2D HC is turbulent. But because of symptoms (c) and (d), a 2D flow – +no matter how erratically time dependent the velocity – cannot be turbulent (Taylor & Green +1937). +The 2D and 3D HC solutions coincide up to a critical Ra at which 3D instabilities first +appear (Gayen et al. 2014; Tsai et al. 2016; Passaggia et al. 2017). This bifurcation to 3D flow +is usually viewed as the first step in a transition to turbulence. But the unstable non-turbulent +2D solution is always present and serves as a “comparison flow” for putatively turbulent 3D +solutions at the same Ra, and with the same boundary conditions. The main thrust of the +argument that follows is that 3D HC is neither qualitatively nor quantitatively different from +the non-turbulent 2D comparison flow: the onset of three-dimensionality in HC does not +inflame turbulence symptoms (b) through (e). +A question of interest, especially in oceanographic context, is how much might HC +contribute to total heat flux? Thus, how much can Nu be enhanced in turbulent HC has +been a motivation for HC research. Symptom (b), emphasized by Scotti & White (2011) +and other authors, demands that HC turbulence is accompanied by a large increase in the +horizontal transport of heat. But in figure 3 and table 1, the 3D Nu is only 20% greater than +that of the non-turbulent 2D comparison flow. Symptom (b) demands much more than a 20% +enhancement in Nu between a turbulent 3D flow and a non-turbulent 2D comparison flow. +Moreover figure 3 and table 1 show that the viscous boundary condition (FS versus NS) has +a larger quantitative effect on Nu than does dimensionality: even after the transition to unsteady +3D flow (𝑅𝑎 ⩾ 6.4 × 108), the 3DNS solutions transport less buoyancy than the non-turbulent +2DFS solutions at the same Ra. If 3DNS transitions to turbulence at some Ra > 3.2 × 1011 +then, no matter the boundary condition, the turbulent 3D flow should transport more heat +than the non-turbulent 2D flow. There is no indication of this hypothetical crossover between +3DNS and 2DFS in figure 3. +Table 3 summarizes gross measures of the departures from the 2D spanwise-averaged +circulation defined in (4.3). For both 3DNS and 3DFS, about two-thirds of the kinetic energy +is in the spanwise averaged flow. In the third column the component of buoyancy gradient +in the spanwise direction (𝑏𝑦) contributes less than 2% to the buoyancy dissipation 𝜒. The +only statistic that is dominated by departures from the spanwise average is mechanical energy +dissipation, 𝜀, in the fourth column. Thus table 3, and particularly the third column, supports +the view that the 3D 𝑁𝑢, even at Ra = 3.20 × 1011, is largely determined by the 2D spanwise +averaged circulation, rather than by robust 3D turbulence characterized by (b) through (e). +Figure 9(a) shows that there is no inertial cascade in the interior, i.e., (c) through (e) do not +apply to this 3DFS solution (nor to the 3DNS solution). An inertial cascade is characterized by +a kinetic energy spectrum ∼ 𝜀2/3𝑘−5/3, or a vorticity spectrum ∼ 𝜀2/3𝑘+1/3. The ultra-violet +vorticity divergence is cut-off at a wavenumber of order 𝜂−1 +K . But in contradiction to (e) the +snapshot in figure 9(a) shows that vorticity is concentrated on length scales very much larger +than 𝜂K. +We do not have an estimate of the length scale of the vorticity fluctuations in figure 9 (nor +for the core radius of the 2D vortices in figure 2). These vorticity length scales are rather +less than the domain scales (ℓ𝑥, ℓ𝑦, ℎ), but very much greater than 𝜂K in (6.3). There is, +however, a simple result for the magnitude of the vorticity 𝝎 = ∇×𝒖. With a well known +identity, the kinetic energy dissipation can be written as 𝜀 = 𝜈⟨|𝝎|2⟩ and the mechanical + +Horizontal convection Nu scaling +21 +⟨| ˆ𝒖|2⟩/⟨|𝒖|2⟩ +⟨𝑣2⟩/⟨|𝒖|2⟩ +𝜅⟨𝑏2𝑦⟩/𝜒 +𝜈⟨𝜁2⟩/𝜀 +3DFS +0.662 +0.112 +0.013 +0.187 +3DNS +0.690 +0.092 +0.019 +0.342 +Table 3: Statistics for the 3D solutions at Ra = 3.20 × 1011. The ratios above were computed +from a single snapshot at the final time, i.e., without the benefit of time averaging. These +ratios decrease monotonically to zero as Ra is lowered to the critical value for the onset of +3D motion. In the first column, ˆ𝒖 = (−𝜓𝑧, 0, 𝜓𝑥) is the spanwise averaged velocity in (4.4) +and in the final column 𝜁 = −∇2𝜓 is the vorticity of the spanwise-averaged flow. +power integral (2.8) rewritten as +⟨|𝝎|2⟩ +𝑏★/ℎ = −𝜅 +𝜈 +¯𝑏(0) +𝑏★ +, +(7.1) +⩽ 𝜅 +𝜈 . +(7.2) +In passing from the exact equality (7.1) to the rigorous inequality in (7.2) we have used the +extremum principle for buoyancy. +What is the significance of the bound (7.2)? It is remarkable that (7.2) is independent of +aspect ratio ℓ𝑦/ℎ. Thus at high Ra, with ¯𝑏(0) close to −𝑏★, 3D HC must have almost the +same ⟨|𝝎|2⟩ as that of the non-turbulent 2D comparison flow. Near equality of ⟨|𝝎|2⟩ in 2D +and 3D is incompatible with turbulence vorticity amplification symptom (c). We can safely +deduce that even a wider domain (increasing ℓ𝑦/ℎ) would not produce a large increase in Nu +resulting from turbulent transport. Moreover, if the Kolmogorov and Batchelor length scales +in (6.3) control the thickness of the transport-determining BL then indeed ℓ𝑦/ℎ is irrelevant. +To reinforce and illustrate the conclusion above, note that ¯𝑏(0) of the Ra = 6.4 × 1013 +2D solution in figure 2(d) is within 15% of the minimum −𝑏★. Thus the root-mean-square +(RMS) vorticity of this 2D flow is within 8% of the maximum +√︁ +𝑏★/ℎ implied by (7.2). (In +this example 𝜅/𝜈 = 1.) It follows that a 3DFS HC solution at Ra = 6.4 × 1013, and Pr = 1, +must have an RMS vorticity within 8% of the RMS vorticity of the 2D comparison flow +in figure 2(d). (The 3D flow has more RMS vorticity than the 2D flow, but less than the +maximum permitted by (7.2).) Moreover, all numerical and experimental results indicate that +¯𝑏(0) → −𝑏★ as Ra → ∞. Thus increasing Ra above 6.4 × 1013 likely reduces the already +small 8% difference between the RMS vorticity of the 3D flow and the 2D comparison flow. +The small enhancement of 3D RMS vorticity implies that vortex stretching (c) does not +effectively operate in 3D HC. +As a concluding illustration of the vorticity bound (7.2), notice that in figures 1 and 2 the +vorticity is scaled with +√︁ +𝑏★/ℎ. With this scaling the same colorbar applies even as Ra is +varied by a factor 106. The vorticity of the 3D flow in figure 9 is also scaled with +√︁ +ℎ/𝑏★. This +simple estimate of the RMS vorticity applies across all 2D and 3D solutions reported here. +8. Conclusion +We have conducted a numerical study of the Ra–Nu relation with Pr = 1 and four cases +corresponding to either no-slip or free-slip boundary conditions, in both 2D (ℓ𝑦/ℎ = 0) or +3D (ℓ𝑦/ℎ = 1) geometries. In all four cases, with Ra in the range 106 to 1010, we find that +heat flux obeys Rossby scaling, that is, Nu ∼ Ra1/5. In the 2D cases, with maximum Rayleigh + +22 +N. C. Constantinou, C. B. Rocha, S. G. L. Smith, and W. R. Young +Figure 9: Panels (a) and (b) show a 𝑦-slice of snapshots of 3DFS HC at Ra = 3.20 × 1011. +The spanwise averages of the snapshots of vorticity and buoyancy are shown in panels (c) +and (d). In all panels, the contours are streamlines computed from a spanwise averaged +snapshot at the final time. At the top surface −1 ⩽ 𝑏/𝑏★ ⩽ +1; the narrow range of the +buoyancy color scale makes the small interior variations visible. +number of order 1014, we found that a scaling regime with Nu ∼ Ra1/4 replaces the Rossby +scaling for Ra beyond 1011; see also Tsai et al. (2020). +The scaling arguments for the Nu–Ra relation of HC reviewed in section 4 do not depend +very much, if at all, on the distinction between 2D and 3D HC. Nor do these arguments +identify the spanwise aspect ratio ℓ𝑦/ℎ as an important parameter. Thus it is informative to +conduct parallel numerical studies of 2D and 3D HC and compare corresponding Nu’s. This +comparison of 2D with 3D HC, extending to Ra = 3.2 × 1011, shows that 3D HC has only a +slight 10 or 20% enhancement of heat transport over non-turbulent 2D HC. It is difficult to +believe that there is only a slight enhancement at significantly higher Ra – otherwise relatively +inexpensive 2D numerical solutions would provide useful estimates of 3D HC heat transport. + +(a) +vorticity, (azu- äxw) Vh/b* at y= 0 +1.0 +1.5 +1.0 +0.5 +0 0.5 +0.0 +-0.5 +1.0 +1.5 +0.0 +(b) +buoyancy, b/b* +1.0 +-0.822 +-0.824 +0 0.5 +-0.826 +-0.828 +0.0 +x/h +(c) +vorticity, (azu- axw) h/b* +1.0 +1.5 +1.0 +0.5 +0 0.5 +0.0 +-0.5 +-1.0 +-1.5 +0.0 +(d) +buoyancy, b/b +1.0 +-0.822 +-0.824 +%0.5 +-0.826 +-0.828 +0.0 - +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +x/hHorizontal convection Nu scaling +23 +It is likely that the 2D Nu is less than (or equal to) the 3D Nu. Proving this plausible conjecture +is an open challenge. +Our results, numerical and theoretical, reinforce the view that HC does not express all of +the characteristics of turbulence (Paparella & Young 2002). +Acknowledgments +Without implying their endorsement, we thank Basile Gallet and Ross Griffiths for discussions +of horizontal convection. Computational resources were provided by the Australian National +Computational Infrastructure at the Australian National University, which is supported by +the Commonwealth of Australia. NCC was supported by the Australian Research Council +DECRA Fellowship DE210100749. CBR was supported by National Aeronautics and Space +Administration Award NNX16AO5OH. SGLS was partially supported by National Science +Foundation Awards OCE-1829919 WRY was supported by National Science Foundation +Awards OCE-1657041 and OCE-2048583. +Appendix A. The low Rayleigh number regime +If the Rayleigh number is sufficiently small then one can employ a straightforward expansion +in powers of Ra to show that the Nusselt number is +Nu = 1 + 𝐶2Ra2 + ord�Ra4� . +(A 1) +In the expansion (A 1), 𝐶2 is a function of the aspect ratio, 𝐴𝑥 = ℓ𝑥/ℎ, but not the Prandtl +number Pr. In this appendix we summarize the calculation of 𝐶2 for horizontal convection +forced with the sinusoidal 𝑏s in (2.4). This calculation is more interesting than one might +anticipate because 𝐶2 turns out to be a very small number for all values of the aspect ratio +𝐴𝑥. Consequently with the aspect ratio 𝐴𝑥 = 4 used in this work “sufficiently small” means +Rayleigh numbers of order 104 (see the inset in figure 4). +Using the streamfunction formulation, with (𝑢, 𝑤) = (−𝜓𝑧, 𝜓𝑥), and scaling lengths with +the depth ℎ and time with ℎ2/𝜅 the steady Boussinesq equations are +𝜓𝑥∇2𝜓𝑧 − 𝜓𝑧∇2𝜓𝑥 = 𝑏𝑥 + Pr ∇4𝜓 , +(A 2) +𝜓𝑥𝑏𝑧 − 𝜓𝑧𝑏𝑥 = ∇2𝑏 , +(A 3) +where here ∇2 = 𝜕2 +𝑥 + 𝜕2 +𝑧 is the two-dimensional Laplacian; the spanwise vorticity is +𝑢𝑧 − 𝑤𝑥 = −∇2𝜓. The surface boundary condition is 𝑏(𝑥, 1) = 𝜖 cos 𝑚𝑥 where +𝑚 +def= 𝑘ℎ = 2𝜋/𝐴𝑥 , +and +𝜖 +def= PrRa/𝐴3 +𝑥 = PrRa (𝑚/2𝜋)3 . +(A 4) +We expand all variables in powers of the small parameter 𝜖 +(𝑏, 𝜓) = 𝜖(𝑏1, 𝜓1) + 𝜖2(𝑏2, 𝜓2) + · · · . +(A 5) +The first-order equations are +Pr∇4𝜓1 = −𝑏1𝑥 , +and +∇2𝑏1 = 0 . +(A 6) +The solution (A 6) is +𝜓1 = sin 𝑚𝑥 𝑃(𝑧) , +and +𝑏1 = cos 𝑚𝑥 𝐵(𝑧) , +(A 7) +where 𝐵(𝑧) +def= sech 𝑚 cosh 𝑚𝑧. In 𝜓1(𝑥, 𝑧) we have the free-slip function +𝑃𝐹𝑆(𝑧) = +𝐵(𝑧) +8𝑚2Pr +� +(𝑚 coth 𝑚 + 1 − 𝑧) tanh 𝑚𝑧 − 𝑚𝑧(2 − 𝑧) +� +, +(A 8) + +24 +N. C. Constantinou, C. B. Rocha, S. G. L. Smith, and W. R. Young +and the no-slip function +𝑃𝑁 𝑆(𝑧) = +1 +8𝑚Pr (sinh2 𝑚 − 𝑚2) +� +(sinh2 𝑚 − 𝑚2)(𝑧2 − 𝑧)𝐵(𝑧) ++ (tanh 𝑚 − 𝑚) 𝑧 sinh 𝑚(1 − 𝑧) + (sinh 𝑚 − 𝑚 sech 𝑚) (1 − 𝑧) sinh 𝑚𝑧 +� +. +(A 9) +At second order in 𝜖 we must solve +∇2𝑏2 = 𝜓1𝑥𝑏1𝑧 − 𝜓1𝑧𝑏1𝑥 , +(A 10) += 1 +2𝑚(𝑃𝐵)′ +�������������� +𝐽0 ++ 1 +2𝑚(𝑃𝐵′ − 𝑃′𝐵) +�������������������������������� +𝐽2 +cos 2𝑚𝑥 . +(A 11) +The solution of (A 11) has the form +𝑏2 = 𝐵20(𝑧) + 𝐵22(𝑧) cos 2𝑚𝑥 , +(A 12) +where 𝐵20 and 𝐵22 are determined by solving +𝐵′′ +20 = 𝐽0 , +(A 13) +𝐵′′ +22 − 4𝑚2𝐵22 = 𝐽2 . +(A 14) +Forming ⟨𝐵22(A 14)⟩ we find the shortcut used below in passing from (A 16) to (A 17): +� +𝐵′ +22 +2 + 4𝑚2𝐵2 +22 +� += − ⟨𝐵22𝐽2⟩ . +(A 15) +The expressions for 𝐵22 and 𝐵20, obtained with Mathematica, are complicated and are not +explicitly presented. Mercifully, to obtain the coefficient 𝐶2 in (A 1), we do not need 𝜓2. +Multiplying ∇2𝑏1 = 0 by 𝑏𝑛, with 𝑛 ⩾ 2, and noting that all these 𝑏𝑛’s have homogeneous +boundary conditions at 𝑧 = 0 and 1, we see that ⟨∇𝑏𝑛 · ∇𝑏1⟩ = 0. Consequently the expansion +of the buoyancy variance dissipation is +𝜒 = 𝜖2 ⟨|∇𝑏1|2⟩ +������������ +𝜒2= 1 +2 𝑚 tanh 𝑚 ++ 𝜖4 ⟨|∇𝑏2|2⟩ +������������ +𝜒4 ++ ord�𝜖6� . +(A 16) +Recalling the definition of 𝜖 in (A 4), the Nusselt number is +Nu = 1 + +𝑚2� +(𝑃𝐵)2� +− 2⟨𝐵22𝐽2⟩ +2𝑚 tanh 𝑚 +������������������������������������������������������ +𝜒4/𝜒2 +�𝑚3Pr +8𝜋3 +�2 +Ra2 + · · · . +(A 17) +Because 𝑃, 𝐵22, and 𝐽2 are all proportional to Pr−1, the Prandtl number Pr cancels out of the +coefficient of Ra2 in (A 17). +With FS boundary conditions, the expression for 𝐶2 in (A 1) is +𝐶𝐹𝑆 +2 += +� +690 − 1920𝑚4 cosech2 𝑚 + 20(3 + 4𝑚2)2 sech 2𝑚 ++ 𝑚(1024𝑚4 − 80𝑚2 − 6195) cosech 𝑚 sech3 𝑚 ++ 5(352𝑚4 − 624𝑚2 + 1065) sech2 𝑚 +�� +41 943 040 𝜋6 . +(A 18) + +Horizontal convection Nu scaling +25 +Limiting values in the FS case are +lim +𝑚→0 𝐶𝐹𝑆 +2 += +31 𝑚8 +30 965 760𝜋6 + ord�𝑚10� , +(A 19) +and +lim +𝑚→∞ 𝐶𝐹𝑆 +2 += +69 +4 194 304 𝜋6 . +(A 20) +We admire the frequently occurring integer 4 194 304 = 222 in the formulas above and below. +With NS boundary conditions, we find +𝐶 𝑁 𝑆 +2 += +1 +41 943 040 𝜋6 +Ξ(𝑚) +� cosh 2𝑚 − 2𝑚2 − 1�2 , +(A 21) +where +Ξ(𝑚) = 85 cosh 4𝑚 − 320𝑚(33 + 8𝑚2) sinh 2𝑚 + 10�845 + 716𝑚2� cosh 2𝑚 +− 5(5123 + 18304𝑚2 + 13720𝑚4 + 2912𝑚6) +− 4𝑚5(7215 + 100𝑚2 − 64𝑚4) coth 𝑚 ++ 20(829 + 4402𝑚2 + 5289𝑚4 + 836𝑚6 − 64𝑚8) sech2 𝑚 +− 4𝑚(4785 + 8010𝑚2 + 1549𝑚4 − 212𝑚6 + 64𝑚8) tanh 𝑚 sech2 𝑚 ++ 20(25 − 184𝑚2 − 80𝑚4 − 16𝑚6) sech 2𝑚 ++ 4𝑚(8865 + 21640𝑚2 + 17719𝑚4 + 740𝑚6 − 64𝑚8) tanh 𝑚 ++ 160𝑚(15 − 7𝑚2 − 4𝑚4) tanh 2𝑚 . +(A 22) +Limiting values in the NS case are +lim +𝑚→0 𝐶 𝑁 𝑆 +2 += +𝑚8 +30 965 760𝜋6 + ord�𝑚10� , +(A 23) +and +lim +𝑚→∞ 𝐶 𝑁 𝑆 +2 += +17 +4 194 304𝜋6 . +(A 24) +It is notable that the small-𝑚 𝐶 𝑁 𝑆 +2 +in (A 23) is smaller by a factor of exactly 31 than 𝐶𝐹𝑆 +2 +in (A 19). +Both 𝐶2’s are very much less than one for all aspect ratios. In the numerical solutions +summarized in figure 4, the aspect ratio is 𝐴𝑥 = 4, corresponding to 𝑚 = 𝑘ℎ = 𝜋/2. The results +in (3.3) and (3.4) follow by evaluating the formulas (A 18) through (A 22) with 𝑚 = 𝜋/2. +REFERENCES +Benzi, R, Patarnello, S & Santangelo, P 1987 On the statistical properties of two-dimensional decaying +turbulence. Europhys. Lett. 3 (7), 811. +Benzi, R, Patarnello, S & Santangelo, P 1988 Self-similar coherent structures in two-dimensional +decaying turbulence. J. Phys. A 21 (5), 1221. +Burns, K. J., Vasil, G. M., Oishi, J. S., Lecoanet, D. & Brown, B. P. 2020 Dedalus: A flexible framework +for numerical simulations with spectral methods. Phys. Rev. Res. 2, 023068. +Carnevale, G. F., McWilliams, J. C., Pomeau, Y., Weiss, J. B. & Young, W. R. 1991 Evolution of vortex +statistics in two-dimensional turbulence. Phys. Rev. Lett. 66 (21), 2735. +Chiu-Webster, S., Hinch, E. J. & Lister, J. R. 2008 Very viscous horizontal convection. J. Fluid Mech. +611, 395–426. + +26 +N. C. Constantinou, C. B. Rocha, S. G. L. Smith, and W. R. Young +Coman, M. A., Griffiths, R. W. & Hughes, G. O. 2006 Sandström’s experiments revisited. J. Mar. Res. +64 (6), 783–796(14). +Dritschel, D. G., Scott, R. K., Macaskill, C., Gottwald, G. A. & Tran, C. V. 2008 Unifying scaling +theory for vortex dynamics in two-dimensional turbulence. Phys. Rev. Lett. 101 (9), 094501. +Gayen, B., Griffiths, R. W. & Hughes, G. O. 2014 Stability transitions and turbulence in horizontal +convection. J. Fluid Mech. 751, 698–724. +Gayen, B., Griffiths, R. W., Hughes, G. O. & Saenz, J. A. 2013 Energetics of horizontal convection. +J. Fluid Mech. 716, R10. +Hughes, G. O. & Griffiths, R. W. 2008 Horizontal convection. Annu. Rev. Fluid Mech. 40, 185–208. +Ilicak, M. & Vallis, G. K. 2012 Simulations and scaling of horizontal convection. Tellus A 64 (1), 18377. +Kuhlbrodt, T. 2008 On Sandström’s inferences from his tank experiments: a hundred years later. Tellus A +60 (5), 819–836. +McWilliams, J. C 1984 The emergence of isolated, coherent vortices in turbulent flow. J. Fluid Mech. 146, +21–43. +McWilliams, J. C 1990 The vortices of two-dimensional turbulence. J. Fluid Mech. 219, 361–385. +Mullarney, J. C., Griffiths, R. W. & Hughes, G. O. 2004 Convection driven by differential heating at a +horizontal boundary. J. Fluid Mech. 516, 181–209. +NCFMF 1968 Turbulence (Educational film from 1968). https://youtu.be/AeBsiEYWZUY. +NCFMF 1972 Illustrated Experiments in Fluid Mechanics: The NCFMF Book of Film Notes. National +Committee for Fluid Mechanics Films . MIT Press. +Paparella, F. & Young, W. R. 2002 Horizontal convection is non-turbulent. J. Fluid Mech. 466, 205–214. +Passaggia, Pierre-Yves, Scotti, Alberto & White, Brian 2017 Transition and turbulence in horizontal +convection: linear stability analysis. J. Fluid Mech. 821, 31–58. +Rocha, C. B., Bossy, T., Llewellyn Smith, S. G. & Young, W. R. 2020a Improved bounds on horizontal +convection. J. Fluid Mech. 883, A41. +Rocha, C. B., Constantinou, N. C., Llewellyn Smith, S. G. & Young, W. R. 2020b The Nusselt numbers +of horizontal convection. J. Fluid Mech. 894, A24. +Rossby, H. T. 1965 On thermal convection driven by non-uniform heating from below: an experimental study. +Deep Sea Res. 12 (1), 9–10, IN9–IN14, 11–16. +Rossby, H. T. 1998 Numerical experiments with a fluid heated non-uniformly from below. Tellus A 50 (2), +242–257. +Sandström, J. W. 1908 Dynamische versuche mit meerwasser. Annals in Hydrodynamic Marine Meteorology +36, 6–23. +Scotti, A. & White, B. 2011 Is horizontal convection really “non-turbulent?”. Geophys. Res. Lett. 38 (21), +L21609. +Sheard, G. J. & King, M. P. 2011 Horizontal convection: effect of aspect ratio on Rayleigh number scaling +and stability. Appl. Math. Modelling 35 (4), 1647–1655. +Shishkina, O., Grossmann, S. & Lohse, D. 2016 Heat and momentum transport scalings in horizontal +convection. Geophys. Res. Lett. 43 (3), 1219–1225. +Siggers, J. H., Kerswell, R. R. & Balmforth, N. J. 2004 Bounds on horizontal convection. J. Fluid Mech. +517, 55–70. +Taylor, G. I. 1938 Production and dissipation of vorticity in a turbulent fluid. Proc. R. Soc. Lond. A +164 (916), 15–23. +Taylor, G. I. & Green, A. E. 1937 Mechanism of the production of small eddies from large ones. Proc. R. +Soc. Lond. A 158 (895), 499–521. +Tsai, T., Hussam, W. K, Fouras, A. & Sheard, G. J. 2016 The origin of instability in enclosed horizontally +driven convection. Int. J. Heat and Mass Transfer 94, 509–515. +Tsai, T., Hussam, W. K., King, M. P. & Sheard, G. J. 2020 Transitions and scaling in horizontal convection +driven by different temperature profiles. Int. J. Thermal Sci. 148, 106166. +Wang, W. & Huang, R. X. 2005 An experimental study on thermal convection driven by horizontal +differential heating. J. Fluid Mech. 540, 49–73. +Winters, K. B. & Young, W. R. 2009 Available potential energy and buoyancy variance in horizontal +convection. J. Fluid Mech. 629, 221–230. + diff --git a/F9E1T4oBgHgl3EQfXASj/content/tmp_files/load_file.txt b/F9E1T4oBgHgl3EQfXASj/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fcde6de7fe9bd1e3ea3d94e30e75fc50732a826d --- /dev/null +++ b/F9E1T4oBgHgl3EQfXASj/content/tmp_files/load_file.txt @@ -0,0 +1,1699 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf,len=1698 +page_content='Under consideration for publication in J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 1 Nusselt number scaling in horizontal convection: boundary conditions and dimensionality Navid C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Constantinou1†, Cesar B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Rocha2, Stefan G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Llewellyn Smith3, 4, and William R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Young4 1Research School of Earth Sciences & ARC Centre of Excellence for Climate Extremes, Australian National University, Canberra, ACT 2601, Australia 2Department of Marine Sciences, University of Connecticut, Groton, CT 06340, USA 3Department of Mechanical and Aerospace Engineering, University of California San Diego, La Jolla, CA 92093-0411, USA 4Scripps Institution of Oceanography, University of California San Diego, La Jolla, CA 92093-0213, USA (Received xx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' revised xx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' accepted xx) We conduct a numerical study of horizontal convection (HC) at Prandtl number Pr = 1, with both with no-slip and free-slip boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' We obtain 2D and 3D solutions and determine the relation between the Rayleigh number Ra and the Nusselt number Nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' In 2D we vary Ra between 0 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4 × 1013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' In the range 106 ⪅ Ra ⪅ 1010 the Nu–Ra relation is, apart from minor departures, in agreement with Rossby’s scaling Nu ∼ Ra1/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' With Ra greater than about 1011 we find a 2D regime with Nu ∼ 𝑅𝑎1/4 over three decades, up to the highest 2D Ra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' In 3D, with maximum Ra = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='2 × 1011, we only find Rossby scaling regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' These results apply to both viscous boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The Nu ∼ 𝑅𝑎1/4 regime has a double boundary layer (BL): there is a thin BL with thickness ∼ Ra−1/4 inside a thicker BL with thickness ∼ Ra−1/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The Ra−1/4 BL thickness, which determines Nu, coincides with the Kolmogorov and Batchelor scales of HC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Our numerical and theoretical results indicate that 3D HC is qualitatively and quantitatively similar to 2D HC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' At the same Ra, the 3D Nu exceeds the non-turbulent 2D Nu by only 20%, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=', there is very little 3D enhancement of heat transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Boundary conditions are more important than dimensionality: the non-turbulent 2D free-slip solutions have larger Nu than 3D no-slip solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The mechanical energy power integral of HC implies that mean square vorticity of 3D HC is nearly equal to that of 2D HC at the same Ra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Thus vorticity amplification by vortex stretching does not operate in 3D HC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Introduction Horizontal convection (HC) is convection driven by imposing non-uniform heating and cooling along a single horizontal surface, such as the top of a rectangular enclosure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' there is no flux of heat through the other boundaries (Hughes & Griffiths 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Oceanography is an important motivation for consideration of HC (Sandström 1908;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Rossby 1965;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Coman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Kuhlbrodt 2008), and in that connection the dependence of horizontal heat transport on the strength of buoyancy forcing applied at the ocean surface is a prime question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Buoyancy forcing is quantified via the horizontal-convective Rayleigh number, Ra, and horizontal heat transport by a suitably defined Nusselt number Nu (Rocha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' HC is an interesting counterpoint to Rayleigh-Bénard convection because HC buoyancy transport in the interior of the domain cannot be easily interpreted as the vertical motion of thermal plumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Instead, heat enters the fluid where the non-uniform heated surface is † Email address for correspondence: navid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='constantinou@anu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='au arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='03122v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='flu-dyn] 8 Jan 2023 2 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Constantinou, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Rocha, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Smith, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Young Figure 1: Snapshots of 2D free-slip HC at Ra = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4 × 107 in panels (a) and (b) and Ra = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4 × 109 in (c) and (d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Pr = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Contours are streamlines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' At the top surface −1 ⩽ 𝑏/𝑏★ ⩽ +1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' the narrow range of the buoyancy color scale in (b) and (d) makes the small interior buoyancy variations visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The sinusoidal surface buoyancy in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4) defines the buoyancy scale 𝑏★;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' ℎ is the layer depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' hotter than average and exits where it is colder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' This horizontal transport is associated with a prominent boundary layer (BL) adjacent to the non-uniform surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The oldest result for the high-Ra variation of horizontal-convective Nu is the scaling law of Rossby (1965) Nu ∼ Pr0 Ra1/5 , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='1) where Pr is the Prandtl number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (Parameters Ra, Pr, and Nu are defined in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=') Figure 1 shows two numerical solutions in the Rossby-scaling regime (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Rossby (1965) was motivated by experiments, mainly with Pr ∼ 103 and 104, and their reasoning leading to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='1) (reviewed in section 4) assumes visco-diffusive balances in the BL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (a) Ra=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4 × 107 vorticity, (azu - axw) Vh/b* 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 3 2 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5 0 1 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 (b) buoyancy, b/b* 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='64 %0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 x/h (c) Ra=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4 × 109 vorticity, (azu - axw) Vh/b* 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 3 2 1 %0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5 0 1 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 (d) buoyancy, b/b* 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='74 %0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 x/hHorizontal convection Nu scaling 3 At moderately large Ra, the exponent 1/5 is supported by both laboratory work and numerical solutions (Rossby 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Mullarney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Siggers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Wang & Huang 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Chiu-Webster et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Sheard & King 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Ilicak & Vallis 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Perhaps because of Rossby’s assumption that the boundary layer is visco-diffusive, the scaling (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='1) is sometimes characterized as applying to steady laminar HC, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=', the flow in figure 1(a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' But (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='1) has much wider applicability: some studies find that the one-fifth law (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='1) applies in unsteady and three-dimensional (3D) HC regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' For example, Gayen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (2014), with Pr = 5, find the exponent 1/5 in a steady 2D flow regime with Ra ⩽ 1010, and also in a turbulent 3D regime with Ra ⩾ 1012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' In the transition, 1010 ⩽ Ra ⩽ 1012, the constant multiplying Ra1/5 increases from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='37 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Gayen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (2014) make the important point that the exponent 1/5 also arises with a non-Rossby balance between inertia and buoyancy in the surface BL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' In section 3 we provide further examples of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='1) in unsteady three-dimensional regimes with Pr = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The tenacity of 1/5 as the primary exponent is striking, and rationalizes (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='1) in cases far removed from Rossby’s visco-diffusive scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Increasing Ra, with Pr fixed, results in qualitative changes in the structure of HC: compare figure 2 with figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' It is unlikely that a single Nu–Ra power-law can persist across the six decades of Ra spanned by these illustrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Tsai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (2020) have investigated 2D HC with Pr = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='14 and no-slip boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' At moderately large Ra, say 108 < Ra < 1010, Tsai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (2020) find Rossby’s scaling law (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' But in the range 1010 < Ra < 1014, and provided that the imposed surface buoyancy varies linearly with the horizontal coordinate 𝑥, Tsai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (2020) report the scaling Nu ∼ Ra1/4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='2) In section 3, using the sinusoidal surface buoyancy in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4), we also find the scaling (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='2) in a suite of 2D solutions at Pr = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' We show that in 2D, with both no-slip and free-slip boundary conditions, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='2) applies from about Ra = 1011 to at least 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4 × 1013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' In 3D, however, we achieve maximum Ra = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='2 × 1011 and there is no numerical evidence for the one-fourth scaling (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The vortices evident in figure 2 indicate that the interior buoyancy fluctuations are so small that the interior dynamics is essentially that of a vortex gas characteristic of freely evolving 2D turbulence (Benzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 1987, 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Carnevale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Dritschel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' McWilliams 1984, 1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Tsai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (2020) note that the exponent one-fourth in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='2) corresponds to one of the exponents proposed by Shishkina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (2016) in their phase diagram of the (Ra, Pr) parameter plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' We review the proposal of Shishkina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (2016) in sections 4 and 5, and conclude that this correspondence is accidental: diagnosis of the solutions in figure 2 does not confirm essential features of the one-fourth Ra–Nu scaling regimes in the phase diagram of Shishkina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' In section 6 we suggest instead that scaling (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='2) follows if the thinnest BL has a thickness scaling as the Kolmogorov length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' In section 7 we argue that our numerical results indicate that HC does not exhibit defining features of turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Formulation of the horizontal convection problem We consider a Boussinesq fluid with density 𝜌 = 𝜌0(1 − 𝑔−1𝑏), where 𝜌0 is a constant reference density and 𝑏 is the “buoyancy”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' If, for example, the fluid is stratified by temperature variations then 𝑏 = 𝑔𝛼(𝑇 − 𝑇0), where 𝑇0 is a reference temperature and 𝛼 is the thermal 4 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Constantinou, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Rocha, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Smith, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Young Figure 2: Snapshots of 2D free-slip solutions at Ra = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4 × 1011 in panels (a) and (b) and Ra = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4 × 1013 in (c) and (d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Pr = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Contours are streamlines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' This illustration uses true aspect ratio so that axisymmetric vortices look circular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The buoyancy color scale is narrow in order to reveal the small interior buoyancy variations, which are localized within the cores of axisymmetric vortices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Across the six-decade range of Ra in figures 1 and 2 vorticity scales with √︁ 𝑏★/ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' expansion coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The Boussinesq equations of motion are 𝒖𝑡 + 𝒖 · ∇𝒖 + ∇𝑝 = 𝑏 ˆ𝒛 + 𝜈∇2𝒖 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='1) 𝑏𝑡 + 𝒖 · ∇𝑏 = 𝜅∇2𝑏 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='2) ∇ · 𝒖 = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='3) The kinematic viscosity is 𝜈 and the thermal diffusivity is 𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (a) Ra=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4 × 1011 vorticity, (azu - axw) h/b* 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 3 2 L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 (b) buoyancy, b/b* 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='735 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='745 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='755 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 xih vorticity, (azu- axw) Vh/b* (c) Ra=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4 × 1013 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 (d) buoyancy,b/b 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='846 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='848 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='850 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='852 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 x/hHorizontal convection Nu scaling 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Horizontal convective boundary conditions and control parameters We suppose the fluid occupies a domain with depth ℎ, length ℓ𝑥, width ℓ𝑦;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' we assume periodicity in the 𝑥- and 𝑦-directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' At the bottom surface (𝑧 = 0) and top surface (𝑧 = ℎ) the primary boundary conditions on the velocity, 𝒖 = (𝑢, 𝑣, 𝑤), is that 𝑤 = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' the viscous boundary condition is either no slip (NS hereafter), 𝑢 = 𝑣 = 0, or free slip (FS hereafter), 𝑢𝑧 = 𝑣𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' At the bottom 𝑧 = 0 the buoyancy boundary condition is no flux, 𝜅𝑏𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' At the top, 𝑧 = ℎ, the boundary condition is 𝑏 = 𝑏s(𝑥), where the top surface buoyancy 𝑏s is a prescribed function of 𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' As a surface buoyancy field we use 𝑏s(𝑥) = 𝑏★ cos 𝑘𝑥 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4) where 𝑘 = 2𝜋/ℓ𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' As an idealization of conditions at the sea surface, FS is better than NS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' But the main reason for considering different viscous boundary conditions is to test scaling arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' We find only minor quantitative differences in the Nu–Ra scaling between the two boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Thus scaling arguments that rely on special properties of NS, such as analogies with the Blasius BL, should be reconsidered: in the numerical solutions described below the main features of the Nu–Ra scaling relation are independent of the viscous boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The problem is characterized by four non-dimensional parameters: the Rayleigh and Prandtl numbers Ra def= ℓ3 𝑥𝑏★ 𝜈𝜅 , and Pr def= 𝜈 𝜅 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5) and the aspect ratios 𝐴𝑥 def= ℓ𝑥/ℎ and 𝐴𝑦 def= ℓ𝑦/ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' With periodic boundary conditions in 𝑦 (no side walls), 2D HC is the special case 𝐴𝑦 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Mechanical energy dissipation We use an overbar to denote an average over 𝑥, 𝑦, and 𝑡, taken at any fixed 𝑧 and angle brackets ⟨ ⟩ to denote a total volume average over 𝑥, 𝑦, 𝑧, and 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Using this notation, we recall some results from Paparella & Young (2002) that are used below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Horizontally averaging the buoyancy equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='2) we obtain the zero-flux constraint 𝑤𝑏 − 𝜅 ¯𝑏𝑧 = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='6) Taking ⟨𝒖 · (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='1)⟩, we obtain the kinetic energy power integral 𝜀 = ⟨𝑤𝑏⟩ , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='7) where 𝜀 def= 𝜈⟨|∇𝒖|2⟩ is the rate of dissipation of kinetic energy and ⟨𝑤𝑏⟩ is rate of conversion between potential and kinetic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Vertically integrating (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='6) from 𝑧 = 0 to ℎ, and using the fact that 𝑏s = 0, we obtain another expression for ⟨𝑤𝑏⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' substituting this into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='7) we find 𝜀 = −𝜅 ¯𝑏(0) ℎ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='8) In (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='8), ¯𝑏(0) is the (𝑥, 𝑦, 𝑡)-average of the buoyancy at the bottom 𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The Nusselt number of horizontal convection Following Rocha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (2020b), we use the dissipation of buoyancy variance, 𝜒 def= 𝜅⟨|∇𝑏|2⟩ , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='9) 6 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Constantinou, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Rocha, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Smith, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Young to define the Nusselt number as Nu def= 𝜒�𝜒diff .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='10) Above, 𝜒diff def= 𝜅⟨|∇𝑏diff|2⟩ is the buoyancy dissipation of the diffusive solution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=', 𝜅∇2𝑏diff = 0 with 𝑏diff satisfying the same boundary conditions as 𝑏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Application of variational methods to HC (Siggers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Winters & Young 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Rocha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 2020a) results in bounds on 𝜒 taking the form Nu ≲ Ra1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The exponent 1/3 is safely larger than the exponents 1/5 and 1/4 reported in numerical studies of HC, including this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Rocha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (2020b) show that there is also a “surface Nusselt number” Nus def= 𝑏s 𝜅𝑏𝑧(ℎ) � 𝑏s 𝜅𝑏diff 𝑧(ℎ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='11) Above, 𝜅𝑏𝑧(ℎ) is the buoyancy flux through the top surface 𝑧 = ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' With sufficient temporal averaging Nu = Nus;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' in physical terms the interior entropy production, 𝜒, is balanced by entropy flux through the surface 𝑧 = ℎ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Nus is the non-dimensional entropy flux though the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' In numerical solutions described below, in which the temporal average is computed over a finite time interval, Nu ≈ Nus is a check on the estimated Nusselt number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' A numerical study of horizontal convection with Pr = 1 In this section we present the results of a numerical study directed at characterizing the variation of the Nusselt number Nu in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='10) as a function of Ra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Computations are performed using Dedalus, a spectral framework for solving partial differential equations (Burns et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 2020, www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='dedalus-project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' We use Fourier bases in the horizontal, periodic directions and a Chebyshev basis in the vertical;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' the equations are time stepped using a fourth-order implicit-explicit Runge–Kutta scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' We limit attention to Pr = 1 and the sinusoidal surface buoyancy forcing 𝑏s(𝑥) in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' We discuss both NS and FS boundary conditions and consider 2D solutions with aspect ratios ℓ𝑥/ℎ = 4 , ℓ𝑦/ℎ = 0 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='1) and 3D solutions with ℓ𝑥/ℎ = 4 , ℓ𝑦/ℎ = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='2) Thus we have four solution suites: 2DFS, 3DFS, 2DNS, and 3DNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The resulting estimates of Nusselt number are summarized in table 1 and figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The low-Ra regime Analysis of the low-Ra regime in appendix A shows that with ℓ𝑥/ℎ = 4 the first variation of the Nusselt number away from unity is Nu𝐹𝑆 = 1 + � Ra 21 567.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5 �2 + ord�Ra4� , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='3) and Nu𝑁 𝑆 = 1 + � Ra 87 789.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='8 �2 + ord�Ra4� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4) The low-Ra regime means that the Ra2 term in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='3) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4) is less than one, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=', that the convective buoyancy transport is a weak enhancement of the diffusive transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' For FS low Ra means that Ra is somewhat less than about 104 and for NS low Ra means that Ra is somewhat Horizontal convection Nu scaling 7 Free-slip Nu No-slip Nu Highest resolution Ra 2D ◦ 3D • 2D □ 3D ■ 𝑛𝑥, 𝑛𝑧 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='28e03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='00† 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='00† 128, 32 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='20e03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='02† 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='00† 128, 32 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='48e03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='04† 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='00† 128, 32 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='40e03 1.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='48∗ 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='28∗ 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='88∗ 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='61∗ 512, 128 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='40e10 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='08∗ 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='94∗ 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='50∗ 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='68∗ 512, 128 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='28e11 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='85∗ 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='74∗ 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='87∗ 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='67∗ 1024, 256 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='60e11 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='80∗ 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='02∗ 1024, 256 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='20e11 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='71∗ 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='91∗ 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='45∗ 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='01∗ 1024, 256 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='40e11 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='36∗ 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='23∗ 1024, 256 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='60e12 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='43∗ 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='65∗ 1024, 256 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='20e12 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='26∗ 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='97∗ 2048, 512 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='40e12 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='32∗ 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='86∗ 4096, 1024 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='60e13 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='44∗ 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='66∗ 4096, 1024 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='40e13 178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='55∗ 133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='16∗ 4096, 1024 Table 1: Nu–Ra data for HC DNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' All runs have 𝑃𝑟 = 1 and ℓ𝑥/ℎ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 3D runs have ℓ𝑦/ℎ = 1 and 𝑛𝑦 = 𝑛𝑧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The surface buoyancy is the sinusoid in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Unsteady solutions are indicated by a superscript ∗ on Nu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' strictly 2D solutions (no 𝑦-dependence and 𝑣 = 0) of 3D computations are marked by a superscript †.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The four NS runs with superscript ♯ are 3D but steady.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' less than about 4 × 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' These analytic results are compared with numerical solutions in the insert of figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' In table 1 all values of Nu are rounded to 2 decimal places, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=', the 2DFS solution at Ra = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4 has Nu − 1 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='8 × 10−8 and at Ra = 640, Nu − 1 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='8 × 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (These boring runs are not reported in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=') 8 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Constantinou, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Rocha, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Smith, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Young 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 10 10 10 11 10 12 10 13 10 14 Ra 1 2 5 10 20 40 100 150 250 Nu Ra 1/5 Ra 1/4 3D 2D free-slip no-slip 0 1 2 3 4 Ra × 10 −4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='1 Nu Figure 3: Variation of Nusselt number Nu with Rayleigh number Ra using the data from table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The inset compares the low-Ra numerical results with the low-Ra analytic results (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='3) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Note that some solid markers fall on top of open markers, indicating that the 3D solutions evolve to become 2D, or that the three dimensionality is weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The four vertical grey line segments mark Ra’s of the solutions in figures 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Nu–Ra scaling regimes: one-fifth and one-fourth Between Ra ∼ 104 and 105 we do not see a simple relation between Nu and Ra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' But once Ra is greater than about 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4 × 105 we find Rossby’s scaling, Nu ∼ 𝐾1/5Ra1/5 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5) in all four solution suites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' See section 4 for a discussion of Rossby’s scaling argument, and for more recent arguments that also predict the exponent 1/5 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5) (Gayen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Shishkina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Starting at around Ra ∼ 1011 in the 2DNS suite and 1012 in the 2DFS suite there is a transition from the one-fifth regime (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5) to the one-fourth regime, Nu ∼ 𝐾1/4 Ra1/4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='6) The one-fourth regime with NS has been documented previously by Tsai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' In figure 4 we show the data from figure 3, replotted using the compensated Nusselt number Ra−1/5Nu in panel (a) and Ra−1/4Nu in panel (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Table 2 summarizes the exponents determined by least-squares fitting the Ra–Nu data over selected ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Least-squares exponents are broadly in agreement with the scaling regimes determined by visual inspection of figure 4 and other compensated plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' We use least-squares because it is objective and reproducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Least squares also assesses the sensitivity of estimated exponents to the points at the beginning and end of a putative scaling range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Horizontal convection Nu scaling 9 10 4 10 5 10 6 10 7 10 8 10 9 10 10 10 11 10 12 10 13 10 14 Ra 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='35 Ra −1/5 Nu 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='215 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='3 3D 2D (a) 10 4 10 5 10 6 10 7 10 8 10 9 10 10 10 11 10 12 10 13 10 14 Ra 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='14 Ra −1/4 Nu 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='065 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='050 (b) free-slip no-slip Figure 4: Variation of “compensated Nusselt numbers” (a) Ra−1/5Nu and (b) Ra−1/4Nu with Rayleigh number Ra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The four vertical grey line segments mark Ra’s of the solutions in figures 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Discussion of the no-slip solutions We begin with easiest case, which is the 2DNS solution suite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The one-fifth scaling (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5) is found across the four-decade range in row 1 of table 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' this is the plateau at 𝐾1/5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='17 in figure 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Least-squares estimates of exponent and prefactor, 𝐾1/5, are robust to changes in the range, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=', row 2 of table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The 2DNS suite transitions to the one-fourth scaling (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='6) at around Ra = 1011 and forms the plateau at 𝐾1/4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='05 in figure 4(b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' see rows 4 through 6 of table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The 3DNS solution suite is more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' With moderate Ra (rows 7 through 9 of table 2) the 3DNS solutions coincide with their 2DNS partners and the scaling is again (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5) with 𝐾1/5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' At a critical Ra, roughly 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='2 × 107, the 3DNS suite becomes unstable to 3D perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' With further increases in Ra the 3DNS solutions have larger Nu than their 2DNS colleagues: the four steady 3DNS solutions in the interval 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='20 × 107 ⩽ Ra ⩽ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='20 × 108, are here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' One might hope that development of 3D flow, albeit steady 3D flow, signals the beginning of a new scaling regime, with an exponent greater than one-fifth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' But alas, this is the transition discovered by Gayen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (2014): at about Ra = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='60 × 109 the 3DNS solutions enter a new one-fifth regime: see rows 10 through 12 of table 2 and the 3DNS plateau at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='215 in figure 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' With maximum Ra = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='20 × 1011, we did not find convincing evidence of the one-fourth scaling (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='6) in the 3DNS solution suite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The NS computations of Gayen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (2014) used Pr = 5 and a piecewise constant surface 10 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Constantinou, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Rocha, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Smith, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Young row suite range points least squares Nu 1 2DNS 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='40 × 105 ⩽ Ra ⩽ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='40 × 109 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='177 Ra0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='198 2 2DNS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='60 × 106 ⩽ Ra ⩽ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='20 × 109 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='178 Ra0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='197 4 2DNS 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='40 × 1010 ⩽ Ra ⩽ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='40 × 1013 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='062 Ra0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='242 5 2DNS 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='40 × 1010 ⩽ Ra ⩽ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='60 × 1013 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='056 Ra0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='246 6 2DNS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='28 × 1011 ⩽ Ra ⩽ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='60 × 1013 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='055 Ra0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='246 7 3DNS 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='40 × 105 ⩽ Ra ⩽ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='20 × 107 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='173 Ra0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='199 8 3DNS 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='40 × 105 ⩽ Ra ⩽ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='40 × 106 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='167 Ra0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='202 9 3DNS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='20 × 106 ⩽ Ra ⩽ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='20 × 107 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='179 Ra0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='197 10 3DNS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='60 × 109 ⩽ Ra ⩽ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='20 × 1011 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='195 Ra0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='204 11 3DNS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='60 × 109 ⩽ Ra ⩽ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='60 × 1010 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='191 Ra0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='205 12 3DNS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='20 × 1010 ⩽ Ra ⩽ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='20 × 1011 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='180 Ra0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='208 20 2DFS 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='40 × 105 ⩽ Ra ⩽ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='60 × 1011 18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='253 Ra0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='199 21 2DFS 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='40 × 105 ⩽ Ra ⩽ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='20 × 108 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='312 Ra0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='186 22 2DFS 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='40 × 108 ⩽ Ra ⩽ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='60 × 1011 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='215 Ra0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='206 23 2DFS 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='40 × 1011 ⩽ Ra ⩽ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='40 × 1013 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='074 Ra0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='245 24 2DFS 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='40 × 1011 ⩽ Ra ⩽ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='20 × 1012 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='082 Ra0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='241 25 2DFS 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='40 × 1012 ⩽ Ra ⩽ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='40 × 1013 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='073 Ra0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='245 26 3DFS 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='40 × 105 ⩽ Ra ⩽ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='20 × 108 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='308 Ra0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='187 27 3DFS 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='40 × 1010 ⩽ Ra ⩽ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='20 × 1011 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='358 Ra0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='193 Table 2: Summary of least-squares fits to various scaling regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Where possible, we assess the sensitivity of the exponent by varying the range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' buoyancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Instead of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='10), Gayen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (2014) defined Nu based on the buoyancy flux through the destabilized portion of the non-uniformly heated surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Despite these differences, Gayen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (2014) document analogous 3DNS behavior within the one-fifth scaling regime: the constant 𝐾1/5 takes different values on either side of a smooth transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Discussion of the free-slip solutions Turning to the 2DFS solutions, the most generous identification of the one-fifth regime in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5) is the five-decade range in row 20 of table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' These 18 points correspond to the plateau 𝐾1/5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='25 in figure 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' We are concerned, however, by 9 points in the first half of this range, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=', row 21 of table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' These 9 points undulate around the 𝐾1/5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='25 plateau with an amplitude of about ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='01 and the least-squares exponent 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='186 is uncomfortably different from 1/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' These wayward points, at only moderately large Ra, correspond to solutions that are either steady, or weakly time dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Thus insufficient time-averaging in the estimate of Nu is not an issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Moreover in this range the 2D and 3D solutions coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' We conducted several tests by changing the spatial resolution and found no significant variation in the numerical estimate of Nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' If one views the exponent 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='186 as close to 1/5 then the wayward points are the lower end of a five-decade 2DFS scaling regime: the undulation is a pre-asymptotic imperfection in the first half of this regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' A more cautious interpretation is that the 2DFS Horizontal convection Nu scaling 11 one-fifth regime begins only at about Ra = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='40 × 108 and consists of the 9 points in row 22 of table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The 2DFS suite transitions to the one-fourth scaling (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='6) at around Ra = 1012 and forms the plateau at 𝐾1/4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='065 in figure 4(b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' see rows 23 through 25 of table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The 3DFS solutions depart significantly from their 2DFS colleagues first at about Ra = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='20 × 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' There is no evidence for a one-fourth scaling in the 3DFS suite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Instead, the three highest Ra 3DFS solutions (row 27 of table 2) indicate a second one-fifth regime e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='3 plateau in figure 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' We speculate that the 3DFS suite is recapitulating the phenomenology seen in the 3DNS suite: two one-fifth scaling regimes separated by a smooth transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' We caution, however, that this speculation is based on three solutions in row 27 spanning less than one decade variation in Ra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Review of Nu ∼ Ra1/5 scaling arguments Rossby (1965) proposed a visco-diffusive balance in the boundary layer adjacent to the non-uniformly heated surface and so arrived at the one-fifth scaling in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Rossby identified the length scale 𝛿1/5 def= Ra−1/5ℎ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='1) as the thickness of the surface BL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' In the following discussion we also need the length 𝛿1/4 def= Ra−1/4ℎ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='2) At Ra = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4 × 1013, the ratio of these two BL scales is 𝛿1/5/𝛿1/4 ≈ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Central to Rossby’s argument is the assumption that BL buoyancy forces are balanced by viscosity and that BL inertia is subdominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' At moderately large Ra, the exponent 1/5 has been supported by subsequent laboratory work and by numerical studies (Rossby 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Mullarney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Siggers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Wang & Huang 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Sheard & King 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Ilicak & Vallis 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' With Pr = ∞, and with both no-slip and free-slip boundary conditions, Chiu-Webster et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (2008) provide a compelling confirmation that Nu ∼ Ra1/5 as Ra → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' We emphasize that the scaling (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5), and the associated BL thickness 𝛿1/5, does not, however, require that Pr ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' For example, the experiments of Mullarney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (2004) and Wang & Huang (2005) present evidence of Rossby scaling in unsteady flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The 2D solutions shown in figure 1 – including the unsteady solution in panels (c) and (d) – are well within the 𝐾1/5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='25 regime of figure 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Our unsteady 2DFS solutions exhibit the one-fifth scaling (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5) over at least three decades of Ra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' To further complicate the situation, 2D solutions in the one-fourth regime (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='6) still express the BL scale 𝛿1/5: figure 5 shows a progressively expanded view of the structure of HC near the upper surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' This 2DNS solution is in the non-Rossby scaling regime (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Nonetheless, panel (d) of figure 5 indicates that 𝛿1/5 is a useful BL length scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' We conclude that at sufficiently high Ra there is a double BL: there is a thin-𝛿1/4 layer nestled with a thicker 𝛿1/5-layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' We discuss this double BL further in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (The 2DNS solution in figure 5(a) exhibits the vortex-gas phenomenology noted previously in the 2DFS solutions shown in figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' At high Ra, no matter the viscous boundary condition, the interior of 2D HC is characterized as a vortex gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=') As an alternative to Rossby scaling, Shishkina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (2016) proposed a set of scaling arguments summarized in a phase diagram of the (Ra, Pr)-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' This diagram shows high-Pr regions denoted I∗ ℓ, I∞ and III∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' these three high-Pr regions have Nu ∼ Ra𝜉 with exponent 𝜉 = 1/4 in I∗ ℓ and III∞ and 1/6 in I∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' This tripartite proposal cannot be reconciled with the high-Pr results of Rossby (1965) and Chiu-Webster et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Instead, in the phase diagram of Shishkina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (2016), the one-fifth scaling (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5) is found only in the low-Pr region Iℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' We discuss the Shishkina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (2016) Iℓ regime in more detail below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 12 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Constantinou, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Rocha, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Smith, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Young Figure 5: (a) A snapshot of vorticity in the 2DNS solution at Ra = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4 × 1013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' this solution is in the non-Rossby Nu ∼ Ra1/4 scaling regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Panels (b)-(d) depict the boundary-layer structure by progressively zooming in to the top surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Green rectangles in panels (a), (b) and (c) indicate the regions in panels (b), (c) and (d) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' In panels (b), (c) and (d), both axes are measured in units of 𝛿1/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The dashed grey line in panel (d) indicates the distance 2𝛿1/4 below the top surface 𝑧 = ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The contours in all panels are streamlines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The spanwise average To identify the various processes in the BL, we begin taking a spanwise 𝑦-average of the equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Denote this spanwise average with a hat so that 𝑏(𝑥, 𝑦, 𝑧, 𝑡) = 1 ℓ𝑦 ∫ ℓ𝑦 0 𝑏(𝑥, 𝑦, 𝑧, 𝑡) d𝑦 ������������������������������������������������ def= ˆ𝑏(𝑥,𝑧,𝑡) + 𝑏′(𝑥, 𝑦, 𝑧, 𝑡) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='3) (a) Ra=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4 × 1013 vorticity, (azu - axw)Vh/b* 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 x/h (b) 0 20 z)/h 10 40 0 80 10 Ra 20 700 800 900 1000 1100 Ra1/5x/h (c) 0 20 z)/h 10 10 10 20 Ra 20 1000 1020 1040 1060 1080 Ra1/5x/h (d) 0 30 Ra 1/5 (h - z) / h 15 2 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4 15 30 1075 1080 1085 1090 Ra 1/5x/ hHorizontal convection Nu scaling 13 Above, 𝑏′(𝑥, 𝑦, 𝑧, 𝑡) is the three-dimensional departure from the spanwise average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Taking the spanwise average of the 3D continuity equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='3), we obtain a 2D “overturning stream function” 𝜓(𝑥, 𝑧, 𝑡), such that ( ˆ𝑢 , ˆ𝑤) = (−𝜓𝑧 , 𝜓𝑥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' With this notation the 3D velocity is written as (𝑢, 𝑣, 𝑤) = (−𝜓𝑧 , 0, 𝜓𝑥) + (𝑢′, 𝑣′, 𝑤′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4) The spanwise-average of the buoyancy equation is ˆ𝑏𝑡 + 𝜓𝑥 ˆ𝑏𝑧 − 𝜓𝑧 ˆ𝑏𝑥 + 𝜕𝑥 � 𝑢′𝑏′ + 𝜕𝑧 � 𝑤′𝑏′ = 𝜅∇2 ˆ𝑏 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5) and the spanwise average of the spanwise vorticity equation is 𝜁𝑡 + 𝜓𝑥𝜁𝑧 − 𝜓𝑧𝜁𝑥 ������������������������������������ inertia + ˆ𝑏𝑥 ���� buoyancy torque + (𝜕2 𝑧 − 𝜕2 𝑥) � 𝑢′𝑤′ + 𝜕𝑥𝜕𝑧 �� 𝑢′2 − � 𝑤′2� �������������������������������������������������������������������������������� Reynolds stress torque = 𝜈∇2𝜁 ���� viscosity , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='6) where 𝜁 def= −∇2𝜓 is the spanwise-averaged spanwise vorticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The power integral (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='8) becomes 𝜀 = 𝜈 � 𝜁2� + 𝜈⟨|∇𝒖′|2⟩ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='7) = −𝜅 ¯𝑏(0)/ℎ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='8) The 2D equations of motion are recovered by suppressing the spanwise averages of quadratic fluctuations in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='6), and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' A review of Nu ∼ Ra1/5 scaling arguments Following Shishkina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (2016), we assume that there is a BL with thickness 𝛿𝑏 in buoyancy and 𝛿𝑢 in momentum and vorticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The Reynolds number is Re def= 𝑈ℎ 𝜈 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='9) where 𝑈 def= √︁ ⟨|𝒖|2⟩ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='10) is the typical flow velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' We use the depth ℎ as representative of the domain dimensions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=', (ℓ𝑥, ℓ𝑦) ∼ ℎ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' these three length scales are roughly comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Scale analysis of the surface Nusselt number in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='11), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=', ˆ𝑏𝑧(ℎ) ∼ 𝑏★/𝛿𝑏, shows that Nu ∼ ℎ 𝛿𝑏 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='11) One reaches the same conclusion via scale analysis of the 𝜒-based Nusselt number in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='9): although |∇𝑏|2 ∼ 𝑏2 ★/𝛿2 𝑏, the 𝜒-BL occupies only a fraction 𝛿𝑏/ℎ of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Thus (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='11) follows because of the volume average ⟨ ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Now apply scale analysis to the buoyancy equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Using results such as 𝜓𝑧 ˆ𝑏𝑥 ∼ 𝜓𝑥 ˆ𝑏𝑧 ∼ 𝑈𝑏★/ℎ and 𝜅∇2 ˆ𝑏 ∼ 𝜅𝑏★/𝛿2 𝑏 one has 𝑈 ∼ 𝜅ℎ 𝛿2 𝑏 , or in non-dimensional form Nu ∼ (RePr)1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='12) To estimate the viscous dissipation 𝜀 on the right of the power integral (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='7), one assumes that an order-one fraction of 𝜀 is concentrated in the BL, and this BL occupies a fraction 𝛿𝑢/ℎ of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' One can either neglect 𝜈⟨|∇𝒖′|2⟩, or assume that both terms on the right 14 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Constantinou, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Rocha, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Smith, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Young of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='7) scale in the same way, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=', as 𝜈𝜁2 ∼ 𝜈(𝑈/𝛿𝑢)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' In either case 𝜀 ∼ 𝜈𝑈2 𝛿𝑢ℎ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='13) Scale analysis of the right of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='8) assumes that the bottom buoyancy, ¯𝑏(0), is an order- one fraction of the minimum buoyancy, −𝑏★, on the top surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (The stronger result that ¯𝑏(0) → −𝑏★ as Ra → ∞ is likely true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=') Thus 𝜀 ∼ 𝜅𝑏★ ℎ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='14) Combining (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='13) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='14) 𝑈2 ∼ 𝜅 𝜈 𝛿𝑢𝑏★ , or in non-dimensional form (RePr)2 ∼ 𝛿𝑢 ℎ Ra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='15) Eliminating RePr between (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='12) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='15), and then using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='11) to get rid of ℎ, one finds Nu5 ∼ 𝛿𝑢 𝛿𝑏 Ra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='16) The final step to obtain the dependence of Nu on Ra and Pr is to express the ratio 𝛿𝑢/𝛿𝑏 on the right of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='16) in terms Ra and Pr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' There are three arguments in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The scaling of Rossby (1965).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Taking 𝛿𝑢 = 𝛿𝑏 one obtains from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='16) Nu ∼ Pr0 Ra1/5 and Re ∼ Pr−1Ra2/5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='17) Rossby’s 1965 argument did not employ the power integral and its consequence (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Instead, Rossby assumes ab initio that 𝛿𝑏 = 𝛿𝑢 and balances buoyancy torque with viscosity in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='6), leading to 𝑈 ∼ 𝑏★𝛿3 𝑢/ℓ𝜈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Combining these results with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='11) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='12) one again finds (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Rossby’s balance between buoyancy torque and viscosity applies to both FS and NS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' In the FS case, the velocity BL results from the vorticity source ˆ𝑏𝑥 in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='6): this rationalization of Rossby’s assumption that 𝛿𝑢 = 𝛿𝑏 also applies to NS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The scaling of Gayen, Griffith & Hughes (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' In the vorticity equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='6), balance buoyancy torque with either inertia or Reynolds stress torques, leading to 𝑈2 ∼ 𝛿𝑢𝑏★, and follow Rossby by assuming that 𝛿𝑏 = 𝛿𝑢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Combining these results with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='11) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='12) one finds Nu ∼ Pr1/5Ra1/5 and Re ∼ Pr−3/5Ra2/5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='18) This argument does not use the power integral and it is not consistent with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='16) unless Pr is order unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Using the scaling assumptions above to estimate 𝜀 in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='7) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='8) we find 𝜈 � 𝜁2� ∼ Ra (𝜅𝜈2/ℎ4) and − 𝜅 ¯𝑏(0)/ℎ ∼ Ra (𝜅2𝜈/ℎ4) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='19) The two terms in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='19) differ by a factor of Pr: this is a problem if Pr is either very large or very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' But with Pr of order unity – and here we consider Pr = 1 – there is no problem closing the mechanical energy budget and thus the scaling of Gayen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (2014) is a valid alternative to that of Rossby.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The scaling of Shishkina, Grossman & Lohse (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' In the vorticity equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='6), balance inertia with viscosity, leading to 𝑈 ∼ 𝜈ℎ/𝛿2 𝑢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Eliminating 𝑈 with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='12) one finds 𝛿𝑢 = Pr1/2 𝛿𝑏, and substituting into the power-integral (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='16) Nu ∼ Pr1/10 Ra1/5 , and Re ∼ Pr−4/5Ra2/5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='20) A distinctive feature of this scaling argument is that buoyancy torque ˆ𝑏𝑥 in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='6) does not appear in the leading-order BL vorticity balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' This is justified by requiring that Pr ≪ 1, so Horizontal convection Nu scaling 15 that 𝛿𝑢 = Pr1/2 𝛿𝑏 ≪ 𝛿𝑏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' In other words, this visco-inertial BL is so thin that both viscosity and inertia are much greater than the buoyancy torque ˆ𝑏𝑥 ∼ 𝑏★/ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Despite different physical assumptions, the three arguments summarized above are in agreement that Nu ∼ Ra1/5: all differences lie in the dependence of Nu on Pr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' In this respect scaling (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='20) – corresponding to region Iℓ in the phase diagram of Shishkina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (2016) – needs clarification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Region Iℓ, with Pr ≪ 1, is referred to by Shishkina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (2016) as “Rossby scaling”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Although the exponent one-fifth is the same as that of Rossby, the dependence on Pr in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='20) differs from that of Rossby in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Moreover Rossby was concerned with 𝑃𝑟 ≫ 1, while scaling (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='20) ostensibly applies provided that Pr ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Thus referring to Iℓ as Rossby scaling is a misnomer: the phase diagram does not contain a region corresponding to the original Rossby scaling in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Nested boundary layers and the Nu ∼ Ra1/4 scaling Tsai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (2020) investigate 2DNS HC with Pr = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' With Ra > 1010, and provided that the imposed surface buoyancy varies linearly with the horizontal coordinate 𝑥, Tsai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (2020) report the one-fourth scaling (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='6) extending over four decades of Ra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Here, using the sinusoidal surface buoyancy (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4), we also find the one-fourth scaling (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='6) in the 2DFS and 2DNS solution suites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Tsai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (2020) speculate that their one-fourth scaling might correspond to a regime proposed by Shishkina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (2016) in their phase diagram of the (Ra, Pr) parameter plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' In this scheme the (Ra, Pr)-plane is partitioned into seven regions and the exponent 1/4 is located in regions III∞, IV𝑢, and I∗ ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' But we now show that defining features of III∞, IV𝑢, and I∗ ℓ do not agree with the numerical solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' We conclude that the exponent one-fourth found here, and likely in the regime identified by Tsai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (2020), is not in agreement with any region of the phase diagram of Shishkina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Partitioning of buoyancy dissipation 𝜒 between BL and interior A main characteristic distinguishing the various regimes by Shishkina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (2016) is the partitioning of kinetic energy dissipation, 𝜀, and buoyancy variance dissipation, 𝜒, between the BL and the interior of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' To quantify the partitioning of 𝜒 we introduce the function 𝐹𝜒(𝑧) def= 𝜅 ℎ ∫ 𝑧 0 |∇𝑏|2 d𝑧′�𝜒 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='1) where the overbar denotes an (𝑥, 𝑦, 𝑡)-average;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' see figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 𝐹𝜒(𝑧) increases monotonically from 0 to 1 with 𝑧/ℎ and indicates the fraction of buoyancy-variance dissipation below the level 𝑧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' In the 2DFS case, figure 6(a) shows that 𝜒 is increasingly localized within a BL as Ra → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Examination of 𝐹𝜒(𝑧) for the 2DNS solutions indicates no significant differences from the 2DFS results in figure 6(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' With both FS and NS, 𝜒 is increasingly concentrated within a BL as Ra → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' A main characteristic of regions III∞ and IV𝑢 in the phase diagram of Shishkina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (2016) is that 𝜒 is dominantly in the interior of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Thus figure 6(a) disqualifies regions III∞ and IV𝑢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The remaining possibility with exponent 1/4 is the Pr ≫ 1 region I∗ ℓ, characterized by a momentum BL that is much thicker than the buoyancy diffusion BL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' But region I∗ ℓ is located at moderate values of Ra in the phase diagram so that 1/4 is the first exponent encountered if Pr is fixed and Ra is increased from small values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' In our 2D solutions, however, we first find the 1/5 scaling (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5), which is replaced at higher Ra by 1/4 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' This is also the case in the study of Tsai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (2020): first 1/5 and then, at higher Ra, 1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' We conclude that the exponent 1/4 in the 2D solution suites is not related to regions 16 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Constantinou, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Rocha, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Smith, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Young 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 free-slip Fχ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='30 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='25 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='20 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='00 (z − h)/h (a) Ra 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4 × 106 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4 × 107 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4 × 108 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4 × 109 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4 × 1010 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4 × 1011 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4 × 1012 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4 × 1013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 free-slip Fχ −6 −5 −4 −3 −2 −1 0 Ra1/5 (z − h)/h (b) 107 108 109 1010 1011 1012 1013 1014 Ra 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='7 free-slip Ra1/5 δχ/h (c) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4 no-slip Ra1/5 δχ/h 2D free-slip 2D no-slip Figure 6: (a) The function 𝐹𝜒(𝑧) defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='1) for the 2DFS solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The vertical axis is distance from the top surface at 𝑧 = ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' A BL thickness, 𝛿𝜒, is defined as the distance from the top at which 𝐹𝜒(𝑧) = 1/2 (dashed vertical line in panel (a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (b) Same as (a) but with the vertical axis rescaled with Ra1/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Horizontal grey dashed lines indicate the distance 2𝛿1/4 below the top surface 𝑧 = ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (c) The compensated BL thickness, Ra1/5𝛿𝜒/ℎ, as a function of Ra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The four vertical grey line segments mark Ra’s of the solutions in figures 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' III∞, IV𝑢 and I∗ ℓ of the phase diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' In section 6 we seek an alternative explanation for the one-fourth scaling regimes in figure 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' To extract more information from 𝐹𝜒(𝑧), we scale the 𝑧-axis with 𝛿1/5 and re-plot the results from figure 6(a) in figure 6(b): the curves now fall largely on top of each other, indicating that the function 𝐹𝜒(𝑧) expresses the BL thickness 𝛿1/5, even if the Nu ∼ Ra1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' To quantify this, we define a BL thickness, 𝛿𝜒, by determining the level at which 𝐹𝜒(𝑧) = 1/2: see the dashed vertical line in figure 6(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The compensated plot in figure 6(c) then shows that 𝛿𝜒 ≈ 𝐾𝜒𝛿1/5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='2) The constant 𝐾𝜒 in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='2) is about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5 for the FS solutions and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='8 for the NS solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The Ra−1/5 scaling in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='2) applies in both the one-fifth regime (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5) and the one-fourth regime (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Yet the reasoning in section 4, leading to Nu ∼ ℎ 𝛿𝑏 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='3) underpins all scaling arguments and seems inescapable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' It must be that in the one-fourth regime (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='6), the buoyancy BL has a double-layer structure: there is a thin BL, with thickness 𝛿𝑏 = 𝛿1/4, embedded within the thicker 𝛿𝜒-BL in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (The scatter of 𝐾𝜒 in figure 6(c) might be considered uncomfortably large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Note, however, Horizontal convection Nu scaling 17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='00 b1 −6 −4 −2 0 Ra1/5 (z − h)/h (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='50 Ra1/5 b1z h/b * (b) Ra 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4 × 109 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='3 × 1011 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4 × 1011 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='6 × 1012 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='2 × 1012 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4 × 1012 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='6 × 1013 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4 × 1013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 Ra2/5 b1zz h2/b * (c) 2D free-slip Figure 7: The structures of (a) 𝑏1(𝑧), (b) 𝑏′ 1(𝑧), and (c) 𝑏′′ 1 (𝑧) for 2DFS solutions at various Ra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Note that 𝑏1(𝑧) was obtained here just from the final snapshot of each simulation without any time-averaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The vertical axis is distance from the surface 𝑧 = ℎ measured in units of 𝛿1/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Horizontal grey dashed lines indicate the distance 2𝛿1/4 below the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' that Ra is varied by seven decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' This large range encompasses the transition from steady to strongly time-dependent 2D flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' At Ra = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4 × 1013, Ra−1/4 is smaller than Ra−1/5 by a factor of 5, which is much greater than the ±20% scatter in figure 6(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=') 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The 𝛿1/4-boundary layer To identify the 𝛿1/4-BL in our solutions, and show consistency with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='3) in the one-fourth regime, we notice that with the sinusoidal surface buoyancy in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4), the surface Nusselt number (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='11) is Nus = 𝑏′ 1(ℎ) � 𝑏′ diff1(ℎ) , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4) where above the prime denotes a 𝑧-derivative and 𝑏1(𝑧) def= 2 cos 𝑘𝑥 𝑏(𝑥, 𝑦, 𝑧, 𝑡) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5) We define the thickness, 𝛿s, of this surface BL in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4) by 𝛿s def= 𝑏★/𝑏′ 1(ℎ) , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='6) where 𝑏★ is the amplitude of the sinusoidal surface buoyancy in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4) The numerator in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='6) is appropriate since 𝑏★ = 𝑏1(ℎ) = 𝑏diff1(ℎ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Figure 7 shows 𝑏1(𝑧), and the first two derivatives of this averaged field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The overline in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5) indicates both a horizontal and temporal average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Unfortunately we did not collect a time series of 𝑏1 and thus figure 7 is based on the horizontal average of single snapshots of the buoyancy field at the final time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The inner BL, with thickness 𝛿1/4, is not visible in 𝑏1(𝑧) in figure 7(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' But the higher derivatives of 𝑏1(𝑧) in the panels (b) and (c) reveal the scale 𝛿1/4 in the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' In particular, the one-fourth scaling (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='6) results from the increase in 𝑏′ 1(ℎ), evident in figure 7(b) as Ra increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The maximum of 𝑏′′ 1 (𝑧) in figure 7(c) appears only in the one-fourth regime (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Figure 8(a) shows 𝛿s, diagnosed from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='6), and compensated by Ra−1/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' In the one-fifth scaling regime (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5), with Ra less than about 1011, Ra−1/5𝛿s/ℎ varies between about 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='6 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' In these cases both 𝛿s and 𝛿𝜒 are ∼ 𝛿1/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Figure 8(b) shows that at the four or five highest values of Ra, Ra−1/4𝛿s/ℎ varies between about 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5 and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' We conclude that in these cases 18 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Constantinou, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Rocha, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Smith, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Young 107 108 109 1010 1011 1012 1013 1014 Ra 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='8 Ra1/5 δs (a) 107 108 109 1010 1011 1012 1013 1014 Ra 6 7 8 9 10 11 Ra1/4 δs (b) Figure 8: The compensated surface BL thickness, 𝛿s in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='6), for the 2DFS suite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' In (a) 𝛿s is compensated by Ra−1/5 and in (b) by Ra−1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The transition between the one-fifth scaling (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5) and the one-fourth scaling (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='6) is at about 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4 × 1011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Vertical grey line segments mark the Ra values corresponding to the solutions in figures 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Vertical grey line segments mark the Ra values corresponding to the solutions in figures 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 𝛿s ∼ 𝛿1/4, but 𝛿𝜒 ∼ 𝛿1/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Via (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='3), the Nusselt number is determined by the 𝛿1/4 inner BL, resulting in the one-fourth regime (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' A scaling argument for Nu ∼ Ra1/4 In this section we present a scaling argument applicable to the one-fourth regime of horizontal convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Although our numerical solutions revealed the one-fourth scaling regime only in the 2D cases, we still hold hope that the one-fourth regime might also emerge in 3D at sufficiently high Ra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' With the 3D case in mind, we propose an overarching explanation for the one-fourth scaling – independent of boundary conditions, dimensionality and 2D vortex-gas phenomenology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The one-fourth regime requires an inner buoyancy BL with thickness 𝛿𝑏 ∼ 𝛿1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' In discussing this inner BL it is helpful to keep figure 5(d) in mind: the 𝛿1/4-BL is identified by the dashed grey line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Think of this inner BL as a laminar sub-layer, stirred by the outer flow in the much thicker 𝛿1/5-BL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The overarching explanation alluded to above is that the thickness of the laminar sub-layer is related to the Kolmogorov and Batchelor length scales 𝜂K = � 𝜈3 𝜀 �1/4 and 𝜂B = � 𝜅2𝜈 𝜀 �1/4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='1) These length scales are identified as the smallest scales of fluctuations in momentum and buoyancy that can survive before the damping by viscosity 𝜈 and diffusion 𝜅 is overwhelming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' By analogy, the HC laminar sub-layer with thickness 𝛿𝑏, is the thinnest BL that can survive in a horizontal-convective flow that is supplied with kinetic energy at a rate 𝜀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Horizontal convection Nu scaling 19 (In the arguments of Kolmogorov and Batchelor the viscous dissipation rate 𝜀 is also the energy cascade rate in a 3D inertial range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' This interpretation of 𝜀 cannot apply to 2D HC: there is no vortex stretching in a 2D flow and therefore no forward cascade of energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' We argue instead that the laminar sub-layer thickness is determined by 𝜀 as the most basic measure of forcing strength and by the molecular parameters 𝜈 and 𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Thus 𝛿𝑏 ∼ (𝜈 𝑝𝜅𝑞/𝜀)1/4, with 𝑝 + 𝑞 = 3, is dimensionally acceptable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 𝜂K and 𝜂B are the most prominent members of this family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Saying more would would require varying Pr which is beyond our scope here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=') Following the scaling arguments reviewed section 4, we assume that ¯𝑏(0) ≈ −𝑏★.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Then, once again, the energy power integral (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='8) implies that 𝜀 ∼ 𝜅𝑏★ ℎ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='2) With 𝜀 in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='2), 𝜂K and 𝜂B in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='1) can be written as 𝜂K ℎ = Pr1/2Ra−1/4 and 𝜂B ℎ = Pr0Ra−1/4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='3) Thus if the laminar sub-layer has thickness 𝛿𝑏 ∼ 𝜂K, or perhaps 𝛿𝑏 ∼ 𝜂B, then Nu ∼ Ra1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Discussion: is 3D horizontal convection turbulent?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Paparella & Young (2002) showed that as a consequence of the mechanical energy power integral (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='2), HC does not satisfy the zeroth law of turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Paparella & Young also noted that the zeroth law is not a universally accepted as part of a definition of turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' For example, Scotti & White (2011) argued that the zeroth law is irrelevant because “HC can transport very large quantities of heat and sustain large amounts of diapycnal mixing with a surprisingly small amount of dissipation”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Similar sentiments, reinforced by arguments involving exchange between available potential energy and kinetic energy, are expressed by Gayen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (2013, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (Note that even the very form of zeroth law of turbulence appropriate to HC is also controversial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Paparella & Young (2002) state that 𝜀 should be non-zero as (𝜈, 𝜅) → (0, 0) with Pr = 𝜈/𝜅 fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' In their “clarification” of the zeroth law, Shishkina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (2016) prefer a different limit in which 𝜈 → 0 with 𝜅 fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' In this case, by inspection of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='8), the zeroth law applies in the high-inertial limit in which Ra → ∞ and simultaneously Pr → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=') A common response to the question “What is turbulence?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (for example given by Google or by chatbot ChatGPT), is that “Turbulence is a fluctuation or disturbance in a fluid (such as air or water) that is characterized by chaotic and irregular movements.” However, there is more to turbulence than this, else the 2D solutions in figure 2 are turbulent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' In addition to “chaotic and irregular movements” a transition to 3D flow is viewed as essential (Gayen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Tsai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 2016, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' It is only in 3D that “the high average vorticity which is known to exist in turbulent motion" can be produced by “extension of vortex filaments in an eddying fluid” (Taylor 1938).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' In this section we use the four solution suites from section 3 to revisit the question of whether HC is “turbulent”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' We show that the 3D solutions, with maximum Ra = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='2 × 1011, cannot be considered turbulent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' We also present theoretical arguments indicating that 3D HC cannot become turbulent at any Ra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' We set aside the zeroth law of turbulence and focus instead on accepted characteristics of 3D hydrodynamic turbulence: (a) chaotic, disordered and irregular fluid motions, irreproducible in detail;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (b) greatly enhanced transport of momentum and heat;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (c) strong vorticity amplification by strain mediated 3D vortex stretching;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (d) a direct cascade of energy in an inertial range;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 20 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Constantinou, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Rocha, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Smith, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Young (e) termination of the inertial range at the viscous length scale 𝜂K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' According to Stewart in the educational film “Turbulence” (NCFMF 1968, 1972) these phenomena “give a defining syndrome, or set of symptoms, for turbulence”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (Stewart listed (a), (b), and (c);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' here we have added (d) and (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=') Chaotic and disordered flow described by symptom (a) applies to both 2D and 3D time- dependent solutions at moderately high Ra, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=', beyond 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Just based on symptom (a) one would conclude that 2D HC is turbulent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' But because of symptoms (c) and (d), a 2D flow – no matter how erratically time dependent the velocity – cannot be turbulent (Taylor & Green 1937).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The 2D and 3D HC solutions coincide up to a critical Ra at which 3D instabilities first appear (Gayen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Tsai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Passaggia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' This bifurcation to 3D flow is usually viewed as the first step in a transition to turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' But the unstable non-turbulent 2D solution is always present and serves as a “comparison flow” for putatively turbulent 3D solutions at the same Ra, and with the same boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The main thrust of the argument that follows is that 3D HC is neither qualitatively nor quantitatively different from the non-turbulent 2D comparison flow: the onset of three-dimensionality in HC does not inflame turbulence symptoms (b) through (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' A question of interest, especially in oceanographic context, is how much might HC contribute to total heat flux?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Thus, how much can Nu be enhanced in turbulent HC has been a motivation for HC research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Symptom (b), emphasized by Scotti & White (2011) and other authors, demands that HC turbulence is accompanied by a large increase in the horizontal transport of heat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' But in figure 3 and table 1, the 3D Nu is only 20% greater than that of the non-turbulent 2D comparison flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Symptom (b) demands much more than a 20% enhancement in Nu between a turbulent 3D flow and a non-turbulent 2D comparison flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Moreover figure 3 and table 1 show that the viscous boundary condition (FS versus NS) has a larger quantitative effect on Nu than does dimensionality: even after the transition to unsteady 3D flow (𝑅𝑎 ⩾ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4 × 108), the 3DNS solutions transport less buoyancy than the non-turbulent 2DFS solutions at the same Ra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' If 3DNS transitions to turbulence at some Ra > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='2 × 1011 then, no matter the boundary condition, the turbulent 3D flow should transport more heat than the non-turbulent 2D flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' There is no indication of this hypothetical crossover between 3DNS and 2DFS in figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Table 3 summarizes gross measures of the departures from the 2D spanwise-averaged circulation defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' For both 3DNS and 3DFS, about two-thirds of the kinetic energy is in the spanwise averaged flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' In the third column the component of buoyancy gradient in the spanwise direction (𝑏𝑦) contributes less than 2% to the buoyancy dissipation 𝜒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The only statistic that is dominated by departures from the spanwise average is mechanical energy dissipation, 𝜀, in the fourth column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Thus table 3, and particularly the third column, supports the view that the 3D 𝑁𝑢, even at Ra = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='20 × 1011, is largely determined by the 2D spanwise averaged circulation, rather than by robust 3D turbulence characterized by (b) through (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Figure 9(a) shows that there is no inertial cascade in the interior, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=', (c) through (e) do not apply to this 3DFS solution (nor to the 3DNS solution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' An inertial cascade is characterized by a kinetic energy spectrum ∼ 𝜀2/3𝑘−5/3, or a vorticity spectrum ∼ 𝜀2/3𝑘+1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The ultra-violet vorticity divergence is cut-off at a wavenumber of order 𝜂−1 K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' But in contradiction to (e) the snapshot in figure 9(a) shows that vorticity is concentrated on length scales very much larger than 𝜂K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' We do not have an estimate of the length scale of the vorticity fluctuations in figure 9 (nor for the core radius of the 2D vortices in figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' These vorticity length scales are rather less than the domain scales (ℓ𝑥, ℓ𝑦, ℎ), but very much greater than 𝜂K in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' There is, however, a simple result for the magnitude of the vorticity 𝝎 = ∇×𝒖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' With a well known identity, the kinetic energy dissipation can be written as 𝜀 = 𝜈⟨|𝝎|2⟩ and the mechanical Horizontal convection Nu scaling 21 ⟨| ˆ𝒖|2⟩/⟨|𝒖|2⟩ ⟨𝑣2⟩/⟨|𝒖|2⟩ 𝜅⟨𝑏2𝑦⟩/𝜒 𝜈⟨𝜁2⟩/𝜀 3DFS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='662 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='112 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='187 3DNS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='690 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='092 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='019 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='342 Table 3: Statistics for the 3D solutions at Ra = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='20 × 1011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The ratios above were computed from a single snapshot at the final time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=', without the benefit of time averaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' These ratios decrease monotonically to zero as Ra is lowered to the critical value for the onset of 3D motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' In the first column, ˆ𝒖 = (−𝜓𝑧, 0, 𝜓𝑥) is the spanwise averaged velocity in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4) and in the final column 𝜁 = −∇2𝜓 is the vorticity of the spanwise-averaged flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' power integral (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='8) rewritten as ⟨|𝝎|2⟩ 𝑏★/ℎ = −𝜅 𝜈 ¯𝑏(0) 𝑏★ , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='1) ⩽ 𝜅 𝜈 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='2) In passing from the exact equality (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='1) to the rigorous inequality in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='2) we have used the extremum principle for buoyancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' What is the significance of the bound (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='2)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' It is remarkable that (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='2) is independent of aspect ratio ℓ𝑦/ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Thus at high Ra, with ¯𝑏(0) close to −𝑏★, 3D HC must have almost the same ⟨|𝝎|2⟩ as that of the non-turbulent 2D comparison flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Near equality of ⟨|𝝎|2⟩ in 2D and 3D is incompatible with turbulence vorticity amplification symptom (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' We can safely deduce that even a wider domain (increasing ℓ𝑦/ℎ) would not produce a large increase in Nu resulting from turbulent transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Moreover, if the Kolmogorov and Batchelor length scales in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='3) control the thickness of the transport-determining BL then indeed ℓ𝑦/ℎ is irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' To reinforce and illustrate the conclusion above, note that ¯𝑏(0) of the Ra = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4 × 1013 2D solution in figure 2(d) is within 15% of the minimum −𝑏★.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Thus the root-mean-square (RMS) vorticity of this 2D flow is within 8% of the maximum √︁ 𝑏★/ℎ implied by (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (In this example 𝜅/𝜈 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=') It follows that a 3DFS HC solution at Ra = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4 × 1013, and Pr = 1, must have an RMS vorticity within 8% of the RMS vorticity of the 2D comparison flow in figure 2(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (The 3D flow has more RMS vorticity than the 2D flow, but less than the maximum permitted by (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=') Moreover, all numerical and experimental results indicate that ¯𝑏(0) → −𝑏★ as Ra → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Thus increasing Ra above 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4 × 1013 likely reduces the already small 8% difference between the RMS vorticity of the 3D flow and the 2D comparison flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The small enhancement of 3D RMS vorticity implies that vortex stretching (c) does not effectively operate in 3D HC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' As a concluding illustration of the vorticity bound (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='2), notice that in figures 1 and 2 the vorticity is scaled with √︁ 𝑏★/ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' With this scaling the same colorbar applies even as Ra is varied by a factor 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The vorticity of the 3D flow in figure 9 is also scaled with √︁ ℎ/𝑏★.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' This simple estimate of the RMS vorticity applies across all 2D and 3D solutions reported here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Conclusion We have conducted a numerical study of the Ra–Nu relation with Pr = 1 and four cases corresponding to either no-slip or free-slip boundary conditions, in both 2D (ℓ𝑦/ℎ = 0) or 3D (ℓ𝑦/ℎ = 1) geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' In all four cases, with Ra in the range 106 to 1010, we find that heat flux obeys Rossby scaling, that is, Nu ∼ Ra1/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' In the 2D cases, with maximum Rayleigh 22 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Constantinou, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Rocha, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Smith, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Young Figure 9: Panels (a) and (b) show a 𝑦-slice of snapshots of 3DFS HC at Ra = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='20 × 1011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The spanwise averages of the snapshots of vorticity and buoyancy are shown in panels (c) and (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' In all panels, the contours are streamlines computed from a spanwise averaged snapshot at the final time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' At the top surface −1 ⩽ 𝑏/𝑏★ ⩽ +1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' the narrow range of the buoyancy color scale makes the small interior variations visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' number of order 1014, we found that a scaling regime with Nu ∼ Ra1/4 replaces the Rossby scaling for Ra beyond 1011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' see also Tsai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The scaling arguments for the Nu–Ra relation of HC reviewed in section 4 do not depend very much, if at all, on the distinction between 2D and 3D HC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Nor do these arguments identify the spanwise aspect ratio ℓ𝑦/ℎ as an important parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Thus it is informative to conduct parallel numerical studies of 2D and 3D HC and compare corresponding Nu’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' This comparison of 2D with 3D HC, extending to Ra = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='2 × 1011, shows that 3D HC has only a slight 10 or 20% enhancement of heat transport over non-turbulent 2D HC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' It is difficult to believe that there is only a slight enhancement at significantly higher Ra – otherwise relatively inexpensive 2D numerical solutions would provide useful estimates of 3D HC heat transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (a) vorticity, (azu- äxw) Vh/b* at y= 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 (b) buoyancy, b/b* 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='822 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='824 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='826 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='828 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 x/h (c) vorticity, (azu- axw) h/b* 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 (d) buoyancy, b/b 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='822 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='824 %0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='826 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='828 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='0 x/hHorizontal convection Nu scaling 23 It is likely that the 2D Nu is less than (or equal to) the 3D Nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Proving this plausible conjecture is an open challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Our results, numerical and theoretical, reinforce the view that HC does not express all of the characteristics of turbulence (Paparella & Young 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Acknowledgments Without implying their endorsement, we thank Basile Gallet and Ross Griffiths for discussions of horizontal convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Computational resources were provided by the Australian National Computational Infrastructure at the Australian National University, which is supported by the Commonwealth of Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' NCC was supported by the Australian Research Council DECRA Fellowship DE210100749.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' CBR was supported by National Aeronautics and Space Administration Award NNX16AO5OH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' SGLS was partially supported by National Science Foundation Awards OCE-1829919 WRY was supported by National Science Foundation Awards OCE-1657041 and OCE-2048583.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The low Rayleigh number regime If the Rayleigh number is sufficiently small then one can employ a straightforward expansion in powers of Ra to show that the Nusselt number is Nu = 1 + 𝐶2Ra2 + ord�Ra4� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (A 1) In the expansion (A 1), 𝐶2 is a function of the aspect ratio, 𝐴𝑥 = ℓ𝑥/ℎ, but not the Prandtl number Pr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' In this appendix we summarize the calculation of 𝐶2 for horizontal convection forced with the sinusoidal 𝑏s in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' This calculation is more interesting than one might anticipate because 𝐶2 turns out to be a very small number for all values of the aspect ratio 𝐴𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Consequently with the aspect ratio 𝐴𝑥 = 4 used in this work “sufficiently small” means Rayleigh numbers of order 104 (see the inset in figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Using the streamfunction formulation, with (𝑢, 𝑤) = (−𝜓𝑧, 𝜓𝑥), and scaling lengths with the depth ℎ and time with ℎ2/𝜅 the steady Boussinesq equations are 𝜓𝑥∇2𝜓𝑧 − 𝜓𝑧∇2𝜓𝑥 = 𝑏𝑥 + Pr ∇4𝜓 , (A 2) 𝜓𝑥𝑏𝑧 − 𝜓𝑧𝑏𝑥 = ∇2𝑏 , (A 3) where here ∇2 = 𝜕2 𝑥 + 𝜕2 𝑧 is the two-dimensional Laplacian;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' the spanwise vorticity is 𝑢𝑧 − 𝑤𝑥 = −∇2𝜓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The surface boundary condition is 𝑏(𝑥, 1) = 𝜖 cos 𝑚𝑥 where 𝑚 def= 𝑘ℎ = 2𝜋/𝐴𝑥 , and 𝜖 def= PrRa/𝐴3 𝑥 = PrRa (𝑚/2𝜋)3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (A 4) We expand all variables in powers of the small parameter 𝜖 (𝑏, 𝜓) = 𝜖(𝑏1, 𝜓1) + 𝜖2(𝑏2, 𝜓2) + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (A 5) The first-order equations are Pr∇4𝜓1 = −𝑏1𝑥 , and ∇2𝑏1 = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (A 6) The solution (A 6) is 𝜓1 = sin 𝑚𝑥 𝑃(𝑧) , and 𝑏1 = cos 𝑚𝑥 𝐵(𝑧) , (A 7) where 𝐵(𝑧) def= sech 𝑚 cosh 𝑚𝑧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' In 𝜓1(𝑥, 𝑧) we have the free-slip function 𝑃𝐹𝑆(𝑧) = 𝐵(𝑧) 8𝑚2Pr � (𝑚 coth 𝑚 + 1 − 𝑧) tanh 𝑚𝑧 − 𝑚𝑧(2 − 𝑧) � , (A 8) 24 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Constantinou, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Rocha, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Smith, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Young and the no-slip function 𝑃𝑁 𝑆(𝑧) = 1 8𝑚Pr (sinh2 𝑚 − 𝑚2) � (sinh2 𝑚 − 𝑚2)(𝑧2 − 𝑧)𝐵(𝑧) + (tanh 𝑚 − 𝑚) 𝑧 sinh 𝑚(1 − 𝑧) + (sinh 𝑚 − 𝑚 sech 𝑚) (1 − 𝑧) sinh 𝑚𝑧 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (A 9) At second order in 𝜖 we must solve ∇2𝑏2 = 𝜓1𝑥𝑏1𝑧 − 𝜓1𝑧𝑏1𝑥 , (A 10) = 1 2𝑚(𝑃𝐵)′ �������������� 𝐽0 + 1 2𝑚(𝑃𝐵′ − 𝑃′𝐵) �������������������������������� 𝐽2 cos 2𝑚𝑥 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (A 11) The solution of (A 11) has the form 𝑏2 = 𝐵20(𝑧) + 𝐵22(𝑧) cos 2𝑚𝑥 , (A 12) where 𝐵20 and 𝐵22 are determined by solving 𝐵′′ 20 = 𝐽0 , (A 13) 𝐵′′ 22 − 4𝑚2𝐵22 = 𝐽2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (A 14) Forming ⟨𝐵22(A 14)⟩ we find the shortcut used below in passing from (A 16) to (A 17): � 𝐵′ 22 2 + 4𝑚2𝐵2 22 � = − ⟨𝐵22𝐽2⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (A 15) The expressions for 𝐵22 and 𝐵20, obtained with Mathematica, are complicated and are not explicitly presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Mercifully, to obtain the coefficient 𝐶2 in (A 1), we do not need 𝜓2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Multiplying ∇2𝑏1 = 0 by 𝑏𝑛, with 𝑛 ⩾ 2, and noting that all these 𝑏𝑛’s have homogeneous boundary conditions at 𝑧 = 0 and 1, we see that ⟨∇𝑏𝑛 · ∇𝑏1⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Consequently the expansion of the buoyancy variance dissipation is 𝜒 = 𝜖2 ⟨|∇𝑏1|2⟩ ������������ 𝜒2= 1 2 𝑚 tanh 𝑚 + 𝜖4 ⟨|∇𝑏2|2⟩ ������������ 𝜒4 + ord�𝜖6� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (A 16) Recalling the definition of 𝜖 in (A 4), the Nusselt number is Nu = 1 + 𝑚2� (𝑃𝐵)2� − 2⟨𝐵22𝐽2⟩ 2𝑚 tanh 𝑚 ������������������������������������������������������ 𝜒4/𝜒2 �𝑚3Pr 8𝜋3 �2 Ra2 + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (A 17) Because 𝑃, 𝐵22, and 𝐽2 are all proportional to Pr−1, the Prandtl number Pr cancels out of the coefficient of Ra2 in (A 17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' With FS boundary conditions, the expression for 𝐶2 in (A 1) is 𝐶𝐹𝑆 2 = � 690 − 1920𝑚4 cosech2 𝑚 + 20(3 + 4𝑚2)2 sech 2𝑚 + 𝑚(1024𝑚4 − 80𝑚2 − 6195) cosech 𝑚 sech3 𝑚 + 5(352𝑚4 − 624𝑚2 + 1065) sech2 𝑚 �� 41 943 040 𝜋6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (A 18) Horizontal convection Nu scaling 25 Limiting values in the FS case are lim 𝑚→0 𝐶𝐹𝑆 2 = 31 𝑚8 30 965 760𝜋6 + ord�𝑚10� , (A 19) and lim 𝑚→∞ 𝐶𝐹𝑆 2 = 69 4 194 304 𝜋6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (A 20) We admire the frequently occurring integer 4 194 304 = 222 in the formulas above and below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' With NS boundary conditions, we find 𝐶 𝑁 𝑆 2 = 1 41 943 040 𝜋6 Ξ(𝑚) � cosh 2𝑚 − 2𝑚2 − 1�2 , (A 21) where Ξ(𝑚) = 85 cosh 4𝑚 − 320𝑚(33 + 8𝑚2) sinh 2𝑚 + 10�845 + 716𝑚2� cosh 2𝑚 − 5(5123 + 18304𝑚2 + 13720𝑚4 + 2912𝑚6) − 4𝑚5(7215 + 100𝑚2 − 64𝑚4) coth 𝑚 + 20(829 + 4402𝑚2 + 5289𝑚4 + 836𝑚6 − 64𝑚8) sech2 𝑚 − 4𝑚(4785 + 8010𝑚2 + 1549𝑚4 − 212𝑚6 + 64𝑚8) tanh 𝑚 sech2 𝑚 + 20(25 − 184𝑚2 − 80𝑚4 − 16𝑚6) sech 2𝑚 + 4𝑚(8865 + 21640𝑚2 + 17719𝑚4 + 740𝑚6 − 64𝑚8) tanh 𝑚 + 160𝑚(15 − 7𝑚2 − 4𝑚4) tanh 2𝑚 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (A 22) Limiting values in the NS case are lim 𝑚→0 𝐶 𝑁 𝑆 2 = 𝑚8 30 965 760𝜋6 + ord�𝑚10� , (A 23) and lim 𝑚→∞ 𝐶 𝑁 𝑆 2 = 17 4 194 304𝜋6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' (A 24) It is notable that the small-𝑚 𝐶 𝑁 𝑆 2 in (A 23) is smaller by a factor of exactly 31 than 𝐶𝐹𝑆 2 in (A 19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Both 𝐶2’s are very much less than one for all aspect ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' In the numerical solutions summarized in figure 4, the aspect ratio is 𝐴𝑥 = 4, corresponding to 𝑚 = 𝑘ℎ = 𝜋/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' The results in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='3) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='4) follow by evaluating the formulas (A 18) through (A 22) with 𝑚 = 𝜋/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' REFERENCES Benzi, R, Patarnello, S & Santangelo, P 1987 On the statistical properties of two-dimensional decaying turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Europhys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 3 (7), 811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Benzi, R, Patarnello, S & Santangelo, P 1988 Self-similar coherent structures in two-dimensional decaying turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' A 21 (5), 1221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Burns, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=', Vasil, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=', Oishi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=', Lecoanet, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' & Brown, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 2020 Dedalus: A flexible framework for numerical simulations with spectral methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 2, 023068.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Carnevale, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=', McWilliams, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=', Pomeau, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=', Weiss, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' & Young, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 1991 Evolution of vortex statistics in two-dimensional turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 66 (21), 2735.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Chiu-Webster, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=', Hinch, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' & Lister, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 2008 Very viscous horizontal convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 611, 395–426.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 26 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Constantinou, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Rocha, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Smith, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Young Coman, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=', Griffiths, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' & Hughes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 2006 Sandström’s experiments revisited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Mar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 64 (6), 783–796(14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Dritschel, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=', Scott, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=', Macaskill, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=', Gottwald, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' & Tran, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 2008 Unifying scaling theory for vortex dynamics in two-dimensional turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 101 (9), 094501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Gayen, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=', Griffiths, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' & Hughes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 2014 Stability transitions and turbulence in horizontal convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 751, 698–724.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Gayen, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=', Griffiths, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=', Hughes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' & Saenz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 2013 Energetics of horizontal convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 716, R10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Hughes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' & Griffiths, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 2008 Horizontal convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Annu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 40, 185–208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Ilicak, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' & Vallis, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 2012 Simulations and scaling of horizontal convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Tellus A 64 (1), 18377.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Kuhlbrodt, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 2008 On Sandström’s inferences from his tank experiments: a hundred years later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Tellus A 60 (5), 819–836.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' McWilliams, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' C 1984 The emergence of isolated, coherent vortices in turbulent flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 146, 21–43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' McWilliams, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' C 1990 The vortices of two-dimensional turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 219, 361–385.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Mullarney, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=', Griffiths, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' & Hughes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 2004 Convection driven by differential heating at a horizontal boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 516, 181–209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' NCFMF 1968 Turbulence (Educational film from 1968).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' https://youtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='be/AeBsiEYWZUY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' NCFMF 1972 Illustrated Experiments in Fluid Mechanics: The NCFMF Book of Film Notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' National Committee for Fluid Mechanics Films .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' MIT Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Paparella, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' & Young, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 2002 Horizontal convection is non-turbulent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 466, 205–214.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Passaggia, Pierre-Yves, Scotti, Alberto & White, Brian 2017 Transition and turbulence in horizontal convection: linear stability analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 821, 31–58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Rocha, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=', Bossy, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=', Llewellyn Smith, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' & Young, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 2020a Improved bounds on horizontal convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 883, A41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Rocha, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=', Constantinou, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=', Llewellyn Smith, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' & Young, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 2020b The Nusselt numbers of horizontal convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 894, A24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Rossby, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 1965 On thermal convection driven by non-uniform heating from below: an experimental study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Deep Sea Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 12 (1), 9–10, IN9–IN14, 11–16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Rossby, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 1998 Numerical experiments with a fluid heated non-uniformly from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Tellus A 50 (2), 242–257.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Sandström, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 1908 Dynamische versuche mit meerwasser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Annals in Hydrodynamic Marine Meteorology 36, 6–23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Scotti, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' & White, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 2011 Is horizontal convection really “non-turbulent?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 38 (21), L21609.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Sheard, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' & King, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 2011 Horizontal convection: effect of aspect ratio on Rayleigh number scaling and stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Modelling 35 (4), 1647–1655.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Shishkina, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=', Grossmann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' & Lohse, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 2016 Heat and momentum transport scalings in horizontal convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 43 (3), 1219–1225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Siggers, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=', Kerswell, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' & Balmforth, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 2004 Bounds on horizontal convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 517, 55–70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Taylor, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 1938 Production and dissipation of vorticity in a turbulent fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Lond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' A 164 (916), 15–23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Taylor, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' & Green, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 1937 Mechanism of the production of small eddies from large ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Lond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' A 158 (895), 499–521.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Tsai, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=', Hussam, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' K, Fouras, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' & Sheard, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 2016 The origin of instability in enclosed horizontally driven convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Heat and Mass Transfer 94, 509–515.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Tsai, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=', Hussam, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=', King, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' & Sheard, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 2020 Transitions and scaling in horizontal convection driven by different temperature profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Thermal Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 148, 106166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Wang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' & Huang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 2005 An experimental study on thermal convection driven by horizontal differential heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 540, 49–73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Winters, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' & Young, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 2009 Available potential energy and buoyancy variance in horizontal convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} +page_content=' 629, 221–230.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfXASj/content/2301.03122v1.pdf'} diff --git a/G9E3T4oBgHgl3EQfWQru/content/tmp_files/2301.04468v1.pdf.txt b/G9E3T4oBgHgl3EQfWQru/content/tmp_files/2301.04468v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d41df5cea88c9ec4da11e08c3774912a2d0371ac --- /dev/null +++ b/G9E3T4oBgHgl3EQfWQru/content/tmp_files/2301.04468v1.pdf.txt @@ -0,0 +1,1211 @@ +arXiv:2301.04468v1 [math.OC] 7 Jan 2023 +Tabu Search and Simulated Annealing +metaheuristic algorithms applied to the RoRo +vessel stowage problem +Eghbal Hosseini +Department of People and Technology Roskilde University, Denmark +kseghbalhosseini@gmail.com, hosseini@ruc.dk +Abstract. The search heuristics Tabu search and Simulated annealing +are commonly used meta-heuristics. The two heuristics have different +ways of ensuring diversification. The heuristics can be implemented for +solving the stowage planning problem. The stowage planning problem +occurs every time a vessel is loaded. The idea is to stow the cargo in an +optimal manner satisfying a set of constraints and specifically the stabil- +ity constraints. An optimal plan can be to assign the cargo to spots on +the vessel so that the vessel trim and draft are optimize ensuring a low +fuel consumption. Here the problem considered is the stowage planning +of trailers on a Roll-on/Roll-off vessel. Roll-on/Roll-off vessels carries +vehicles or trailers on the different levels of the vessel. When making a +stowage plan of a vessel ballast tanks can be adjusted to improve sta- +bility and to change draft. This leads to stability constraints which can +be very complex to model and solve using a mixed integer model and +solver. Complicating constraints occur in many different applications and +different forms and are the cause of the popularity of search algorithms, +such as tabu search and simulated annealing, for solving real-life applica- +tions. Results of the two meta-heuristics are shown for real-life stowage +planning cases for Roll-on/Roll-off vessels. +Keywords: Stowage; Tabu Search; Simulated Annealing; Laying Chicken +Algorithm; Volcano Eruption Algorithm; Multiverse Algorithm. +1 +Introduction +2 +Modelling +The problem is stowage planning for a ship with some unusable spaces. Cargos +should be loaded in defined ports and unloaded in others. The objective of the +problem is minimizing total time of load and unload of cargos. The problem +has been modeled as a mixed-integer linear programming (MILP) problem as +follows: +min +k +� +i=1 +m×n +� +j=1 +tij +(1) + +2 +Eghbal Hosseini +k +� +i=1 +xij ≤ 1, +(2) +m×n +� +j=1 +xij = 1,for i=1,2,...,k, +(3) +k +� +i=1 +yip · wi ≤ wp,for p=1,2,...,l, +(4) +tij = +� +T +i,j belong to a same category +T + Q +Otherwise +(5) +xij = +� +1 Cargo i set in the cell j +0 +Otherwise +(6) +yip = +� +1 Cargo i set in the deck p +0 +Otherwise +(7) +The objective of the model, (1), minimises the total time of unloading cargos. +Constraints (2) ensure that each cargo is loaded exactly once at a cell. Con- +straints (3) make sure that each cell will only have at most one cargo loaded. +Constraints (4) make sure that the total weight of cargoes loaded on each deck +does not exceed the maximum weight limit per deck. wi is the weight of the ith +cargo. +3 +LCA, VEA, SA, and TS Algorithms +3.1 +Simulating Annealing (SA) +There are several heuristic and meta-heuristic algorithms in the literature [1- +23]. Simulated annealing (SA) is a meta-heuristic technique for approximating +the global optimum of a given optimization problem. Specifically, it is a proba- +bilistic method to approximate global optimization in a big search region for an +optimization problem. It is often used for discrete optimization problem. In the +Simulated Annealing case, the equation has been altered to the following: +p = +� +1 +∆c ≤ 0 +e +−∆c +t +∆c > 0 +(8) +Where the delta c is the change in cost and the t is the current temperature. +The P calculated in this case is the probability that we should accept the new +solution. In our implementation the formulation of SA has been little changed +based on the problem: +p = +� +e +1 +∆c ∆c ≤ 0 +e +−∆c +t +M +∆c > 0 +(9) + +Survey on Stowage Planning +3 +Which M > 1. The process of the modified SA are as follows: +1. The algorithm starts with an Initial Solution (IS). +2. The population is generated close to initial solution. +3. Each solution of population gets a probability based on (2). +4. The solution with the biggest probability will be selected as the best solution +of current population. +5. Set IS = best solution and go back to step 2. +Algorithm 1 SA Procedure for Stowage Planning Problem +1: n: Number of solutions +2: N: Number of Iterations +3: ∆ck: Difference of objective functions between solution k and initial solution +4: α: A given positive number (less than size of the problem) +5: Generate a random initial feasible solution X0 +6: Generate initial population near initial solution +7: for i ← 1 to N do +8: +for k ← 1 to n do +9: +if Xk is better than X0 then +10: Pk = e +1 +∆ck +11: +else if +12: +thenPk = e +−∆c +t +M +13: +end if +14: +end for +15: Find Maximum of Pk +16: Pmax = MaximumofPk and r = k +17: Xbest = Xr +18: end for +19: X0 = Xbest and back to 6 +3.2 +Tabu Search (TS) +Tabu Search (TS) is a metaheuristic search method employing local search meth- +ods used for mathematical optimization. Local searches take a feasible solution +to the optimization problem and check its very close neighbors to find a better +solution. Local search methods tend to become stuck in local optimal solutions. +TS modifies the performance of local search by relaxing its basic rule. At each +step of TS algorithm, changing of moves can be accepted if better solution is not +available such as when the search is stuck at a strict local optimal. In addition, +prohibitions (tabu) are introduced to discourage the search from coming back +to previous visited solutions. + +4 +Eghbal Hosseini +Algorithm 2 TS Procedure for Stowage Planning Problem +1: n: Number of solutions +2: s: Size of the problem +3: Rand: Random integer number between 1 and s +4: λ: A given integer positive number +5: Generate initial population +6: for t ← 1 to λ do +7: +for k ← 1 to n do +8: Xk = Xk + Rand ∗ +Xk +||Xk|| (Distribution of solutions in feasible region) +9: +end for +10: Find best solution +11: Let Xt = xbest +12: end for +13: for t ← 1 to λ do +14: +for i ← 1 to n do +15: Xi = Xt + Rand ∗ +Xk +||Xk|| (Distribution of best solutions) +16: +end for +17: Find best solution +18: Let Y t = xbest +19: end for +3.3 +Laying Chicken Algorithm (LCA) +Laying Chicken Algorithm has been inspired from behavior of laying hens. It fo- +cuses on finding an answer to how does the hen convert the egg to the chicken? +LCA converts the feasible solutions to the optimal solution, same as what laying +hen does from the eggs to the chickens. In fact, each feasible solution of a con- +tinuous programming problem displays an egg and the optimal solution of the +problem is a chicken. Hens try to warm their eggs; this concept has inspired by +LCA to change and improve the feasible solutions. There are the following steps +to formulate of the behavior the hen in the LCA optimizer [4]: +1. The first egg which displays initial solution. +2. More eggs displays initial population close to the initial solution. +3. Improve solutions of population inspiring from warming eggs. +4. Little mutation of solutions inspiring of rotation of eggs. +3.4 +Volcano Eruption Algorithm (VEA) +The Volcano Eruption Algorithm has been inspired from the nature of a volcano +eruption. VEA optimizer imitates the process of volcano eruption, which is a +hole on the earth’s surface. This phenomenon acts as a vent for release of pres- +surized gases, molten rock or magma deep beneath the surface of earth. Magma +is passed through a channel from deep underground called the volcanic pipe. +Magma erupts out of the earth’s surface when it reaches the hole on the surface. +There are the following steps leading to formation of a volcano to VEA optimizer +[5]: + +Survey on Stowage Planning +5 +Algorithm 3 LCA Procedure for Stowage Planning Problem +1: n: Number of solutions +2: N: Number of Iterations +3: α: A given positive number (less than size of the problem) +4: Generate a random initial feasible solution X0 +5: Generate initial population near initial solution +6: for i ← 1 to N do +7: +for k ← 1 to n do +8: +if Xk is not better than X0 then +9: Xk = X0 + α ∗ ( +Xk +||Xk||) +10: +end if +11: Xk = Xk + +Xk +||Xk|| +12: +end for +13: end for +1. Rise of magma through the volcanic pipe. +2. Volcanic eruption by rising of magma to the surface of the earth. +3. Lava’s cooling down and therefore formation of a crust. +4. Repetition of this process over time leading to several layers of rock that +builds up over time resulting in a volcano. +Algorithm 4 VEA Procedure for Stowage Planning Problem +1: n: Number of solutions +2: s: Size of the problem +3: Rand: Random integer number between 1 and s +4: λ: A given integer positive number +5: Generate initial population +6: for t ← 1 to λ do +7: +for k ← 1 to n do +8: Xk = Xk + Rand ∗ +Xk +||Xk|| +9: +end for +10: Find best solution +11: Let Xt = xbest +12: end for +13: for t ← 1 to λ do +14: +for i ← 1 to n do +15: Xi = Xt + Rand ∗ +Xk +||Xk|| +16: +end for +17: Find best solution +18: Let Y t = xbest +19: end for + +6 +Eghbal Hosseini +Algorithm 5 MVA Procedure for Stowage Planning Problem +1: N: Number of solutions +2: s: Size of the problem +3: Rand: Random integer number between 1 and s +4: λ: A given integer positive number +5: m: Number of solutions for the next population +6: Generate initial population +7: for t ← 1 to λ do +8: +for l ← 1 to N do +9: +for k ← 1 to m do +10: Xk = Xk + Rand ∗ +Xk +||Xk|| (Distribution of solutions in feasible region) +11: +end for +12: Find best solution +13: Let Xt = xbest +14: +end for +15: end for +16: for t ← 1 to λ do +17: +for i ← 1 to m do +18: Xi = Xt + Rand ∗ +Xk +||Xk|| (Distribution of best solutions) +19: +end for +20: Find best solution +21: Let yt = xbest +22: end for +3.5 +Multiverse Algorithm (MVA) +Algorithms 1-5 show the pseudocodes of SA, TS, LCA, VEA, and MVA algo- +rithms respectively. +4 +Computational Results +In this section, four algorithms: Tabu Search (TS), Simulated Annealing (SA), +Laying Chicken Algorithm (LCA) and Volcano Eruption Algorithm (VEA) are +implemented to solve some benchmarks of the problem. + +Survey on Stowage Planning +7 +5 +10 +15 +20 +25 +30 +35 +20 +40 +60 +80 +100 +(a) Unusable space (dark blue) before +loading +5 +10 +15 +20 +25 +30 +35 +20 +40 +60 +80 +100 +(b) Initial solution by LCA +Fig. 1. Matrix format of ship before loading and initial solution by LCA +5 +10 +15 +20 +25 +30 +35 +20 +40 +60 +80 +100 +(a) Problem 1 +5 +10 +15 +20 +25 +30 +35 +20 +40 +60 +80 +100 +(b) Problem 2 +5 +10 +15 +20 +25 +30 +35 +20 +40 +60 +80 +100 +(c) Problem 3 +5 +10 +15 +20 +25 +30 +35 +20 +40 +60 +80 +100 +(d) Problem 4 +5 +10 +15 +20 +25 +30 +35 +20 +40 +60 +80 +100 +(e) Problem 5 +5 +10 +15 +20 +25 +30 +35 +20 +40 +60 +80 +100 +(f) Problem 6 +Fig. 2. Optimal Solutions problems 1-6 + +8 +Eghbal Hosseini +5 +10 +15 +20 +25 +30 +35 +20 +40 +60 +80 +100 +(a) 10th Iteration-Prob.1 +5 +10 +15 +20 +25 +30 +35 +20 +40 +60 +80 +100 +(b) Best Solution- Prob.1 +5 +10 +15 +20 +25 +30 +35 +20 +40 +60 +80 +100 +(c) 10th Iteration-Prob.2 +5 +10 +15 +20 +25 +30 +35 +20 +40 +60 +80 +100 +(d) Best Solution- Prob.2 +5 +10 +15 +20 +25 +30 +35 +20 +40 +60 +80 +100 +(e) 10th Iteration-Prob.3 +5 +10 +15 +20 +25 +30 +35 +20 +40 +60 +80 +100 +(f) Best Solution- Prob.3 +5 +10 +15 +20 +25 +30 +35 +20 +40 +60 +80 +100 +(g) 10th Iteration-Prob.4 +5 +10 +15 +20 +25 +30 +35 +20 +40 +60 +80 +100 +(h) Best Solution- Prob.4 +5 +10 +15 +20 +25 +30 +35 +20 +40 +60 +80 +100 +(i) 10th Iteration-Prob.5 +5 +10 +15 +20 +25 +30 +35 +20 +40 +60 +80 +100 +(j) Best Solution- Prob.5 +5 +10 +15 +20 +25 +30 +35 +20 +40 +60 +80 +100 +(k) 10th Iteration-Prob.6 +5 +10 +15 +20 +25 +30 +35 +20 +40 +60 +80 +100 +(l) Best Solution- Prob.6 +Fig. 3. Behavior of LCA for benchmarks + +Survey on Stowage Planning +9 +5 +10 +15 +20 +25 +30 +35 +20 +40 +60 +80 +100 +(a) 10th Iteration-Prob.1 +5 +10 +15 +20 +25 +30 +35 +20 +40 +60 +80 +100 +(b) Best Solution- Prob.1 +5 +10 +15 +20 +25 +30 +35 +20 +40 +60 +80 +100 +(c) 10th Iteration-Prob.2 +5 +10 +15 +20 +25 +30 +35 +20 +40 +60 +80 +100 +(d) Best Solution- Prob.2 +5 +10 +15 +20 +25 +30 +35 +20 +40 +60 +80 +100 +(e) 10th Iteration-Prob.3 +5 +10 +15 +20 +25 +30 +35 +20 +40 +60 +80 +100 +(f) Best Solution- Prob.3 +5 +10 +15 +20 +25 +30 +35 +20 +40 +60 +80 +100 +(g) 10th Iteration-Prob.4 +5 +10 +15 +20 +25 +30 +35 +20 +40 +60 +80 +100 +(h) Best Solution- Prob.4 +5 +10 +15 +20 +25 +30 +35 +20 +40 +60 +80 +100 +(i) 10th Iteration-Prob.5 +5 +10 +15 +20 +25 +30 +35 +20 +40 +60 +80 +100 +(j) Best Solution- Prob.5 +5 +10 +15 +20 +25 +30 +35 +20 +40 +60 +80 +100 +(k) 10th Iteration-Prob.6 +5 +10 +15 +20 +25 +30 +35 +20 +40 +60 +80 +100 +(l) Best Solution- Prob.6 +Fig. 4. Behavior of VEA for benchmarks + +10 +Eghbal Hosseini +Table 1. Results of SA for Benchmarks +Problems +Initial Solution +5 Iterations +10 Iterations Best Solution +Inst0 +31650 +31350 +10360 +2570 +Inst1 +27440 +27730 +11230 +1980 +Inst2 +27970 +27980 +9610 +2380 +Inst3 +27780 +27220 +12760 +3450 +Inst4 +27930 +26540 +9780 +1210 +Inst5 +27790 +27990 +9110 +2340 +Table 2. Results of TS for Benchmarks +Problems +Initial Solution +5 Iterations +10 Iterations Best Solution +Inst0 +31570 +31570 +31260 +1030 +Inst1 +31630 +31630 +31260 +1210 +Inst2 +31640 +31600 +31220 +1920 +Inst3 +32070 +31880 +31210 +1880 +Inst4 +32020 +31920 +31230 +2100 +Inst5 +31770 +31420 +31120 +1750 +References +1. Hosseini E, Reinhardt L, Rawat DB. Optimizing Gradient Methods for IoT Appli- +cations. IEEE Internet of Things Journal. 2022 Jan 11. +2. Hosseini E, Reinhardt L, Ghafoor KZ, Rawat DB. Implementation and Comparison +of Four Algorithms on Transportation Problem. InInternational Summit Smart +City 360° 2022 (pp. 422-433). Springer, Cham. +3. Hosseini E, Ghafoor KZ, Emrouznejad A, Sadiq AS, Rawat DB. Novel metaheuris- +tic based on multiverse theory for optimization problems in emerging systems. +Applied Intelligence. 2021 Jun;51(6):3275-92. +4. Hosseini E. Cost-Flow Summation Algorithm Based on Table Form to Solve Min- +imum Cost-Flow Problem. arXiv preprint arXiv:2101.01103. 2020 Dec 29. +5. Hosseini E, Ghafoor KZ, Sadiq AS, Guizani M, Emrouznejad A. Covid-19 op- +timizer algorithm, modeling and controlling of coronavirus distribution process. +IEEE Journal of Biomedical and Health Informatics. 2020 Jul 28;24(10):2765-75. +6. Hosseini E, Sadiq AS, Ghafoor KZ, Rawat DB, Saif M, Yang X. Volcano eruption +algorithm for solving optimization problems. Neural Computing and Applications. +2021 Apr;33(7):2321-37. +7. Ghafoor KZ, Kong L, Rawat DB, Hosseini E, Sadiq AS. Quality of service aware +routing protocol in software-defined internet of vehicles. IEEE Internet of Things +Journal. 2018 Oct 11;6(2):2817-28. +8. Hosseini E. Presentation and solving non-linear quad-level programming problem +utilizing a heuristic approach based on Taylor theorem. Journal of Optimization +in Industrial Engineering. 2018 Mar 1;11(1):91-101. +9. Hosseini E. Three new methods to find initial basic feasible solution of transporta- +tion problems. Applied Mathematical Sciences. 2017;11(37):1803-14. +10. Hosseini E. Solving linear tri-level programming problem using heuristic method +based on bi-section algorithm. Asian J. Sci. Res. 2017;10:227-35. + +Survey on Stowage Planning +11 +Table 3. Results of LCA for Benchmarks +Problems +Initial Solution +5 Iterations +10 Iterations Best Solution +Inst0 +31710 +31310 +9690 +1740 +Inst1 +31600 +31230 +9720 +2370 +Inst2 +31720 +31290 +9760 +660 +Inst3 +31180 +30950 +9750 +510 +Inst4 +26830 +24120 +9630 +2060 +Inst5 +27130 +26720 +9680 +930 +Table 4. Results of VEA for Benchmarks +Problems +Initial Solution +5 Iterations +10 Iterations Best Solution +Inst0 +30590 +28710 +9690 +1620 +Inst1 +26380 +23560 +9200 +1630 +Inst2 +27420 +24670 +7420 +620 +Inst3 +24330 +21450 +6740 +440 +Inst4 +26690 +22680 +9390 +1730 +Inst5 +26430 +25270 +9310 +680 +11. Hosseini E. Laying chicken algorithm: a new meta-heuristic approach to solve con- +tinuous programming problems. J Appl Comput Math. 2017;6(1):1-8. +12. Hosseini E. Big bang algorithm: A new meta-heuristic approach for solving opti- +mization problems. Asian Journal of Applied Sciences. 2017;10(3):134-44. +13. Hosseini E, Kamalabadi IN, Daneshfar F. Solving Non-Linear Bi-Level Program- +ming Problem Using Taylor Algorithm. InHandbook of Research on Modern Op- +timization Algorithms and Applications in Engineering and Economics 2016 (pp. +797-810). IGI Global. +14. Hosseini E, Kamalabadi IN. Line search and genetic approaches for solving linear +tri-level programming problem. Int J Manag Acc Econ. 2015;1(4). +15. Hosseini E, Nakhai Kamalabadi I. Two approaches for solving non-linear bi-level +programming problem. Advances in Research. 2015;3(5):512-25. +16. Hosseini E, Kamalabadi IN. Bi-section algorithm for solving linear Bi-level pro- +gramming problem. Int. J. Sci. Eng. 2015;1:101-7. +17. Hosseini E, Kamalabadi IN. Smoothing and solving linear quad-level programming +problem using mathematical theorems. Int. J. Math. Comput. Sci. 2015;1:116-26. +18. Hosseini E, Kamalabadi IN. A modified simplex method for solving linear- +quadratic and linear fractional bi-level programming problem. Global Journal of +Advanced Research. 2015;1(2):142-54. +19. Nakhai Kamalabadi I, Hosseini E, Fathi M. Enhancing the solution method of lin- +ear Bi–level programming problem based on enumeration method and dual method. +Journal of Advanced Mathematical Modeling. 2014 Aug 23;4(1):27-53. +20. Hosseini E, Kamalabadi IN. Line search and genetic approaches for solving linear +tri-level programming problem. Int J Manag Acc Econ. 2015;1(4). +21. Hosseini E, Kamalabadi IN. Solving linear bi-level programming problem using +two new approaches based on line search and taylor methods. Int J Manage Sci +Education. 2014 Nov;2(6):243-52. + +12 +Eghbal Hosseini +Table 5. Results of MVA for Benchmarks +Problems +Initial Solution +5 Iterations +10 Iterations Best Solution +Inst0 +31900 +31510 +7420 +1300 +Inst1 +30300 +29880 +8760 +1150 +Inst2 +28720 +28210 +8560 +380 +Inst3 +30180 +29850 +990 +360 +Inst4 +26230 +25780 +9880 +1010 +Inst5 +28340 +27200 +7430 +780 +22. Hussein E, Kamalabadi I. Taylor approach for solving nonlinear bilevel program- +ming problem. Adv. Comput. Sci., Int. J. 2014;3(5):91-7. +23. Hosseini E, Kamalabadi IN. Solving Linear-Quadratic Bi-Level Programming and +Linear-Fractional Bi-Level Programming Problems Using Genetic Based Algo- +rithm. Applied Mathematics and Computational Intelligence. 2013;2. + +This figure "1-1.jpg" is available in "jpg"� format from: +http://arxiv.org/ps/2301.04468v1 + +5 +10 +15 +20 +25 +30 +35 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 + +This figure "11-1.jpg" is available in "jpg"� format from: +http://arxiv.org/ps/2301.04468v1 + +5 +10 +15 +20 +25 +30 + +5 +10 +15 +20 +25 +30 + +5 +10 +15 +20 +25 +30 + +5 +10 +15 +20 +25 +30 + +This figure "1-2.jpg" is available in "jpg"� format from: +http://arxiv.org/ps/2301.04468v1 + +5 +10 +15 +20 +25 +30 +35 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 + +This figure "11-2.jpg" is available in "jpg"� format from: +http://arxiv.org/ps/2301.04468v1 + +5 +10 +15 +20 +25 +30 + +5 +10 +15 +20 +25 +30 + +This figure "1-3.jpg" is available in "jpg"� format from: +http://arxiv.org/ps/2301.04468v1 + +5 +10 +15 +20 +25 +30 +35 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 + +This figure "11-3.jpg" is available in "jpg"� format from: +http://arxiv.org/ps/2301.04468v1 + +5 +10 +15 +20 +25 +30 + +5 +10 +15 +20 +25 +30 + +5 +10 +15 +20 +25 +30 + +5 +10 +15 +20 +25 +30 + +5 +10 +15 +20 +25 +30 + +This figure "1-4.jpg" is available in "jpg"� format from: +http://arxiv.org/ps/2301.04468v1 + +5 +10 +15 +20 +25 +30 +35 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 + +This figure "11-4.jpg" is available in "jpg"� format from: +http://arxiv.org/ps/2301.04468v1 + +5 +10 +15 +20 +25 +30 + +This figure "1-5.jpg" is available in "jpg"� format from: +http://arxiv.org/ps/2301.04468v1 + +5 +10 +15 +20 +25 +30 +35 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 + +This figure "11-5.jpg" is available in "jpg"� format from: +http://arxiv.org/ps/2301.04468v1 + diff --git a/G9E3T4oBgHgl3EQfWQru/content/tmp_files/load_file.txt b/G9E3T4oBgHgl3EQfWQru/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b881e311814518b789eeb468077c725393c58554 --- /dev/null +++ b/G9E3T4oBgHgl3EQfWQru/content/tmp_files/load_file.txt @@ -0,0 +1,465 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf,len=464 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='04468v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='OC] 7 Jan 2023 Tabu Search and Simulated Annealing metaheuristic algorithms applied to the RoRo vessel stowage problem Eghbal Hosseini Department of People and Technology Roskilde University, Denmark kseghbalhosseini@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='com, hosseini@ruc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='dk Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' The search heuristics Tabu search and Simulated annealing are commonly used meta-heuristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' The two heuristics have different ways of ensuring diversification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' The heuristics can be implemented for solving the stowage planning problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' The stowage planning problem occurs every time a vessel is loaded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' The idea is to stow the cargo in an optimal manner satisfying a set of constraints and specifically the stabil- ity constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' An optimal plan can be to assign the cargo to spots on the vessel so that the vessel trim and draft are optimize ensuring a low fuel consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Here the problem considered is the stowage planning of trailers on a Roll-on/Roll-off vessel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Roll-on/Roll-off vessels carries vehicles or trailers on the different levels of the vessel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' When making a stowage plan of a vessel ballast tanks can be adjusted to improve sta- bility and to change draft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' This leads to stability constraints which can be very complex to model and solve using a mixed integer model and solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Complicating constraints occur in many different applications and different forms and are the cause of the popularity of search algorithms, such as tabu search and simulated annealing, for solving real-life applica- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Results of the two meta-heuristics are shown for real-life stowage planning cases for Roll-on/Roll-off vessels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Keywords: Stowage;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Tabu Search;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Simulated Annealing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Laying Chicken Algorithm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Volcano Eruption Algorithm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Multiverse Algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' 1 Introduction 2 Modelling The problem is stowage planning for a ship with some unusable spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Cargos should be loaded in defined ports and unloaded in others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' The objective of the problem is minimizing total time of load and unload of cargos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' The problem has been modeled as a mixed-integer linear programming (MILP) problem as follows: min k � i=1 m×n � j=1 tij (1) 2 Eghbal Hosseini k � i=1 xij ≤ 1, (2) m×n � j=1 xij = 1,for i=1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=',k, (3) k � i=1 yip · wi ≤ wp,for p=1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=',l, (4) tij = � T i,j belong to a same category T + Q Otherwise (5) xij = � 1 Cargo i set in the cell j 0 Otherwise (6) yip = � 1 Cargo i set in the deck p 0 Otherwise (7) The objective of the model, (1), minimises the total time of unloading cargos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Constraints (2) ensure that each cargo is loaded exactly once at a cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Con- straints (3) make sure that each cell will only have at most one cargo loaded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Constraints (4) make sure that the total weight of cargoes loaded on each deck does not exceed the maximum weight limit per deck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' wi is the weight of the ith cargo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' 3 LCA, VEA, SA, and TS Algorithms 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='1 Simulating Annealing (SA) There are several heuristic and meta-heuristic algorithms in the literature [1- 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Simulated annealing (SA) is a meta-heuristic technique for approximating the global optimum of a given optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Specifically, it is a proba- bilistic method to approximate global optimization in a big search region for an optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' It is often used for discrete optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' In the Simulated Annealing case, the equation has been altered to the following: p = � 1 ∆c ≤ 0 e −∆c t ∆c > 0 (8) Where the delta c is the change in cost and the t is the current temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' The P calculated in this case is the probability that we should accept the new solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' In our implementation the formulation of SA has been little changed based on the problem: p = � e 1 ∆c ∆c ≤ 0 e −∆c t M ∆c > 0 (9) Survey on Stowage Planning 3 Which M > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' The process of the modified SA are as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' The algorithm starts with an Initial Solution (IS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' The population is generated close to initial solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Each solution of population gets a probability based on (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' The solution with the biggest probability will be selected as the best solution of current population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Set IS = best solution and go back to step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='Algorithm 1 SA Procedure for Stowage Planning Problem ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='1: n: Number of solutions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='2: N: Number of Iterations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='3: ∆ck: Difference of objective functions between solution k and initial solution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='4: α: A given positive number (less than size of the problem) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='5: Generate a random initial feasible solution X0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='6: Generate initial population near initial solution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='7: for i ← 1 to N do ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='8: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='for k ← 1 to n do ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='9: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='if Xk is better than X0 then ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='10: Pk = e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='∆ck ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='11: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='else if ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='12: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='thenPk = e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='−∆c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='13: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='end if ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='14: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='end for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='15: Find Maximum of Pk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='16: Pmax = MaximumofPk and r = k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='17: Xbest = Xr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='18: end for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='19: X0 = Xbest and back to 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='2 Tabu Search (TS) Tabu Search (TS) is a metaheuristic search method employing local search meth- ods used for mathematical optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Local searches take a feasible solution to the optimization problem and check its very close neighbors to find a better solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Local search methods tend to become stuck in local optimal solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' TS modifies the performance of local search by relaxing its basic rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' At each step of TS algorithm, changing of moves can be accepted if better solution is not available such as when the search is stuck at a strict local optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' In addition, prohibitions (tabu) are introduced to discourage the search from coming back to previous visited solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='Eghbal Hosseini ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='Algorithm 2 TS Procedure for Stowage Planning Problem ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='1: n: Number of solutions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='2: s: Size of the problem ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='3: Rand: Random integer number between 1 and s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='4: λ: A given integer positive number ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='5: Generate initial population ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='6: for t ← 1 to λ do ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='7: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='for k ← 1 to n do ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='8: Xk = Xk + Rand ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='Xk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='||Xk|| (Distribution of solutions in feasible region) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='9: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='end for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='10: Find best solution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='11: Let Xt = xbest ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='12: end for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='13: for t ← 1 to λ do ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='14: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='for i ← 1 to n do ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='15: Xi = Xt + Rand ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='Xk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='||Xk|| (Distribution of best solutions) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='16: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='end for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='17: Find best solution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='18: Let Y t = xbest ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='19: end for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='3 Laying Chicken Algorithm (LCA) Laying Chicken Algorithm has been inspired from behavior of laying hens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' It fo- cuses on finding an answer to how does the hen convert the egg to the chicken?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' LCA converts the feasible solutions to the optimal solution, same as what laying hen does from the eggs to the chickens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' In fact, each feasible solution of a con- tinuous programming problem displays an egg and the optimal solution of the problem is a chicken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Hens try to warm their eggs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' this concept has inspired by LCA to change and improve the feasible solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' There are the following steps to formulate of the behavior the hen in the LCA optimizer [4]: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' The first egg which displays initial solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' More eggs displays initial population close to the initial solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Improve solutions of population inspiring from warming eggs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Little mutation of solutions inspiring of rotation of eggs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='4 Volcano Eruption Algorithm (VEA) The Volcano Eruption Algorithm has been inspired from the nature of a volcano eruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' VEA optimizer imitates the process of volcano eruption, which is a hole on the earth’s surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' This phenomenon acts as a vent for release of pres- surized gases, molten rock or magma deep beneath the surface of earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Magma is passed through a channel from deep underground called the volcanic pipe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Magma erupts out of the earth’s surface when it reaches the hole on the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='There are the following steps leading to formation of a volcano to VEA optimizer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='[5]: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='Survey on Stowage Planning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='Algorithm 3 LCA Procedure for Stowage Planning Problem ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='1: n: Number of solutions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='2: N: Number of Iterations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='3: α: A given positive number (less than size of the problem) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='4: Generate a random initial feasible solution X0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='5: Generate initial population near initial solution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='6: for i ← 1 to N do ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='7: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='for k ← 1 to n do ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='8: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='if Xk is not better than X0 then ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='9: Xk = X0 + α ∗ ( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='Xk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='||Xk||) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='10: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='end if ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='11: Xk = Xk + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='Xk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='||Xk|| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='12: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='end for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='13: end for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Rise of magma through the volcanic pipe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Volcanic eruption by rising of magma to the surface of the earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Lava’s cooling down and therefore formation of a crust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Repetition of this process over time leading to several layers of rock that builds up over time resulting in a volcano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='Algorithm 4 VEA Procedure for Stowage Planning Problem ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='1: n: Number of solutions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='2: s: Size of the problem ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='3: Rand: Random integer number between 1 and s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='4: λ: A given integer positive number ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='5: Generate initial population ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='6: for t ← 1 to λ do ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='7: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='for k ← 1 to n do ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='8: Xk = Xk + Rand ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='Xk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='||Xk|| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='9: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='end for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='10: Find best solution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='11: Let Xt = xbest ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='12: end for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='13: for t ← 1 to λ do ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='14: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='for i ← 1 to n do ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='15: Xi = Xt + Rand ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='Xk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='||Xk|| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='16: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='end for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='17: Find best solution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='18: Let Y t = xbest ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='19: end for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='Eghbal Hosseini ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='Algorithm 5 MVA Procedure for Stowage Planning Problem ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='1: N: Number of solutions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='2: s: Size of the problem ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='3: Rand: Random integer number between 1 and s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='4: λ: A given integer positive number ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='5: m: Number of solutions for the next population ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='6: Generate initial population ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='7: for t ← 1 to λ do ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='8: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='for l ← 1 to N do ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='9: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='for k ← 1 to m do ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='10: Xk = Xk + Rand ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='Xk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='||Xk|| (Distribution of solutions in feasible region) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='11: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='end for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='12: Find best solution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='13: Let Xt = xbest ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='14: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='end for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='15: end for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='16: for t ← 1 to λ do ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='17: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='for i ← 1 to m do ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='18: Xi = Xt + Rand ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='Xk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='||Xk|| (Distribution of best solutions) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='19: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='end for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='20: Find best solution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='21: Let yt = xbest ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='22: end for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='5 Multiverse Algorithm (MVA) Algorithms 1-5 show the pseudocodes of SA, TS, LCA, VEA, and MVA algo- rithms respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' 4 Computational Results In this section, four algorithms: Tabu Search (TS), Simulated Annealing (SA), Laying Chicken Algorithm (LCA) and Volcano Eruption Algorithm (VEA) are implemented to solve some benchmarks of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Survey on Stowage Planning 7 5 10 15 20 25 30 35 20 40 60 80 100 (a) Unusable space (dark blue) before loading 5 10 15 20 25 30 35 20 40 60 80 100 (b) Initial solution by LCA Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Matrix format of ship before loading and initial solution by LCA 5 10 15 20 25 30 35 20 40 60 80 100 (a) Problem 1 5 10 15 20 25 30 35 20 40 60 80 100 (b) Problem 2 5 10 15 20 25 30 35 20 40 60 80 100 (c) Problem 3 5 10 15 20 25 30 35 20 40 60 80 100 (d) Problem 4 5 10 15 20 25 30 35 20 40 60 80 100 (e) Problem 5 5 10 15 20 25 30 35 20 40 60 80 100 (f) Problem 6 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Optimal Solutions problems 1-6 8 Eghbal Hosseini 5 10 15 20 25 30 35 20 40 60 80 100 (a) 10th Iteration-Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='1 5 10 15 20 25 30 35 20 40 60 80 100 (b) Best Solution- Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='1 5 10 15 20 25 30 35 20 40 60 80 100 (c) 10th Iteration-Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='2 5 10 15 20 25 30 35 20 40 60 80 100 (d) Best Solution- Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='2 5 10 15 20 25 30 35 20 40 60 80 100 (e) 10th Iteration-Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='3 5 10 15 20 25 30 35 20 40 60 80 100 (f) Best Solution- Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='3 5 10 15 20 25 30 35 20 40 60 80 100 (g) 10th Iteration-Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='4 5 10 15 20 25 30 35 20 40 60 80 100 (h) Best Solution- Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='4 5 10 15 20 25 30 35 20 40 60 80 100 (i) 10th Iteration-Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='5 5 10 15 20 25 30 35 20 40 60 80 100 (j) Best Solution- Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='5 5 10 15 20 25 30 35 20 40 60 80 100 (k) 10th Iteration-Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='6 5 10 15 20 25 30 35 20 40 60 80 100 (l) Best Solution- Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='6 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Behavior of LCA for benchmarks Survey on Stowage Planning 9 5 10 15 20 25 30 35 20 40 60 80 100 (a) 10th Iteration-Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='1 5 10 15 20 25 30 35 20 40 60 80 100 (b) Best Solution- Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='1 5 10 15 20 25 30 35 20 40 60 80 100 (c) 10th Iteration-Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='2 5 10 15 20 25 30 35 20 40 60 80 100 (d) Best Solution- Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='2 5 10 15 20 25 30 35 20 40 60 80 100 (e) 10th Iteration-Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='3 5 10 15 20 25 30 35 20 40 60 80 100 (f) Best Solution- Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='3 5 10 15 20 25 30 35 20 40 60 80 100 (g) 10th Iteration-Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='4 5 10 15 20 25 30 35 20 40 60 80 100 (h) Best Solution- Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='4 5 10 15 20 25 30 35 20 40 60 80 100 (i) 10th Iteration-Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='5 5 10 15 20 25 30 35 20 40 60 80 100 (j) Best Solution- Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='5 5 10 15 20 25 30 35 20 40 60 80 100 (k) 10th Iteration-Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='6 5 10 15 20 25 30 35 20 40 60 80 100 (l) Best Solution- Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='6 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Behavior of VEA for benchmarks 10 Eghbal Hosseini Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Results of SA for Benchmarks Problems Initial Solution 5 Iterations 10 Iterations Best Solution Inst0 31650 31350 10360 2570 Inst1 27440 27730 11230 1980 Inst2 27970 27980 9610 2380 Inst3 27780 27220 12760 3450 Inst4 27930 26540 9780 1210 Inst5 27790 27990 9110 2340 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Results of TS for Benchmarks Problems Initial Solution 5 Iterations 10 Iterations Best Solution Inst0 31570 31570 31260 1030 Inst1 31630 31630 31260 1210 Inst2 31640 31600 31220 1920 Inst3 32070 31880 31210 1880 Inst4 32020 31920 31230 2100 Inst5 31770 31420 31120 1750 References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Hosseini E, Reinhardt L, Rawat DB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Optimizing Gradient Methods for IoT Appli- cations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' IEEE Internet of Things Journal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' 2022 Jan 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Hosseini E, Reinhardt L, Ghafoor KZ, Rawat DB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Implementation and Comparison of Four Algorithms on Transportation Problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' InInternational Summit Smart City 360° 2022 (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' 422-433).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Springer, Cham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Hosseini E, Ghafoor KZ, Emrouznejad A, Sadiq AS, Rawat DB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Novel metaheuris- tic based on multiverse theory for optimization problems in emerging systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Applied Intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='11(37):1803-14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Hosseini E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Solving linear tri-level programming problem using heuristic method based on bi-section algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Asian J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='10:227-35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Survey on Stowage Planning 11 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Results of LCA for Benchmarks Problems Initial Solution 5 Iterations 10 Iterations Best Solution Inst0 31710 31310 9690 1740 Inst1 31600 31230 9720 2370 Inst2 31720 31290 9760 660 Inst3 31180 30950 9750 510 Inst4 26830 24120 9630 2060 Inst5 27130 26720 9680 930 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Results of VEA for Benchmarks Problems Initial Solution 5 Iterations 10 Iterations Best Solution Inst0 30590 28710 9690 1620 Inst1 26380 23560 9200 1630 Inst2 27420 24670 7420 620 Inst3 24330 21450 6740 440 Inst4 26690 22680 9390 1730 Inst5 26430 25270 9310 680 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Hosseini E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Laying chicken algorithm: a new meta-heuristic approach to solve con- tinuous programming problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' J Appl Comput Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='6(1):1-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Hosseini E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Big bang algorithm: A new meta-heuristic approach for solving opti- mization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Asian Journal of Applied Sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='10(3):134-44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Hosseini E, Kamalabadi IN, Daneshfar F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Solving Non-Linear Bi-Level Program- ming Problem Using Taylor Algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' InHandbook of Research on Modern Op- timization Algorithms and Applications in Engineering and Economics 2016 (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' 797-810).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' IGI Global.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Hosseini E, Kamalabadi IN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Line search and genetic approaches for solving linear tri-level programming problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Int J Manag Acc Econ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='1(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Hosseini E, Nakhai Kamalabadi I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Two approaches for solving non-linear bi-level programming problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Advances in Research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='3(5):512-25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Hosseini E, Kamalabadi IN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Bi-section algorithm for solving linear Bi-level pro- gramming problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='1:101-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Hosseini E, Kamalabadi IN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Smoothing and solving linear quad-level programming problem using mathematical theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='1:116-26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Hosseini E, Kamalabadi IN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' A modified simplex method for solving linear- quadratic and linear fractional bi-level programming problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Global Journal of Advanced Research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='1(2):142-54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Nakhai Kamalabadi I, Hosseini E, Fathi M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Enhancing the solution method of lin- ear Bi–level programming problem based on enumeration method and dual method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Journal of Advanced Mathematical Modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' 2014 Aug 23;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='4(1):27-53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Hosseini E, Kamalabadi IN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Line search and genetic approaches for solving linear tri-level programming problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Int J Manag Acc Econ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='1(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Hosseini E, Kamalabadi IN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Solving linear bi-level programming problem using two new approaches based on line search and taylor methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Int J Manage Sci Education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' 2014 Nov;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='2(6):243-52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' 12 Eghbal Hosseini Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Results of MVA for Benchmarks Problems Initial Solution 5 Iterations 10 Iterations Best Solution Inst0 31900 31510 7420 1300 Inst1 30300 29880 8760 1150 Inst2 28720 28210 8560 380 Inst3 30180 29850 990 360 Inst4 26230 25780 9880 1010 Inst5 28340 27200 7430 780 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Hussein E, Kamalabadi I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Taylor approach for solving nonlinear bilevel program- ming problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=', Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='3(5):91-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Hosseini E, Kamalabadi IN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Solving Linear-Quadratic Bi-Level Programming and Linear-Fractional Bi-Level Programming Problems Using Genetic Based Algo- rithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' Applied Mathematics and Computational Intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content=' This figure "1-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='jpg" is available in "jpg"� format from: http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E3T4oBgHgl3EQfWQru/content/2301.04468v1.pdf'} +page_content='org/ps/2301.' metadata={'source': 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Bauman,‡ Erdal +Mutlu,† John J. Rehr,¶ Fernando D. Vila,∗,¶ and Karol Kowalski∗,‡ +†Advanced Computing, Mathematics, and Data Division, Pacific Northwest National Laboratory, +Richland, Washington 99354, USA +‡Physical Sciences Division, Pacific Northwest National Laboratory, Richland, Washington +99354, United States +¶Department of Physics, University of Washington, Seattle, Washington 98195, United States +E-mail: himadri.pathak@pnnl.gov; fdv@uw.edu; karol.kowalski@pnnl.gov +Abstract +We report the implementation of the real-time +equation-of-motion +coupled-cluster +(RT-EOM- +CC) cumulant Green’s function method [J. Chem. +Phys. +152, 174113 (2020)] within the Tensor +Algebra for Many-body Methods (TAMM) in- +frastructure. TAMM is a massively parallel het- +erogeneous tensor library designed for utilizing +forthcoming exascale computing resources. The +two-body electron repulsion matrix elements are +Cholesky-decomposed, and we imposed spin- +explicit forms of the various operators when eval- +uating the tensor contractions. Unlike our previ- +ous real algebra Tensor Contraction Engine (TCE) +implementation, the TAMM implementation sup- +ports fully complex algebra. +The RT-EOM-CC +singles (S) and doubles (D) time-dependent am- +plitudes are propagated using a first-order Adams– +Moulton method. This new implementation shows +excellent scalability tested up to 500 GPUs using +the Zn-porphyrin molecule with 655 basis func- +tions, with parallel efficiencies above 90% up to +400 GPUs. +The TAMM RT-EOM-CCSD was +used to study core photo-emission spectra in the +formaldehyde and ethyl trifluoroacetate (ESCA) +molecules. +Simulations of the latter involve as +many as 71 occupied and 649 virtual orbitals. The +relative quasiparticle ionization energies and over- +all spectral functions agree well with available +experimental results. +Introduction +Photoemission spectroscopy (PES) is a widely +used spectroscopic probe, covering a broad energy +range from a few to several thousands of elec- +tronvolts (eV). The ubiquity of this technique is +due, in part, to the variety of instruments avail- +able, ranging from small laboratory-based ones to +synchrotron facilities.1,2 In the UV energy regime, +PES provides access to the valence electronic +structure, where low-energy electrons are the driv- +ing force of many chemical and biological pro- +cesses.3 In the X-ray regime, core or X-ray pho- +toelectron spectroscopy (XPS) is one of the most +commonly used fingerprinting method in materi- +als science, catalysis, and chemical engineering, +where it is used to investigate the composition +1 +arXiv:2301.04249v1 [physics.chem-ph] 10 Jan 2023 + +and chemistry of materials at the atomic level.4–6 +Given the importance of this experimental tech- +nique, complementary developments of accurate +theoretical methods and associated software for +the calculation and interpretation of XPS are cru- +cial. Thus, a variety of methods have been de- +veloped in both the frequency and time domains. +They are often used to predict the position of the +main transition or quasiparticle (QP) peak. How- +ever, only advanced electron correlation meth- +ods can accurately simulate details of core-level +photoemission spectra, particularly for the shake- +up peaks, since these many-body phenomena re- +flect a complex interplay between electron correla- +tion and orbital-reorganization effects. The time- +independent coupled-cluster (CC) method and its +many extensions7–16 have proven their utility in +recovering electron correlation in a variety of +problems for both ground and excited states. The +ground-state CC method is size-extensive at any +level of truncation of the excitation operators, and +scales polynomially with the number of active or- +bitals. +This makes these methods an attractive +choice over other electron correlation methods, +as they provide a balanced trade-off between the +computation cost and desired accuracy. Further- +more, it is possible to improve the results sys- +tematically by incorporating more correlated de- +terminantal spaces. The key feature of CC meth- +ods is the use of an exponential parameterization +of the correlated ground state wavefunction |Ψ⟩: +|Ψ⟩ = eT |Φ⟩, where T is the cluster operator, and +|Φ⟩ is a reference wavefunction, which is usually +but not necessarily a Hartree–Fock wavefunction. +The cluster operator is defined by order of ex- +citation, i.e. T = T1 + T2 + ··· + Tn correspond- +ing to singles, doubles, triples, ..., n-tuples, where +Tn = ( 1 +n!)2 ∑ i1,...,in +a1,...,an +ta1...an +i1...in a† +a1 ...a† +anain ...aii. Here +a† +p and ap are creation and annihilation operators, +respectively, associated with a set of Nso = 2Nbas +orthonormal spin-orbitals +� +φp +� +, with Nbas being +the number of basis functions. The indices in (an) +correspond to orbitals that are occupied (unoccu- +pied) with respect to the reference determinant. +When combined with the Green’s function +(GF) formalism in the frequency domain, the +CC method provides an avenue to treat excited- +state correlation effects that play a crucial role +in accurately simulating quasiparticles and satel- +lite peaks in XPS.17–26 However, for large sys- +tems, time-domain methods offer advantages over +their frequency-domain counterparts by trading off +memory resources for the serialization of the cal- +culation. Therefore, there has been a lot of effort in +developing efficient time-dependent approaches. +Hoodbhoy and Negele,27,28 and Sch¨onhammer +and Gunnarsson29 reported formulations of a +time-dependent CC theory at about the same time. +More recently, Kvaal proposed an orbital-adaptive +time-dependent coupled-cluster method30 rely- +ing on Arponen’s bi-orthogonal formulation of +CC theory,31 considering the complex analytic +action formulation of the time-dependent varia- +tional principle (TDVP). Sato et al. +have also +developed a time-dependent optimized coupled- +cluster method considering the real-action formu- +lation of the TDVP,32,33 to approximately solve +the time-dependent Schr¨odinger equation (TDSE) +as a polynomial cost-scaling alternative to multi- +configuration time-dependent Hartree–Fock meth- +ods.34–36 Both approaches30,32,33 choose to op- +timize the orbitals and ignore the one-body ex- +citation (T1) and de-excitation (Λ1) operators. +This approximation is well suited for strong-field +physics, where consideration of optimal orbitals +is crucial to obtain meaningful results. Such ap- +proaches provide a gauge-invariant description +of the time-dependent properties of interest and +satisfy the Ehrenfest theorem due to the use of +variationally optimized orbitals.37–40 Despite this +advantage, such methods30,32,33 are ill-suited for +large-scale applications, especially when simula- +tions involve core-hole states of chemical systems +containing many active occupied electrons. Other +developments in time-dependent electronic struc- +ture theory41 and time-dependent coupled-cluster +methods39,42–60 are reviewed elsewhere. +Recently, +we +have +developed +a +real-time +equation-of-motion +coupled-cluster +(RT-EOM- +CC) +cumulant +Green’s +function +method61–65 +building on the Sch¨onhammer and Gunnarsson +formulation of the TDCC.29 Subsequently sev- +eral applications to the XPS of small molecules +containing a few electrons in moderate size ba- +sis have been reported.62–65 In this methodology, +as described in more detail below, the Green’s +function has a natural exponential cumulant form, +2 + +which is given by solutions to a set of coupled, +first-order, nonlinear differential equations for the +time-dependent CC amplitudes. While the tradi- +tional cumulant approximation is linear in the one- +particle self-energy, the RT-EOM-CC approach +builds in high-order nonlinear, nonperturbative +contributions. +Even with their inherent memory usage advan- +tage, large-scale time-dependent simulations are +computationally challenging for chemically rele- +vant systems that go beyond a handful of corre- +lated electrons. +However, thanks to recent ad- +vances in high-performance computing techniques +that can take advantage of peta- and eventually +exascale computational resources, such calcula- +tions are no longer insurmountable. +In this ar- +ticle, we report the implementation of the RT- +EOM-CC method with single and double excita- +tions (RT-EOM-CCSD) within the Tensor Alge- +bra for Many-body Methods (TAMM) infrastruc- +ture.66,67 TAMM is a massively parallel heteroge- +neous tensor library designed for developing quan- +tum chemistry applications for forthcoming exas- +cale supercomputers. Our RT-EOM-CC code uses +Cholesky-decomposed two-electron repulsion ma- +trix elements68,69 that aid in reducing the mem- +ory requirements and inter-node communication. +Other speed-ups arise from spin-explicit evalua- +tion of the coupled-cluster amplitudes. +As dis- +cussed in the next section, the CC amplitudes in +RT-EOM-CC are naturally complex valued. +In +contrast, our original implementation based on the +real-valued Tensor Contraction Engine (TCE)62,64 +used separate real-valued data structures to rep- +resent the real and imaginary parts and required +two distinct subroutines to handle the propagation. +Since the new TAMM implementation uses ex- +plicit complex algebra, only a single subroutine is +required to handle the complex data, thus reducing +the coding and data intricacy, even though com- +plex algebra is more floating point operation in- +tensive. +To demonstrate the capabilities of this new im- +plementation, we study the core spectral functions +of the formaldehyde and ethyl trifluoroacetate +(ESCA) molecules and compare them with ex- +perimental spectra. We observe satisfying agree- +ment between the computed and available exper- +imental spectra. In addition to these systems, we +present parallel and storage performance for a few +nominally “large” systems such as Zn-porphyrins, +uracil, and the benzene-ammonia dimer. +Methods +Real-time Equation-of-Motion +Coupled-Cluster +Cumulant +Green’s +Function Method +The many-body Green’s function approach has +proved very useful for the calculation of spec- +tral functions of extended systems.70–73 By com- +bining this approach with the CC method, we +have developed a time-dependent CC cumulant +Green’s function method61 that integrates the ad- +vantages of both. +A detailed derivation of the +complete method can be found elsewhere.62–65 +In this section, we give a brief introduction to +the RT-EOM-CC formulation. +The goal is to +construct the retarded core-hole Green’s func- +tion GR +c (ω) and associated core spectral func- +tion Ac(ω) = (−1/π)ImGR +c (ω), by introduc- +ing +a time-dependent +coupled-cluster ansatz, +eiHτ |Ψ⟩ = |Ψ(τ)⟩ = Nc(τ)eT(τ) |Φ⟩, which is a +formal solution to the time-dependent Schr¨odinger +equation −i ∂ +∂τ |Ψ(τ)⟩ = H |Ψ(τ)⟩. Here τ is time, +and |Ψ(τ)⟩ is the fully correlated wavefunction +for the (N −1)-electron state, H = ∑pq hpqa† +paq + +1 +4 ∑pqrs vrs +pqa† +pa† +qasar is the nonrelativistic elec- +tronic Hamiltonian in second-quantization form, +hpq are the single-particle kinetic and electron- +nuclei spinorbital integrals, and vrs +pq = ⟨pq||rs⟩ are +the usual antisymmetrized two-particle Coulomb +integrals. +Nc(τ) and T(τ) are, respectively, the +time-dependent normalization constant and the +time-dependent coupled-cluster operator. In prin- +ciple, c can be any occupied orbital, but here we +focus on deep core excitations. +As usual, the +p,q,r,s indices indicate generic spin-orbital states. +In the current formulation, the time-independent +reference determinant |Φ⟩ is a single (N − 1)- +electron determinant formed from the N-electron +Hartree–Fock states where the state c has been +annihilated. +Thus, it is important to note that +T acts in the (N − 1)-electron space where c is +now included in the set of unoccupied single- +particle states. Following Ref. 61, the final form +3 + +of the equations of motion (EOMs) for Nc(τ), and +tab... +ij... (τ) are: +−i∂τ lnNc(τ) = ⟨Φ|(HNeT(τ))C|Φ⟩+EHF +N−1(1) +−i∂τ tab... +i j... (τ) = ⟨Φab... +ij... |(HNeT(τ))C|Φ⟩, +(2) +Here HN = ∑pq fpq{a† +paq}′+ 1 +4 ∑pqrs vrs +pq{a† +pa† +qasar}′, +and {}′ indicates that the normal ordering is done +with respect to |Φ⟩ instead of the usual N-electron +Hartree–Fock determinant. EHF +N−1 = ⟨Φ|H|Φ⟩ is +the Hartree–Fock energy of the (N − 1)-electron +systems. The Fock operator matrix elements are +fpq = εpδpq − vqc +pc, where c is the core-hole index +as used above. +The subscript ”C” designates a +connected part of a given operator expression. +Within this CC approximation, the retarded +core-hole GF GR +c (τ) is simply proportional to the +normalization factor Nc(τ), since Eq. 1 has an ex- +ponential solution, and therefore CR +c (τ), the re- +tarded cumulant associated with c, is proportional +to lnNc(τ): +GR +c (τ) += +−iΘ(τ)e−i(εc+Ecorr +N +)τNc(τ) +(3) += +−iΘ(τ)e−i(εc+Ecorr +N +)τeCRc (τ) +(4) +CR +c (τ) += +i +� τ +0 +� +Φ +���(HNeT(τ′))C +���Φ +� +dτ′. (5) +Here εc is the single-particle Hartree–Fock energy +of the core orbital c, and Ecorr +N +corresponds to the +correlation energy of the N-electron closed-shell +ground state. As a consequence, GR +c (τ) can be ex- +pressed as the product of the free particle GF and +the exponential of a cumulant, as expected within +the cumulant approximation. We can now write +down from Eq. 1 a complete expression for the +time derivative of CR +c (τ) within the single and dou- +ble excitations approximation (CCSD): +−i∂τCR +c = ⟨Φ|[HN,T2(τ)]|Φ⟩ ++ 1 +2 ⟨Φ|[[HN,T1(τ)],T1(τ)]|Φ⟩ +(6) +Unlike conventional linear self-energy formula- +tions, it is evident from Eq. 6 that the CCSD cu- +mulant includes nonlinear, nonperturbative contri- +butions. +The EOMs for the amplitudes within the CCSD +Perform N-electron RHF calculation +Perform N-electron closed-shell + CCSD calculation +Reorder orbitals keeping an arbitrary +core hole +Reconstruct the Fock operator in the + N-1 electron Fock-space +Perform time-propagation and compute +the cumulant in each time-step +Construct Green’s function and Fourier +transform it to get the spectral function +Photoemission +Spectra +Figure 1: Real-time equation-of-motion coupled- +cluster workflow. +approximation are obtained from Eq. 2 as +−i∂τta +i (τ) = +� +Φa +i |(HNeT1(τ)+T2(τ))C|Φ +� +(7) +−i∂τtab +i j (τ)) = +� +Φab +i j |(HNeT1(τ)+T2(τ))C|Φ +� +, (8) +where ta +i (τ) and tab +i j (τ)) are time-dependent singly +and doubly excited cluster amplitudes. +Numerical Solution of the EOMs +To construct the core-hole spectral function Ac(ω) +we need to compute the Green’s function GR +c (τ), +which in turn depends on the cumulant CR +c (τ) +over the whole simulation range (Eqs. 5 and 6). +Thus, the main task is to propagate Eqs. 7 and +8 in time, which provide solutions for both the +T1 and T2 amplitudes needed to compute the cu- +4 + +mulant CR +c (τ). Fig. 1 demonstrates a typical RT- +EOM-CC workflow. Given that the RT-EOM-CC +uses ground-state orbitals and the closed-shell CC +ground-state energy, stationary restricted Hartree– +Fock (RHF) and closed-shell coupled-cluster cal- +culations are prerequisites to the time-dependent +simulation. The computational cost of the RHF +calculations is O(N3 +bas) while that for the CC part +varies, depending on the imposed truncation in the +excitation operator. The singles and doubles ap- +proximation used in this work scales as O(O2V 4), +where O and V denote the total number of occu- +pied and virtual orbitals, respectively, and O+V = +Nbas. +The first two ground-state calculation steps are +common to most simulations involving core-hole +excitations, but these are not the main bottlenecks +since they need to be performed only once. Af- +ter these, all calculations involving different core- +hole states are unique and can run simultaneously. +These calculations start by reading the N-electron +closed-shell orbitals into the (N − 1)-electron or- +bital space and then constructing the (N − 1)- +electron Fock matrix. +At this point, the time- +dependent simulation can start, where we propa- +gate the CCSD amplitudes forward for the desired +time, computing the time-derivative of the cumu- +lant at each time step. It is worth pointing out that +given that the time-dependent simulations involve +an initial core hole, this is an open-shell problem +and is roughly three times more expensive than its +closed-shell counterpart in spin-explicit form. +As in our previous implementations, the first- +order coupled nonlinear simultaneous differential +equations for the amplitudes are integrated us- +ing the first-order Adams–Moulton method,74 also +known as the implicit trapezoidal rule. In this ap- +proximation the tab... +ij... (τ) are propagated with: +tab... +i j... (τ +∆τ) = tab... +ij... (τ) ++ i +2∆τ +�� +Φab... +ij... +���(HNeT(τ))C +���Φ +� ++ +� +Φab... +ij... +���(HNeT(τ+∆τ))C +���Φ +�� +, +(9) +where ∆τ is the simulation time-step. The propa- +gation starts with the initial conditions T1 = 0 and +T2 = 0. Since the Adams–Moulton method is im- +Applications +CCSD +CCSD(T) EOMCC +GFCC +DLPNO- +CC +… +Tensor Algebra Interface +Distribution +Memory +Management +Execution +Context +Scheduler +TAMM +Dependencies +Global +Arrays/ +MPI +Vendor +BLAS +(CPU/ +GPU) +LibreTT +HPTT +HDF5(I/O) +Figure 2: Overview of the Tensor Algebra for +Many-body Methods (TAMM) framework. +plicit, the values of the amplitudes at (τ +∆τ) de- +pend on themselves. +Thus, to solve Eq. +9 we +use a fixed-point iteration scheme at each time +step. Other methods of solution that use a vari- +able time step and are more stable than fixed-point +iteration are currently under development. After +completion of the time propagation, the remainder +of the workflow is not compute-intensive, since it +only involves forming the time-dependent Green’s +function from the cumulant, and then Fourier- +transforming to obtain the spectral function in the +frequency domain. +RT-EOM-CC Implementation in Ten- +sor Algebra for Many-body Methods +Infrastructure +The Tensor Algebra for Many-body Methods +(TAMM) library66,80 is a massively parallel, het- +erogeneous tensor algebra library. +It provides +a computational infrastructure (see Fig. 2) that +can achieve scalable performance. Furthermore, +it allows portable implementations of many-body +methods both on existing and forthcoming exas- +cale super-computing platforms. +High-dimensional tensor contractions are the +most compute-intensive components of the RT- +EOM-CC method. Within the single and double +excitations approximation (RT-EOM-CCSD), the +most expensive tensor contraction is of the form +R(V,V,O,O) = α × v(V,V,V,V) × t2(V,V,O,O), +involving four-dimensional tensors, the antisym- +metrized two-body matrix elements v, and the +two-body CC excitation operator t2, α is a scalar +pre-factor. These multi-dimensional tensor con- +5 + +Table 1: Structural features of TAMM, including various used third-party dependency libraries. +Third-party dependencies +Global Arrays,75 BLIS,76 vendor BLAS/LAPACK, +cuBLAS/rocBLAS/oneMKL +HPTT,77 TALSH,78 LibInt279 +Programming Languages +C++17, CUDA, HIP, SYCL, MPI, OpenMP +Precision +Double +Data Types +Real, Complex +Supports Restart Capabilities? +Yes +I/O requirements +Minor +tractions are not only computational- but also +communication-intensive. There have been many +efforts to develop specialized parallel tensor alge- +bra libraries, including automated code generators +and better memory management to facilitate these +demanding tensor contractions.81–84 The TAMM +library is one such effort aiming to achieve scal- +able performance on several heterogeneous archi- +tectures, by delivering a common platform for the +portable implementation of numerous many-body +methods. TAMM provides a variety of features +to users including the ability to specify and ma- +nipulate tensor distribution, memory management, +and scheduling of tensor operations. In addition, +it supports both complex and mixed real-complex +algebra for mathematical operations. +A summary of structural features and third-party +libraries for the TAMM library is shown in Table 1. +TAMM uses Global Arrays75 and Message Pass- +ing Interface (MPI) to achieve scalable paralleliza- +tion on distributed memory platforms, while using +optimized libraries that help efficient intra-node +execution of the tensor operation, both in CPU +kernels and accelerators. TAMM also uses multi- +granular dependence analysis and task-based exe- +cution to execute operations. First, it constructs a +macro operation graph by analyzing the dependen- +cies between various operations. When two oper- +ations share the same data structure, with one of +them writing to the other, they are in conflict and +impossible to execute in parallel. The operation +graph is analyzed to identify and order the non- +parallel operations to minimize the required num- +bers of synchronizations. The possible scheduled +operations are executed in a single program multi- +ple data fashion. Such executions are compatible +with MPI, and their collective executions are per- +formed on a given MPI communicator. Various +tasks constitute an operation, which is produced +using task iterators. Each task performs part of the +computation, usually adding a block of data to the +output tensor. A given task is migratable and can +be scheduled for execution on any compute node +or processor core until its execution begins. The +data needed for a job are transported to its location +once the execution of the process has started. At +this stage, migration of tasks is no longer possible +and they are bound to process. TAMM’s GPU ex- +ecution scheme uses localized summation loops to +minimize the transfer of output blocks from GPUs +to CPUs. This helps to reduce the data transmis- +sion between CPUs and GPUs by keeping the out- +put block that is being updated by several input +tensor blocks on the GPU until all updates are +complete. +Programming Models, Software Dependencies, +I/O, Restart/Checkpoint Capabilities +Memory demands, operation count, and time-to- +solution are three main concerns that limit large- +scale CC calculations. This is further complicated +when extending the CC formalism to the time do- +main. To combat these hindrances, various tech- +niques and features were incorporated during the +implementation of the RT-EOM-CCSD method. +In canonical spin-orbital CC calculations, the +four-dimensional electron repulsion integral (ERI) +tensors are easily the largest memory-demanding +objects. The storage requirement for the ERI ten- +sor in its full spin-orbital form is of order N4 +so, and +they must undergo a tensor transformation from +the atomic-orbital to molecular-orbital basis which +scales as O(N5 +bas). In our RT-EOM-CC implemen- +tation, we employed Cholesky decomposition85–89 +6 + +of the ERI tensors to ease the memory/storage de- +mands and increase the data locality resulting in +reduced communication. We leveraged the same +Cholesky decomposition previously implemented +using the TAMM library,68,69 which is an on- +the-fly pivoting decomposition of the two-body +ERIs. +The resulting Cholesky bases are three- +index quantities that reduce the storage require- +ments from order N4 +so to KN2 +so, where K ∼ O(Nbas) +is the number of Cholesky bases. In addition, the +atomic-orbital to molecular-orbital transformation +is conducted on the Cholesky bases rather than di- +rectly on the ERI, thus reducing the scaling of the +transformation from O(N5 +bas) to O(N4 +bas). The ac- +curacy of the correlation calculations employing +the Cholesky bases in comparison with the canon- +ical results is well-controlled through adjusting the +diagonal cutoff in the Cholesky decomposition. +The next largest memory-demanding objects, af- +ter the ERIs, are the T2 amplitudes. In a naive spin- +orbital implementation, the memory requirement +for all elements of the T2 operator is 16O2V 2. The +declaration includes 16 possible combinations of +spins for the four indices, most of which do not +contribute or are over-specified as they are equiv- +alent through permutational symmetry. In our RT- +EOMCC methodology, we implemented the spin- +integrated form of equations. Only unique spin +combinations of tensors with a non-zero contri- +bution, including intermediates, are programmed. +For the T2 amplitudes, this means only three spin +cases are necessary (tαα +αα , tββ +ββ , and tαβ +αβ ), reduc- +ing the memory requirements to ∼ 3 +16 that of the +full T2 operator. +Memory requirements are re- +duced for other operators as well. +Since non- +contributing spin combinations are removed, and +only unique spin cases are specified, the over- +all operation count is significantly reduced. The +memory requirements of various systems with var- +ious large systems with 760 to 2450 spin-orbital +functions can be seen in Table 2. In this Table, the +memory requirement for the cluster amplitudes re- +flects the sum of the three timeline tensors needs +for the iterative update given by Eq. 9. The com- +bination of Cholesky-decomposed ERIs and spin- +integrated equations allows for simulations with +hundreds of orbitals on moderately sized computer +clusters. +Another challenging aspect of RT-EOM-CC cal- +culations is that it is necessary to propagate for a +sufficiently long time to have well-resolved spec- +tra. In practice, shared computing resource usually +do not allow simulations to propagate for enough +time in a single run to achieve sufficient reso- +lution. +Furthermore, it is important to be able +to track simulations and adjust/optimize the time +propagation parameters promptly, before a long +time has elapsed. For these reasons, checkpoint- +ing and restart capabilities were imperative in our +RT-EOM-CCSD implementation. Restarting the +time-propagation algorithm at any ith step requires +the T amplitudes of the (i−1)th and ith time steps, +in addition to the Fock and Cholesky-decomposed +ERIs. Parallel read and write capabilities in the +TAMM library allow for periodic checkpointing +while minimizing their impact on the overall com- +putational time. +Geometries, Basis Sets, and Computa- +tional Details +In this study, we perform RT-EOM-CC simula- +tions of two molecules, formaldehyde and ethyl +trifluoroacetate (ESCA), in order to compare to +previous theoretical and experimental results. For +formaldehyde, we used the experimental geom- +etry90 and studied both C and O core ioniza- +tions using either the aug-cc-pVDZ,91 aug-cc- +pVTZ,91 or Sapporo-TZP92,93 basis sets for all +atoms in the molecule. +Given that the ESCA +molecule contains four C atoms, we computed ion- +ization spectra from each of them. +These cal- +culations were performed with the Sapporo-TZP +basis set for all the first-row elements, while for +the H atom we used the aug-cc-pVDZ basis set. +Since a complete experimental geometry of the +ESCA molecule is not available, we used the +one obtained from a B3LYP/aug-cc-pVTZ opti- +mization.94 The real-time time-propagation used a +time-step of 0.015 au (∼0.36 as) for formaldehyde +and 0.01 au (∼0.24 as) for the ESCA molecule, +with a convergence cutoff of 10−4 for the fixed- +point micro-iteration solution of the implicit first- +order Adams–Moulton integrator. The total prop- +agation time was 450 au (∼11 fs) for formalde- +hyde and 100 au (∼2.5 fs) for ESCA. All sta- +tionary calculations were performed with a linear +7 + +Table 2: Workflow memory requirements for the TAMM implementation of RT-EOM-CCSD for systems +with Sapporo-TZP basis set for all atoms except H, for which we use aug-cc-pVTZ, using a Cholesky +vectors diagonal cutoff of 10−6, and linear dependence threshold of 10−6. +System +Configuration Space +T2 +Cholesky +# of Cholesky +amplitude +vectors +vectors +Nso +nα +occ +nβ +occ +nα +vir +nβ +vir +(GB) +(GB) +Uracil +760 +29 +28 +351 +352 +3×4.5 +26.2 +2044 +ESCA +880 +36 +35 +404 +405 +3× 9.2 +40.6 +2423 +Benzene-Ammonia +940 +26 +25 +444 +445 +3×5.8 +49.5 +2457 +Zn-porphyrin +2450 +95 +94 +1130 +1131 +3× 510.2 +876.6 +6258 +dependence threshold for the basis sets of 10−6, +SCF convergence cutoff of 10−8 au for the energy, +Cholesky diagonal cutoff of 10−6 and a CCSD +convergence cutoff of 10−8 au. None of the vir- +tual orbitals are frozen in any of our simulations. +Results and Discussion +Performance Analysis +As described in previous sections, a simulation is +divided into a series of “macro-iterations” associ- +ated with each time step, and within each macro- +iteration, many “micro-iterations” are performed +to solve Eq. 9. Thus, the fundamental parallel per- +formance metric for RT-EOM-CC is the “time per +micro-iteration” associated with the calculation of +the matrix elements in Eq. 9. In order to inves- +tigate the performance and scalability of the new +optimized TAMM implementation of RT-EOM- +CCSD, we have performed a series of calculations +on the “pre-production” NERSC Perlmutter sys- +tem (Nvidia A100). If the scaling is ideal, the total +compute time is inversely proportional to the to- +tal number of allocated processors provided the to- +tal number of mathematical operations in each test +case remains the same. Figure 3 shows the scala- +bility of the TAMM implementation of RT-EOM- +CCSD for Zn-porphyrin using the aug-cc-pVDZ +basis set, which results in a total of 655 basis func- +tions (nocc=94, nvir=561) after pruning 122 linear +dependencies. Our time-dependent simulations in- +volve a configuration space of nα +occ = 95, nα +vir = 560, +nβ +occ = 94, nβ +vir = 561. We explore scaling between +200 and 500 GPUs (50 to 125 nodes given that +each node is connected to 4 GPUs), using the per- +200 +250 +300 +350 +400 +450 +500 +# of GPUs +150 +200 +250 +300 +350 +Time (s) +Real +Ideal +Figure 3: Real vs ideal single micro-iteration time +as a function of number of GPUs for Zn-porphyrin +using 655 basis functions. +formance with 200 GPUs as the reference. Figure +3 also shows the ideal theoretical scaling. In this +range, we observe a nearly ideal drop in the com- +putation time for each micro-iteration, very close +to the theoretically limit. The parallel efficiency +remains high (i.e. +higher than 94%) up to 400 +GPUs, after which it drops to 83% for 500 GPUs. +Spectral Function Results +The high quality of the RT-EOM-CC results has +been previously demonstrated.63,65 Thus here we +focus on showcasing the new capabilities of the +RT-EOM-CCSD TAMM implementation for more +complex systems. For this purpose, we have simu- +lated formaldehyde (H2CO) and ethyl trifluoroac- +etate (ESCA). For formaldehyde, we focus on the +satellite region of the C and O core spectral func- +8 + +-30 +-20 +-10 +0 +10 +E - EQP (eV) +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +0.12 +0.14 +Spectral Function (1/eV) +XPS (Expt) +aug-cc-pVDZ +aug-cc-pVTZ +Sapporo-TZP +H2CO +-30 +-20 +-10 +0 +10 +E - EQP (eV) +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +0.12 +0.14 +Spectral Function (1/eV) +XPS (Expt) +aug-cc-pVDZ +aug-cc-pVTZ +Sapporo-TZP +H2CO +Figure 4: Comparison of satellite regions of the +C (top) and O (bottom) RT-EOM-CCSD core +spectral functions to the XPS experiment95 for +formaldehyde (H2CO) as a function of basis set. +The data has been shifted so that the quasiparticle +peak is at 0 eV. +tions (Fig. 4). We find that the theory reproduces +the XPS semi-quantitatively, with the position of +most of the satellite features relative to the quasi- +particle peak in reasonable agreement with exper- +iment. The relative intensities of the peaks are not +as well reproduced, probably due to i) the limita- +tions of the local valence basis set used that misses +the continuum background contribution, and ii) the +limitations of the RT-EOM-CCSD to include all +the relevant excitations in this energy range. +For the case of the ESCA molecule, shown in +Fig. 5, we calculated the C core spectral function +for each of the inequivalent C atoms in the sys- +tem. The agreement with experiment is excellent, +| +| +| +| +| +| +| +| +290 +292 +294 +296 +298 +300 +Binding Energy (eV) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +Spectral Function (1/eV) +C(H2OC) +C(O2C) +C(F3C) +C(H3C) +-0.84 +-0.87 +-0.89 +-0.97 +CH3-CH2-O-CO-CF3 +XPS (Expt) +RT-EOM-CCSD +RT-EOM-CCSD (Shft) +Figure +5: +Comparison +of +the +RT-EOM- +CCSD/Sapporo-TZP C core spectral functions +(red) to the experimental94 XPS (black dots) +for the ethyl trifluoroacetate (ESCA) molecule. +Each peak corresponds to an individual C core +ionization and has been broadened to match the +vibrational experimental broadening. Also shown +are the same results shifted by 0.89 eV (blue). +apart from a nearly-constant overall underestima- +tion of the binding energies. When the underesti- +mation shift is removed, the relative mean absolute +error is only 0.04 eV. We also find that, unlike for +the satellite peaks in H2CO, the relative intensities +of the ESCA quasiparticle peaks are qualitatively +reproduced by the theory. By fitting the experi- +ment to a skew Gaussian distribution (to account +for the vibrational asymmetry), we find that the +intensities of the C(H2OC), C(O2C) and C(F3C) +peaks relative to the C(H3C) one are 0.81, 0.90 and +0.91, respectively, while for the theory the ratios +are 0.94, 0.91 and 0.96. It was speculated94 that +some of the intensity of the different C quasiparti- +cle peaks might originate from underlying satellite +peaks from lower energy cores. We find that for +each of the individual spectral functions the first +satellite peaks appear more than 10 eV above the +quasiparticle, and thus all the intensity observed +for the peaks between 291 and 299 eV can be as- +signed exclusively to the quasiparticle transitions. +9 + +Conclusions +We have successfully implemented the RT-EOM- +CCSD method within the parallel TAMM infras- +tructure using Cholesky decomposed two-body re- +pulsion matrix elements. +This implementation +eliminates the memory bottleneck of the original +approach associated with storing two-electron in- +tegrals. Unlike our earlier TCE-based RT-EOM- +CCSD implementation, which relied only on real +algebra, our new TAMM implementation supports +explicit complex algebra. +This implementation +is also flexible regarding the choice of the ref- +erence function (i.e. where the hole state is lo- +cated), and has checkpointing/restart capabilities +at any stage of the workflow. This is quite impor- +tant since the propagation portion of the workflow +can be very time-consuming and restarts are usu- +ally needed in shared computing systems. More- +over, the TAMM RT-EOM-CCSD shows very +good scalability, paving the way for simulations +of larger, more realistic and chemically relevant +systems, employing larger basis sets in conjunc- +tion with reduced memory requirements. Illustra- +tive calculations for the formaldehyde and ESCA +molecules demonstrate that the predicted positions +for the quasiparticle and satellite peaks are in good +agreement with experimental values. In particu- +lar, the RT-EOM-CCSD reproduced the relative +position of the different core ionizations in the +ESCA molecule, highlighting its capabilities to +study chemical speciation. The method describes +the positions of the satellite peaks without any cor- +rections to the quasiparticle-satellite gap. +Finally, work is in progress on an atomic orbital- +based implementation, coupled to more efficient +solvers for the implicit Adams–Moulton propaga- +tor that should reduce the computational time by at +least an order of magnitude. Other future method- +ological developments include the implementation +of spin-orbit coupling, and the multi-component +coupled-cluster formalism. These extensions will +allow first-principles studies of multielectron dy- +namics in previously unreachable large chemical +systems, such as simulations involving multiple +core holes. +Acknowledgement This work was supported by +the Computational Chemical Sciences Program of +the U.S. Department of Energy, Office of Sci- +ence, BES, Chemical Sciences, Geosciences and +Biosciences Division in the Center for Scalable +and Predictive methods for Excitations and Cor- +related phenomena (SPEC) at PNNL, with com- +putational support from NERSC, a DOE Office +of Science User Facility, under contract no. DE- +AC02-05CH11231. B.P. also acknowledges sup- +port from the Laboratory Directed Research and +Development (LDRD) Program at PNNL. +References +(1) Meirer, F.; Weckhuysen, B. M. Spatial and +temporal exploration of heterogeneous cat- +alysts with synchrotron radiation. Nat. Rev. +Mater. 2018, 3, 324–340. +(2) Qiao, S.; He, Q.; Zhang, P.; Zhou, Y.; +Chen, S.; Song, L.; Wei, S. Synchrotron- +radiation +spectroscopic +identification +to- +wards diverse local environments of single- +atom catalysts. J. Mater. Chem. A 2022, 10, +5771–5791. +(3) Kunin, A.; Neumark, D. M. Time-resolved +radiation chemistry: +femtosecond photo- +electron spectroscopy of electron attach- +ment and photodissociation dynamics in +iodide–nucleobase clusters. Phys. Chem. +Chem. Phys. 2019, 21, 7239–7255. +(4) Seah, M. The quantitative analysis of sur- +faces by XPS: A review. Surf. Interface Anal. +1980, 2, 222–239. +(5) Bagus, P. S.; Ilton, E. S.; Nelin, C. J. The +interpretation of XPS spectra: Insights into +materials properties. Surf. Sci. Rep. 2013, 68, +273–304. +(6) Shchukarev, A. XPS at solid–aqueous so- +lution interface. Adv. Colloid Interface Sci. +2006, 122, 149–157. +(7) Coester, F. Bound states of a many-particle +system. Nucl. Phys. 1958, 7, 421–424. +(8) Coester, F.; K¨ummel, H. Short-range correla- +tions in nuclear wave functions. Nucl. Phys. +1960, 17, 477–485. +10 + +(9) ˇC´ıˇzek, J. On the Correlation Problem in +Atomic and Molecular Systems. Calculation +of Wavefunction Components in Ursell-Type +Expansion Using Quantum-Field Theoretical +Methods. J. Chem. Phys. 1966, 45, 4256– +4266. +(10) Paldus, J.; +ˇC´ıˇzek, J.; +Shavitt, I. Corre- +lation Problems in Atomic and Molecular +Systems. IV. Extended Coupled-Pair Many- +Electron Theory and Its Application to the +BH3 Molecule. Phys. Rev. A 1972, 5, 50–67. +(11) K¨ummel, H. G. A biography of the coupled +cluster method. Int. J. Mod. Phys. B 2003, 17, +5311–5325. +(12) Bartlett, R. J.; Musiał, M. Coupled-cluster +theory in quantum chemistry. Rev. Mod. +Phys. 2007, 79, 291–352. +(13) Mukherjee, D.; Pal, S. Use of cluster ex- +pansion methods in the open-shell correla- +tion problem. Adv. Quantum Chem. 1989, 20, +291–373. +(14) Krylov, A. I. Equation-of-motion coupled- +cluster methods for open-shell and electroni- +cally excited species: the Hitchhiker’s guide +to Fock space. Annu. Rev. Phys. Chem. 2008, +59, 433–462. +(15) Piecuch, P.; Kowalski, K.; Pimienta, I. S.; +Mcguire, M. J. Recent advances in electronic +structure theory: +Method of moments of +coupled-cluster equations and renormalized +coupled-cluster approaches. Int. Rev. Phys. +Chem. 2002, 21, 527–655. +(16) Crawford, T. D.; Schaefer, H. F. An introduc- +tion to coupled cluster theory for computa- +tional chemists. Rev. Comput. Chem. 2000, +14, 33–136. +(17) Nooijen, M.; Snijders, J. G. Coupled cluster +approach to the single-particle Green’s func- +tion. Int. J. Quantum Chem. 1992, 44, 55–83. +(18) Nooijen, M.; Snijders, J. G. Coupled cluster +Green’s function method: Working equations +and applications. Int. J. Quantum Chem. +1993, 48, 15–48. +(19) Nooijen, M.; Snijders, J. G. Second or- +der many-body perturbation approximations +to the coupled cluster Green’s function. J. +Chem. Phys. 1995, 102, 1681–1688. +(20) Meissner, L.; Bartlett, R. J. Electron propa- +gator theory with the ground state correlated +by the coupled-cluster method. Int. J. Quan- +tum Chem. 1993, 48, 67–80. +(21) Bhaskaran-Nair, K.; Kowalski, K.; Shel- +ton, W. A. Coupled cluster Green function: +Model involving single and double excita- +tions. J. Chem. Phys. 2016, 144, 144101. +(22) Peng, B.; +Kowalski, K. Coupled-cluster +Green’s function: +Analysis of properties +originating in the exponential parametriza- +tion of the ground-state wave function. Phys. +Rev. A 2016, 94, 062512. +(23) Peng, B.; Kowalski, K. Properties of ad- +vanced coupled-cluster Green’s function. +Mol. Phys. 2018, 116, 561–569. +(24) Peng, B.; Bauman, N. P.; Gulania, S.; Kowal- +ski, K. Annu. Rep. Comput. Chem.; Elsevier, +2021; Vol. 17; pp 23–53. +(25) Shee, A.; Zgid, D. Coupled Cluster as an im- +purity solver for Green’s function embedding +methods. J. Chem. Theory Comput. 2019, 15, +6010–6024. +(26) Lange, M. F.; Berkelbach, T. C. On the re- +lation between equation-of-motion coupled- +cluster theory and the GW approximation. +J. Chem. Theory Comput. 2018, 14, 4224– +4236. +(27) Hoodbhoy, P.; Negele, J. Time-dependent +coupled-cluster approximation to nuclear dy- +namics. I. Application to a solvable model. +Phys. Rev. C 1978, 18, 2380. +(28) Hoodbhoy, P.; Negele, J. Time-dependent +coupled-cluster approximation to nuclear dy- +namics. II. General formulation. Phys. Rev. C +1979, 19, 1971. +(29) Sch¨onhammer, K.; Gunnarsson, O. Time- +dependent approach to the calculation of +11 + +spectral functions. Phys. Rev. B 1978, 18, +6606. +(30) Kvaal, S. Ab initio quantum dynamics using +coupled-cluster. J. Chem. Phys. 2012, 136, +194109. +(31) Arponen, +J. +Variational +principles +and +linked-cluster exp S expansions for static +and dynamic many-body problems. Ann. +Phys. 1983, 151, 311–382. +(32) Sato, +T.; +Pathak, +H.; +Orimo, +Y.; +Ishikawa, K. L. Communication: +Time- +dependent +optimized +coupled-cluster +method +for +multielectron +dynamics. +J. +Chem. Phys. 2018, 148, 051101. +(33) Pathak, H.; Sato, T.; Ishikawa, K. L. Time- +dependent optimized coupled-cluster method +for multielectron dynamics. II. A coupled +electron-pair approximation. J. Chem. Phys. +2020, 152, 124115. +(34) Caillat, J.; +Zanghellini, J.; +Kitzler, M.; +Koch, O.; Kreuzer, W.; Scrinzi, A. Corre- +lated multielectron systems in strong laser +fields: A multiconfiguration time-dependent +Hartree-Fock approach. Phys. Rev. A 2005, +71, 012712. +(35) Kato, T.; Kono, H. Time-dependent multi- +configuration theory for electronic dynamics +of molecules in an intense laser field. Chem. +Phys. Lett. 2004, 392, 533–540. +(36) Sato, T.; Ishikawa, K. L. Time-dependent +complete-active-space +self-consistent-field +method for multielectron dynamics in in- +tense laser fields. Phys. Rev. A 2013, 88, +023402. +(37) Ishikawa, K. L.; Sato, T. A review on ab ini- +tio approaches for multielectron dynamics. +IEEE J. Sel. Top. Quantum Electron 2015, +21, 1–16. +(38) Sato, +T.; +Pathak, +H.; +Orimo, +Y.; +Ishikawa, +K. +L. +Time-dependent +multi- +configuration self-consistent-field and time- +dependent optimized coupled-cluster meth- +ods for intense laser-driven multielectron +dynamics. arXiv preprint arXiv:2211.10116 +2022, +(39) Pathak, H.; Sato, T.; Ishikawa, K. L. Time- +dependent optimized coupled-cluster method +for multielectron dynamics. III. A second- +order many-body perturbation approxima- +tion. J. Chem. Phys. 2020, 153, 034110. +(40) Pathak, H.; Sato, T.; Ishikawa, K. L. Study +of laser-driven multielectron dynamics of +Ne atom using time-dependent optimised +second-order many-body perturbation the- +ory. Mol. Phys. 2020, 118, e1813910. +(41) Li, X.; +Govind, N.; +Isborn, C.; +De- +Prince III, A. E.; Lopata, K. Real-time time- +dependent electronic structure theory. Chem. +Rev. 2020, 120, 9951–9993. +(42) Monkhorst, H. J. Calculation of Properties +with the Coupled-Cluster Method. Int. J. +Quantum Chem. 1977, 12, 421–432. +(43) Mukherjee, D.; Mukherjee, P. A response- +function approach to the direct calculation of +the transition-energy in a multiple-cluster ex- +pansion formalism. Chem. Phys. 1979, 39, +325–335. +(44) Guha, S.; Mukherjee, D. A multireference +coupled-cluster approach to quantum dy- +namics. Chem. Phys. Lett. 1991, 186, 84–90. +(45) Dalgaard, E.; Monkhorst, H. J. Some aspects +of the time-dependent coupled-cluster ap- +proach to dynamic response functions. Phys. +Rev. A 1983, 28, 1217. +(46) Koch, H.; Jørgensen, P. Coupled cluster re- +sponse functions. J. Chem. Phys. 1990, 93, +3333–3344. +(47) Takahashi, M.; Paldus, J. Time-dependent +coupled cluster approach: +Excitation en- +ergy calculation using an orthogonally spin- +adapted formalism. J. Chem. Phys. 1986, 85, +1486–1501. +(48) Prasad, M. D. Time-dependent coupled clus- +ter method: +A new approach to the cal- +culation of molecular absorption spectra. J. +Chem. Phys. 1988, 88, 7005–7010. +12 + +(49) Sebastian, K. Correlation effects in ion neu- +tralization scattering with the use of a time- +dependent coupled-cluster approach. Phys. +Rev. B 1985, 31, 6976. +(50) Pigg, D. A.; Hagen, G.; Nam, H.; Papen- +brock, T. Time-dependent coupled-cluster +method for atomic nuclei. Phys. Rev. C 2012, +86, 014308. +(51) Nascimento, D. R.; +DePrince III, A. E. +Linear absorption spectra from explicitly +time-dependent equation-of-motion coupled- +cluster theory. J. Chem. Theory Comput. +2016, 12, 5834–5840. +(52) Nascimento, D. R.; +DePrince III, A. E. +Simulation of near-edge X-ray absorption +fine structure with time-dependent equation- +of-motion coupled-cluster theory. J. Phys. +Chem. Lett. 2017, 8, 2951–2957. +(53) Koulias, L. N.; +Williams-Young, D. B.; +Nascimento, D. R.; DePrince III, A. E.; +Li, X. Relativistic real-time time-dependent +equation-of-motion +coupled-cluster. +J. +Chem. Theory Comput. 2019, 15, 6617– +6624. +(54) Nascimento, D. R.; +DePrince III, A. E. +A +general +time-domain +formulation +of +equation-of-motion coupled-cluster theory +for linear spectroscopy. J. Chem. Phys. 2019, +151, 204107. +(55) Pathak, H.; Sato, T.; Ishikawa, K. L. Time- +dependent optimized coupled-cluster method +for multielectron dynamics. IV. Approximate +consideration of the triple excitation ampli- +tudes. J. Chem. Phys. 2021, 154, 234104. +(56) Park, Y. C.; Perera, A.; Bartlett, R. J. Equa- +tion of motion coupled-cluster for core ex- +citation spectra: +Two complementary ap- +proaches. J. Chem. Phys. 2019, 151, 164117. +(57) Huber, C.; Klamroth, T. Explicitly time- +dependent coupled cluster singles doubles +calculations of laser-driven many-electron +dynamics. +J. +Chem. +Phys. +2011, +134, +054113. +(58) Shushkov, P.; Miller III, T. F. Real-time +density-matrix coupled-cluster approach for +closed and open systems at finite tempera- +ture. J. Chem. Phys. 2019, 151, 134107. +(59) White, A. F.; +Chan, G. K.-L. A Time- +Dependent Formulation of Coupled-Cluster +Theory for Many-Fermion Systems at Fi- +nite Temperature. J. Chem. Theory Comput. +2018, 14, 5690–5700. +(60) Pedersen, T. B.; Kvaal, S. Symplectic inte- +gration and physical interpretation of time- +dependent coupled-cluster theory. J. Chem. +Phys. 2019, 150, 144106. +(61) Rehr, J. J.; Vila, F. D.; Kas, J. J.; Hirsh- +berg, N. Y.; Kowalski, K.; Peng, B. Equa- +tion of motion coupled-cluster cumulant ap- +proach for intrinsic losses in x-ray spectra. J. +Chem. Phys. 2020, 152, 174113. +(62) Vila, F. D.; Rehr, J. J.; Kas, J. J.; Kowal- +ski, K.; Peng, B. Real-time coupled-cluster +approach for the cumulant Green’s function. +J. Chem. Theory Comput. 2020, 16, 6983– +6992. +(63) Vila, F. D.; +Kas, J. J.; +Rehr, J. J.; +Kowalski, K.; Peng, B. Equation-of-Motion +Coupled-Cluster Cumulant Green’s Function +for Excited States and X-Ray Spectra. Front. +Chem. 2021, 9, 734945. +(64) Vila, F.; Kowalski, K.; Peng, B.; Kas, J.; +Rehr, +J. +Real-Time +Equation-of-Motion +CCSD Cumulant Green’s Function. J. Chem. +Theory Comput. 2022, 18, 1799–1807. +(65) Vila, F.; Rehr, J.; Pathak, H.; Peng, B.; Pa- +nyala, A.; Mutlu, E.; Bauman, N.; Kowal- +ski, K. Real-time equation-of-motion CC cu- +mulant and CC Green’s function simulations +of photoemission spectra of water and water +dimer. J. Chem. Phys. 2022, 157, 044101. +(66) Mutlu, E.; +Panyala, A.; +Kowalski, K.; +Bauman, N.; +Peng, B.; +Brabec, J.; +Kr- +ishnamoorthy, S. TAMM: Tensor Algebra +for Many-body Methods. arXiv preprint +arXiv:2201.01257 2022, +13 + +(67) Mutlu, E.; +Kowalski, K.; +Krishnamoor- +thy, S. Toward generalized tensor algebra +for ab initio quantum chemistry methods. +Proceedings of the 6th ACM SIGPLAN +International Workshop on Libraries, Lan- +guages and Compilers for Array Program- +ming. 2019; pp 46–56. +(68) Peng, B.; Kowalski, K. Low-rank factoriza- +tion of electron integral tensors and its appli- +cation in electronic structure theory. Chem. +Phys. Lett. 2017, 672, 47–53. +(69) Peng, B.; Kowalski, K. Highly efficient and +scalable compound decomposition of two- +electron integral tensor and its application in +coupled cluster calculations. J. Chem. Theory +Comput. 2017, 13, 4179–4192. +(70) Kas, J. J.; Rehr, J. J.; Reining, L. Cumulant +expansion of the retarded one-electron Green +function. Phys. Rev. B 2014, 90, 085112. +(71) Kas, J. J.; Vila, F. D.; Rehr, J. J.; Cham- +bers, S. A. Real-time cumulant approach for +charge-transfer satellites in x-ray photoemis- +sion spectra. Phys. Rev. B 2015, 91, 121112. +(72) Kas, J. J.; Rehr, J. J.; Curtis, J. B. Particle- +hole cumulant approach for inelastic losses +in x-ray spectra. Phys. Rev. B 2016, 94, +035156. +(73) Rehr, J. J.; Kas, J. J. Strengths of plasmon +satellites in XPS: Real-time cumulant ap- +proach. J. Vac. Sci. Technol. A 2021, 39, +060401. +(74) Quarteroni, A.; Sacco, R.; Saleri, F. Numer- +ical mathematics; Springer Science & Busi- +ness Media, 2010; Vol. 37. +(75) Nieplocha, J.; Palmer, B.; Tipparaju, V.; +Krishnan, M.; Trease, H.; Apr`a, E. Ad- +vances, Applications and Performance of +the Global Arrays Shared Memory Program- +ming Toolkit. The International Journal of +High Performance Computing Applications +2006, 20, 203–231. +(76) Van Zee, F. G.; van de Geijn, R. A. BLIS: A +Framework for Rapidly Instantiating BLAS +Functionality. ACM Transactions on Mathe- +matical Software 2015, 41, 14:1–14:33. +(77) Springer, P.; Su, T.; Bientinesi, P. HPTT: +A High-Performance Tensor Transposition +C++ Library. 2017, 56–62. +(78) Lyakh, D. I. TAL-SH: Tensor Algebra Li- +brary for Shared Memory Computers: Nodes +equipped with multicore CPU, Nvidia GPU, +and Intel Xeon Phi. https://github. +com/DmitryLyakh/TAL_SH, 2019. +(79) Valeev, E. F. Libint - a library for the evalua- +tion of molecular integrals of many-body op- +erators over Gaussian functions. https:// +github.com/evaleev/libint, 2019. +(80) Kowalski, K.; Bair, R.; Bauman, N. P.; +Boschen, J. S.; Bylaska, E. J.; Daily, J.; +de Jong, W. A.; Dunning Jr, T.; Govind, N.; +Harrison, R. J., et al. From NWChem to +NWChemEx: Evolving with the computa- +tional chemistry landscape. Chem. Rev. 2021, +121, 4962–4998. +(81) Gunnels, +J. +A.; +Gustavson, +F. +G.; +Henry, G. M.; Van De Geijn, R. A. FLAME: +Formal linear algebra methods environment. +ACM Trans. Math. Softw. 2001, 27, 422–455. +(82) Solomonik, +E.; +Matthews, +D.; +Ham- +mond, J. R.; Stanton, J. F.; Demmel, J. +A +massively +parallel +tensor +contraction +framework +for +coupled-cluster +computa- +tions. J. Parallel Distrib. Comput. 2014, 74, +3176–3190. +(83) Epifanovsky, E.; +Wormit, M.; +Ku´s, T.; +Landau, A.; +Zuev, D.; +Khistyaev, K.; +Manohar, P.; +Kaliman, I.; +Dreuw, A.; +Krylov, A. I. New implementation of high- +level correlated methods using a general +block tensor library for high-performance +electronic structure calculations. Wiley On- +line Library 2013, +(84) Ibrahim, K. Z.; +Williams, S. W.; +Epi- +fanovsky, E.; Krylov, A. I. Analysis and +tuning of libtensor framework on multicore +14 + +architectures. 2014 21st International Con- +ference on High Performance Computing +(HiPC). 2014; pp 1–10. +(85) Røeggen, I.; +Wisløff-Nilssen, E. On the +Beebe-Linderberg two-electron integral ap- +proximation. Chem. Phys. Lett. 1986, 132, +154–160. +(86) Koch, H.; S´anchez de Mer´as, A.; Peder- +sen, T. B. Reduced scaling in electronic +structure calculations using Cholesky de- +compositions. J. Chem. Phys. 2003, 118, +9481–9484. +(87) Beebe, N. H.; Linderberg, J. Simplifications +in the generation and transformation of two- +electron integrals in molecular calculations. +Int. J. Quantum Chem. 1977, 12, 683–705. +(88) Aquilante, F.; Pedersen, T. B.; Lindh, R. +Low-cost evaluation of the exchange Fock +matrix from Cholesky and density fitting rep- +resentations of the electron repulsion inte- +grals. J. Chem. Phys. 2007, 126, 194106. +(89) Røeggen, I.; Johansen, T. Cholesky decom- +position of the two-electron integral matrix +in electronic structure calculations. J. Chem. +Phys. 2008, 128, 194107. +(90) https://cccbdb.nist.gov/. +(91) Kendall, R. A.; Dunning Jr, T. H.; Harri- +son, R. J. Electron affinities of the first-row +atoms revisited. Systematic basis sets and +wave functions. J. Chem. Phys. 1992, 96, +6796–6806. +(92) Noro, T.; Sekiya, M.; Koga, T. Contracted +polarization functions for the atoms helium +through neon. Theor. Chem. Acc. 1997, 98, +25–32. +(93) Noro, T.; Sekiya, M.; Koga, T. Correlating +basis sets for the H atom and the alkali- +metal atoms from Li to Rb. Theor. Chem. +Acc. 2003, 109, 85–90. +(94) Travnikova, O.; Patanen, M.; Soderstrom, J.; +Lindblad, A.; +Kas, J. J.; +Vila, F. D.; +C´eolin, D.; Marchenko, T.; Goldsztejn, G.; +Guillemin, R., et al. Energy-Dependent Rel- +ative Cross Sections in Carbon 1s Pho- +toionization: +Separation of Direct Shake +and Inelastic Scattering Effects in Single +Molecules. J. Phys. Chem. A 2019, 123, +7619–7636. +(95) Kuramoto, K.; Ehara, M.; Nakatsuji, H.; +Kitajima, M.; Tanaka, H.; De Fanis, A.; +Tamenori, Y.; Ueda, K. C 1s and O 1s photo- +electron spectra of formaldehyde with satel- +lites: +theory and experiment. J. Electron. +Spectros. Relat. Phenomena 2005, 142, 253– +259. +15 + +Graphical TOC Entry +For Table of Contents Only +16 + +C 1s +Leadership-Class +C 1S +RT-EOM-CC ++ +TAMM +System +XPS +0 1s +O 1s +RT-EOM-CC +TAMM \ No newline at end of file diff --git a/ItE2T4oBgHgl3EQf_gmP/content/tmp_files/load_file.txt b/ItE2T4oBgHgl3EQf_gmP/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a6f6f706f1dc526166de2b6de937a90a0d9b1f40 --- /dev/null +++ b/ItE2T4oBgHgl3EQf_gmP/content/tmp_files/load_file.txt @@ -0,0 +1,1361 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf,len=1360 +page_content='Real-time Equation-of-Motion Coupled-Cluster Cumulant Green’s Function Method: Heterogeneous Parallel Implementation Based on the Tensor Algebra for Many-body Methods Infrastructure Himadri Pathak,∗,† Ajay Panyala,† Bo Peng,‡ Nicholas P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Bauman,‡ Erdal Mutlu,† John J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Rehr,¶ Fernando D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Vila,∗,¶ and Karol Kowalski∗,‡ †Advanced Computing, Mathematics, and Data Division, Pacific Northwest National Laboratory, Richland, Washington 99354, USA ‡Physical Sciences Division, Pacific Northwest National Laboratory, Richland, Washington 99354, United States ¶Department of Physics, University of Washington, Seattle, Washington 98195, United States E-mail: himadri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='pathak@pnnl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='gov;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' fdv@uw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='edu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' karol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='kowalski@pnnl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='gov Abstract We report the implementation of the real-time equation-of-motion coupled-cluster (RT-EOM- CC) cumulant Green’s function method [J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 152, 174113 (2020)] within the Tensor Algebra for Many-body Methods (TAMM) in- frastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' TAMM is a massively parallel het- erogeneous tensor library designed for utilizing forthcoming exascale computing resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' The two-body electron repulsion matrix elements are Cholesky-decomposed, and we imposed spin- explicit forms of the various operators when eval- uating the tensor contractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Unlike our previ- ous real algebra Tensor Contraction Engine (TCE) implementation, the TAMM implementation sup- ports fully complex algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' The RT-EOM-CC singles (S) and doubles (D) time-dependent am- plitudes are propagated using a first-order Adams– Moulton method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' This new implementation shows excellent scalability tested up to 500 GPUs using the Zn-porphyrin molecule with 655 basis func- tions, with parallel efficiencies above 90% up to 400 GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' The TAMM RT-EOM-CCSD was used to study core photo-emission spectra in the formaldehyde and ethyl trifluoroacetate (ESCA) molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Simulations of the latter involve as many as 71 occupied and 649 virtual orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' The relative quasiparticle ionization energies and over- all spectral functions agree well with available experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Introduction Photoemission spectroscopy (PES) is a widely used spectroscopic probe, covering a broad energy range from a few to several thousands of elec- tronvolts (eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' The ubiquity of this technique is due, in part, to the variety of instruments avail- able, ranging from small laboratory-based ones to synchrotron facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='1,2 In the UV energy regime, PES provides access to the valence electronic structure, where low-energy electrons are the driv- ing force of many chemical and biological pro- cesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='3 In the X-ray regime, core or X-ray pho- toelectron spectroscopy (XPS) is one of the most commonly used fingerprinting method in materi- als science, catalysis, and chemical engineering, where it is used to investigate the composition 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='04249v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='chem-ph] 10 Jan 2023 and chemistry of materials at the atomic level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='4–6 Given the importance of this experimental tech- nique, complementary developments of accurate theoretical methods and associated software for the calculation and interpretation of XPS are cru- cial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Thus, a variety of methods have been de- veloped in both the frequency and time domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' They are often used to predict the position of the main transition or quasiparticle (QP) peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' How- ever, only advanced electron correlation meth- ods can accurately simulate details of core-level photoemission spectra, particularly for the shake- up peaks, since these many-body phenomena re- flect a complex interplay between electron correla- tion and orbital-reorganization effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' The time- independent coupled-cluster (CC) method and its many extensions7–16 have proven their utility in recovering electron correlation in a variety of problems for both ground and excited states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' The ground-state CC method is size-extensive at any level of truncation of the excitation operators, and scales polynomially with the number of active or- bitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' This makes these methods an attractive choice over other electron correlation methods, as they provide a balanced trade-off between the computation cost and desired accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Further- more, it is possible to improve the results sys- tematically by incorporating more correlated de- terminantal spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' The key feature of CC meth- ods is the use of an exponential parameterization of the correlated ground state wavefunction |Ψ⟩: |Ψ⟩ = eT |Φ⟩, where T is the cluster operator, and |Φ⟩ is a reference wavefunction, which is usually but not necessarily a Hartree–Fock wavefunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' The cluster operator is defined by order of ex- citation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' T = T1 + T2 + ··· + Tn correspond- ing to singles, doubles, triples, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=', n-tuples, where Tn = ( 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' )2 ∑ i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=',in a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=',an ta1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='an i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='in a† a1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='a† anain .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='aii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Here a† p and ap are creation and annihilation operators, respectively, associated with a set of Nso = 2Nbas orthonormal spin-orbitals � φp � , with Nbas being the number of basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' The indices in (an) correspond to orbitals that are occupied (unoccu- pied) with respect to the reference determinant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' When combined with the Green’s function (GF) formalism in the frequency domain, the CC method provides an avenue to treat excited- state correlation effects that play a crucial role in accurately simulating quasiparticles and satel- lite peaks in XPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='17–26 However, for large sys- tems, time-domain methods offer advantages over their frequency-domain counterparts by trading off memory resources for the serialization of the cal- culation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Therefore, there has been a lot of effort in developing efficient time-dependent approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Hoodbhoy and Negele,27,28 and Sch¨onhammer and Gunnarsson29 reported formulations of a time-dependent CC theory at about the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' More recently, Kvaal proposed an orbital-adaptive time-dependent coupled-cluster method30 rely- ing on Arponen’s bi-orthogonal formulation of CC theory,31 considering the complex analytic action formulation of the time-dependent varia- tional principle (TDVP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Sato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' have also developed a time-dependent optimized coupled- cluster method considering the real-action formu- lation of the TDVP,32,33 to approximately solve the time-dependent Schr¨odinger equation (TDSE) as a polynomial cost-scaling alternative to multi- configuration time-dependent Hartree–Fock meth- ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='34–36 Both approaches30,32,33 choose to op- timize the orbitals and ignore the one-body ex- citation (T1) and de-excitation (Λ1) operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' This approximation is well suited for strong-field physics, where consideration of optimal orbitals is crucial to obtain meaningful results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Such ap- proaches provide a gauge-invariant description of the time-dependent properties of interest and satisfy the Ehrenfest theorem due to the use of variationally optimized orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='37–40 Despite this advantage, such methods30,32,33 are ill-suited for large-scale applications, especially when simula- tions involve core-hole states of chemical systems containing many active occupied electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Other developments in time-dependent electronic struc- ture theory41 and time-dependent coupled-cluster methods39,42–60 are reviewed elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Recently, we have developed a real-time equation-of-motion coupled-cluster (RT-EOM- CC) cumulant Green’s function method61–65 building on the Sch¨onhammer and Gunnarsson formulation of the TDCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='29 Subsequently sev- eral applications to the XPS of small molecules containing a few electrons in moderate size ba- sis have been reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='62–65 In this methodology, as described in more detail below, the Green’s function has a natural exponential cumulant form, 2 which is given by solutions to a set of coupled, first-order, nonlinear differential equations for the time-dependent CC amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' While the tradi- tional cumulant approximation is linear in the one- particle self-energy, the RT-EOM-CC approach builds in high-order nonlinear, nonperturbative contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Even with their inherent memory usage advan- tage, large-scale time-dependent simulations are computationally challenging for chemically rele- vant systems that go beyond a handful of corre- lated electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' However, thanks to recent ad- vances in high-performance computing techniques that can take advantage of peta- and eventually exascale computational resources, such calcula- tions are no longer insurmountable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' In this ar- ticle, we report the implementation of the RT- EOM-CC method with single and double excita- tions (RT-EOM-CCSD) within the Tensor Alge- bra for Many-body Methods (TAMM) infrastruc- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='66,67 TAMM is a massively parallel heteroge- neous tensor library designed for developing quan- tum chemistry applications for forthcoming exas- cale supercomputers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Our RT-EOM-CC code uses Cholesky-decomposed two-electron repulsion ma- trix elements68,69 that aid in reducing the mem- ory requirements and inter-node communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Other speed-ups arise from spin-explicit evalua- tion of the coupled-cluster amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' As dis- cussed in the next section, the CC amplitudes in RT-EOM-CC are naturally complex valued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' In contrast, our original implementation based on the real-valued Tensor Contraction Engine (TCE)62,64 used separate real-valued data structures to rep- resent the real and imaginary parts and required two distinct subroutines to handle the propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Since the new TAMM implementation uses ex- plicit complex algebra, only a single subroutine is required to handle the complex data, thus reducing the coding and data intricacy, even though com- plex algebra is more floating point operation in- tensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' To demonstrate the capabilities of this new im- plementation, we study the core spectral functions of the formaldehyde and ethyl trifluoroacetate (ESCA) molecules and compare them with ex- perimental spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' We observe satisfying agree- ment between the computed and available exper- imental spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' In addition to these systems, we present parallel and storage performance for a few nominally “large” systems such as Zn-porphyrins, uracil, and the benzene-ammonia dimer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Methods Real-time Equation-of-Motion Coupled-Cluster Cumulant Green’s Function Method The many-body Green’s function approach has proved very useful for the calculation of spec- tral functions of extended systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='70–73 By com- bining this approach with the CC method, we have developed a time-dependent CC cumulant Green’s function method61 that integrates the ad- vantages of both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' A detailed derivation of the complete method can be found elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='62–65 In this section, we give a brief introduction to the RT-EOM-CC formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' The goal is to construct the retarded core-hole Green’s func- tion GR c (ω) and associated core spectral func- tion Ac(ω) = (−1/π)ImGR c (ω), by introduc- ing a time-dependent coupled-cluster ansatz, eiHτ |Ψ⟩ = |Ψ(τ)⟩ = Nc(τ)eT(τ) |Φ⟩, which is a formal solution to the time-dependent Schr¨odinger equation −i ∂ ∂τ |Ψ(τ)⟩ = H |Ψ(τ)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Here τ is time, and |Ψ(τ)⟩ is the fully correlated wavefunction for the (N −1)-electron state, H = ∑pq hpqa† paq + 1 4 ∑pqrs vrs pqa† pa† qasar is the nonrelativistic elec- tronic Hamiltonian in second-quantization form, hpq are the single-particle kinetic and electron- nuclei spinorbital integrals, and vrs pq = ⟨pq||rs⟩ are the usual antisymmetrized two-particle Coulomb integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Nc(τ) and T(τ) are, respectively, the time-dependent normalization constant and the time-dependent coupled-cluster operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' In prin- ciple, c can be any occupied orbital, but here we focus on deep core excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' As usual, the p,q,r,s indices indicate generic spin-orbital states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' In the current formulation, the time-independent reference determinant |Φ⟩ is a single (N − 1)- electron determinant formed from the N-electron Hartree–Fock states where the state c has been annihilated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Thus, it is important to note that T acts in the (N − 1)-electron space where c is now included in the set of unoccupied single- particle states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Following Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 61, the final form 3 of the equations of motion (EOMs) for Nc(τ), and tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (τ) are: −i∂τ lnNc(τ) = ⟨Φ|(HNeT(τ))C|Φ⟩+EHF N−1(1) −i∂τ tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' i j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (τ) = ⟨Φab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' |(HNeT(τ))C|Φ⟩, (2) Here HN = ∑pq fpq{a† paq}′+ 1 4 ∑pqrs vrs pq{a† pa† qasar}′, and {}′ indicates that the normal ordering is done with respect to |Φ⟩ instead of the usual N-electron Hartree–Fock determinant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' EHF N−1 = ⟨Φ|H|Φ⟩ is the Hartree–Fock energy of the (N − 1)-electron systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' The Fock operator matrix elements are fpq = εpδpq − vqc pc, where c is the core-hole index as used above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' The subscript ”C” designates a connected part of a given operator expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Within this CC approximation, the retarded core-hole GF GR c (τ) is simply proportional to the normalization factor Nc(τ), since Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 1 has an ex- ponential solution, and therefore CR c (τ), the re- tarded cumulant associated with c, is proportional to lnNc(τ): GR c (τ) = −iΘ(τ)e−i(εc+Ecorr N )τNc(τ) (3) = −iΘ(τ)e−i(εc+Ecorr N )τeCRc (τ) (4) CR c (τ) = i � τ 0 � Φ ���(HNeT(τ′))C ���Φ � dτ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (5) Here εc is the single-particle Hartree–Fock energy of the core orbital c, and Ecorr N corresponds to the correlation energy of the N-electron closed-shell ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' As a consequence, GR c (τ) can be ex- pressed as the product of the free particle GF and the exponential of a cumulant, as expected within the cumulant approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' We can now write down from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 1 a complete expression for the time derivative of CR c (τ) within the single and dou- ble excitations approximation (CCSD): −i∂τCR c = ⟨Φ|[HN,T2(τ)]|Φ⟩ + 1 2 ⟨Φ|[[HN,T1(τ)],T1(τ)]|Φ⟩ (6) Unlike conventional linear self-energy formula- tions, it is evident from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 6 that the CCSD cu- mulant includes nonlinear, nonperturbative contri- butions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' The EOMs for the amplitudes within the CCSD Perform N-electron RHF calculation Perform N-electron closed-shell CCSD calculation Reorder orbitals keeping an arbitrary core hole Reconstruct the Fock operator in the N-1 electron Fock-space Perform time-propagation and compute the cumulant in each time-step Construct Green’s function and Fourier transform it to get the spectral function Photoemission Spectra Figure 1: Real-time equation-of-motion coupled- cluster workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' approximation are obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 2 as −i∂τta i (τ) = � Φa i |(HNeT1(τ)+T2(τ))C|Φ � (7) −i∂τtab i j (τ)) = � Φab i j |(HNeT1(τ)+T2(τ))C|Φ � , (8) where ta i (τ) and tab i j (τ)) are time-dependent singly and doubly excited cluster amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Numerical Solution of the EOMs To construct the core-hole spectral function Ac(ω) we need to compute the Green’s function GR c (τ), which in turn depends on the cumulant CR c (τ) over the whole simulation range (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 5 and 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Thus, the main task is to propagate Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 7 and 8 in time, which provide solutions for both the T1 and T2 amplitudes needed to compute the cu- 4 mulant CR c (τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 1 demonstrates a typical RT- EOM-CC workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Given that the RT-EOM-CC uses ground-state orbitals and the closed-shell CC ground-state energy, stationary restricted Hartree– Fock (RHF) and closed-shell coupled-cluster cal- culations are prerequisites to the time-dependent simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' The computational cost of the RHF calculations is O(N3 bas) while that for the CC part varies, depending on the imposed truncation in the excitation operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' The singles and doubles ap- proximation used in this work scales as O(O2V 4), where O and V denote the total number of occu- pied and virtual orbitals, respectively, and O+V = Nbas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' The first two ground-state calculation steps are common to most simulations involving core-hole excitations, but these are not the main bottlenecks since they need to be performed only once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Af- ter these, all calculations involving different core- hole states are unique and can run simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' These calculations start by reading the N-electron closed-shell orbitals into the (N − 1)-electron or- bital space and then constructing the (N − 1)- electron Fock matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' At this point, the time- dependent simulation can start, where we propa- gate the CCSD amplitudes forward for the desired time, computing the time-derivative of the cumu- lant at each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' It is worth pointing out that given that the time-dependent simulations involve an initial core hole, this is an open-shell problem and is roughly three times more expensive than its closed-shell counterpart in spin-explicit form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' As in our previous implementations, the first- order coupled nonlinear simultaneous differential equations for the amplitudes are integrated us- ing the first-order Adams–Moulton method,74 also known as the implicit trapezoidal rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' In this ap- proximation the tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (τ) are propagated with: tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' i j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (τ +∆τ) = tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (τ) + i 2∆τ �� Φab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' ���(HNeT(τ))C ���Φ � + � Φab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' ���(HNeT(τ+∆τ))C ���Φ �� , (9) where ∆τ is the simulation time-step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' The propa- gation starts with the initial conditions T1 = 0 and T2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Since the Adams–Moulton method is im- Applications CCSD CCSD(T) EOMCC GFCC DLPNO- CC … Tensor Algebra Interface Distribution Memory Management Execution Context Scheduler TAMM Dependencies Global Arrays/ MPI Vendor BLAS (CPU/ GPU) LibreTT HPTT HDF5(I/O) Figure 2: Overview of the Tensor Algebra for Many-body Methods (TAMM) framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' plicit, the values of the amplitudes at (τ +∆τ) de- pend on themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Thus, to solve Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 9 we use a fixed-point iteration scheme at each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Other methods of solution that use a vari- able time step and are more stable than fixed-point iteration are currently under development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' After completion of the time propagation, the remainder of the workflow is not compute-intensive, since it only involves forming the time-dependent Green’s function from the cumulant, and then Fourier- transforming to obtain the spectral function in the frequency domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' RT-EOM-CC Implementation in Ten- sor Algebra for Many-body Methods Infrastructure The Tensor Algebra for Many-body Methods (TAMM) library66,80 is a massively parallel, het- erogeneous tensor algebra library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' It provides a computational infrastructure (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 2) that can achieve scalable performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Furthermore, it allows portable implementations of many-body methods both on existing and forthcoming exas- cale super-computing platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' High-dimensional tensor contractions are the most compute-intensive components of the RT- EOM-CC method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Within the single and double excitations approximation (RT-EOM-CCSD), the most expensive tensor contraction is of the form R(V,V,O,O) = α × v(V,V,V,V) × t2(V,V,O,O), involving four-dimensional tensors, the antisym- metrized two-body matrix elements v, and the two-body CC excitation operator t2, α is a scalar pre-factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' These multi-dimensional tensor con- 5 Table 1: Structural features of TAMM, including various used third-party dependency libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Third-party dependencies Global Arrays,75 BLIS,76 vendor BLAS/LAPACK, cuBLAS/rocBLAS/oneMKL HPTT,77 TALSH,78 LibInt279 Programming Languages C++17, CUDA, HIP, SYCL, MPI, OpenMP Precision Double Data Types Real, Complex Supports Restart Capabilities?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Yes I/O requirements Minor tractions are not only computational- but also communication-intensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' There have been many efforts to develop specialized parallel tensor alge- bra libraries, including automated code generators and better memory management to facilitate these demanding tensor contractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='81–84 The TAMM library is one such effort aiming to achieve scal- able performance on several heterogeneous archi- tectures, by delivering a common platform for the portable implementation of numerous many-body methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' TAMM provides a variety of features to users including the ability to specify and ma- nipulate tensor distribution, memory management, and scheduling of tensor operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' In addition, it supports both complex and mixed real-complex algebra for mathematical operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' A summary of structural features and third-party libraries for the TAMM library is shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' TAMM uses Global Arrays75 and Message Pass- ing Interface (MPI) to achieve scalable paralleliza- tion on distributed memory platforms, while using optimized libraries that help efficient intra-node execution of the tensor operation, both in CPU kernels and accelerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' TAMM also uses multi- granular dependence analysis and task-based exe- cution to execute operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' First, it constructs a macro operation graph by analyzing the dependen- cies between various operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' When two oper- ations share the same data structure, with one of them writing to the other, they are in conflict and impossible to execute in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' The operation graph is analyzed to identify and order the non- parallel operations to minimize the required num- bers of synchronizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' The possible scheduled operations are executed in a single program multi- ple data fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Such executions are compatible with MPI, and their collective executions are per- formed on a given MPI communicator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Various tasks constitute an operation, which is produced using task iterators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Each task performs part of the computation, usually adding a block of data to the output tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' A given task is migratable and can be scheduled for execution on any compute node or processor core until its execution begins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' The data needed for a job are transported to its location once the execution of the process has started.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' At this stage, migration of tasks is no longer possible and they are bound to process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' TAMM’s GPU ex- ecution scheme uses localized summation loops to minimize the transfer of output blocks from GPUs to CPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' This helps to reduce the data transmis- sion between CPUs and GPUs by keeping the out- put block that is being updated by several input tensor blocks on the GPU until all updates are complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Programming Models, Software Dependencies, I/O, Restart/Checkpoint Capabilities Memory demands, operation count, and time-to- solution are three main concerns that limit large- scale CC calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' This is further complicated when extending the CC formalism to the time do- main.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' To combat these hindrances, various tech- niques and features were incorporated during the implementation of the RT-EOM-CCSD method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' In canonical spin-orbital CC calculations, the four-dimensional electron repulsion integral (ERI) tensors are easily the largest memory-demanding objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' The storage requirement for the ERI ten- sor in its full spin-orbital form is of order N4 so, and they must undergo a tensor transformation from the atomic-orbital to molecular-orbital basis which scales as O(N5 bas).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' In our RT-EOM-CC implemen- tation, we employed Cholesky decomposition85–89 6 of the ERI tensors to ease the memory/storage de- mands and increase the data locality resulting in reduced communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' We leveraged the same Cholesky decomposition previously implemented using the TAMM library,68,69 which is an on- the-fly pivoting decomposition of the two-body ERIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' The resulting Cholesky bases are three- index quantities that reduce the storage require- ments from order N4 so to KN2 so, where K ∼ O(Nbas) is the number of Cholesky bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' In addition, the atomic-orbital to molecular-orbital transformation is conducted on the Cholesky bases rather than di- rectly on the ERI, thus reducing the scaling of the transformation from O(N5 bas) to O(N4 bas).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' The ac- curacy of the correlation calculations employing the Cholesky bases in comparison with the canon- ical results is well-controlled through adjusting the diagonal cutoff in the Cholesky decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' The next largest memory-demanding objects, af- ter the ERIs, are the T2 amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' In a naive spin- orbital implementation, the memory requirement for all elements of the T2 operator is 16O2V 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' The declaration includes 16 possible combinations of spins for the four indices, most of which do not contribute or are over-specified as they are equiv- alent through permutational symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' In our RT- EOMCC methodology, we implemented the spin- integrated form of equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Only unique spin combinations of tensors with a non-zero contri- bution, including intermediates, are programmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' For the T2 amplitudes, this means only three spin cases are necessary (tαα αα , tββ ββ , and tαβ αβ ), reduc- ing the memory requirements to ∼ 3 16 that of the full T2 operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Memory requirements are re- duced for other operators as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Since non- contributing spin combinations are removed, and only unique spin cases are specified, the over- all operation count is significantly reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' The memory requirements of various systems with var- ious large systems with 760 to 2450 spin-orbital functions can be seen in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' In this Table, the memory requirement for the cluster amplitudes re- flects the sum of the three timeline tensors needs for the iterative update given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' The com- bination of Cholesky-decomposed ERIs and spin- integrated equations allows for simulations with hundreds of orbitals on moderately sized computer clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Another challenging aspect of RT-EOM-CC cal- culations is that it is necessary to propagate for a sufficiently long time to have well-resolved spec- tra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' In practice, shared computing resource usually do not allow simulations to propagate for enough time in a single run to achieve sufficient reso- lution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Furthermore, it is important to be able to track simulations and adjust/optimize the time propagation parameters promptly, before a long time has elapsed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' For these reasons, checkpoint- ing and restart capabilities were imperative in our RT-EOM-CCSD implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Restarting the time-propagation algorithm at any ith step requires the T amplitudes of the (i−1)th and ith time steps, in addition to the Fock and Cholesky-decomposed ERIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Parallel read and write capabilities in the TAMM library allow for periodic checkpointing while minimizing their impact on the overall com- putational time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Geometries, Basis Sets, and Computa- tional Details In this study, we perform RT-EOM-CC simula- tions of two molecules, formaldehyde and ethyl trifluoroacetate (ESCA), in order to compare to previous theoretical and experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' For formaldehyde, we used the experimental geom- etry90 and studied both C and O core ioniza- tions using either the aug-cc-pVDZ,91 aug-cc- pVTZ,91 or Sapporo-TZP92,93 basis sets for all atoms in the molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Given that the ESCA molecule contains four C atoms, we computed ion- ization spectra from each of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' These cal- culations were performed with the Sapporo-TZP basis set for all the first-row elements, while for the H atom we used the aug-cc-pVDZ basis set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Since a complete experimental geometry of the ESCA molecule is not available, we used the one obtained from a B3LYP/aug-cc-pVTZ opti- mization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='94 The real-time time-propagation used a time-step of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='015 au (∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='36 as) for formaldehyde and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='01 au (∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='24 as) for the ESCA molecule, with a convergence cutoff of 10−4 for the fixed- point micro-iteration solution of the implicit first- order Adams–Moulton integrator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' The total prop- agation time was 450 au (∼11 fs) for formalde- hyde and 100 au (∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='5 fs) for ESCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' All sta- tionary calculations were performed with a linear 7 Table 2: Workflow memory requirements for the TAMM implementation of RT-EOM-CCSD for systems with Sapporo-TZP basis set for all atoms except H, for which we use aug-cc-pVTZ, using a Cholesky vectors diagonal cutoff of 10−6, and linear dependence threshold of 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' System Configuration Space T2 Cholesky # of Cholesky amplitude vectors vectors Nso nα occ nβ occ nα vir nβ vir (GB) (GB) Uracil 760 29 28 351 352 3×4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='5 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='2 2044 ESCA 880 36 35 404 405 3× 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='2 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='6 2423 Benzene-Ammonia 940 26 25 444 445 3×5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='8 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='5 2457 Zn-porphyrin 2450 95 94 1130 1131 3× 510.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='2 876.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='6 6258 dependence threshold for the basis sets of 10−6, SCF convergence cutoff of 10−8 au for the energy, Cholesky diagonal cutoff of 10−6 and a CCSD convergence cutoff of 10−8 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' None of the vir- tual orbitals are frozen in any of our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Results and Discussion Performance Analysis As described in previous sections, a simulation is divided into a series of “macro-iterations” associ- ated with each time step, and within each macro- iteration, many “micro-iterations” are performed to solve Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Thus, the fundamental parallel per- formance metric for RT-EOM-CC is the “time per micro-iteration” associated with the calculation of the matrix elements in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' In order to inves- tigate the performance and scalability of the new optimized TAMM implementation of RT-EOM- CCSD, we have performed a series of calculations on the “pre-production” NERSC Perlmutter sys- tem (Nvidia A100).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' If the scaling is ideal, the total compute time is inversely proportional to the to- tal number of allocated processors provided the to- tal number of mathematical operations in each test case remains the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Figure 3 shows the scala- bility of the TAMM implementation of RT-EOM- CCSD for Zn-porphyrin using the aug-cc-pVDZ basis set, which results in a total of 655 basis func- tions (nocc=94, nvir=561) after pruning 122 linear dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Our time-dependent simulations in- volve a configuration space of nα occ = 95, nα vir = 560, nβ occ = 94, nβ vir = 561.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' We explore scaling between 200 and 500 GPUs (50 to 125 nodes given that each node is connected to 4 GPUs), using the per- 200 250 300 350 400 450 500 # of GPUs 150 200 250 300 350 Time (s) Real Ideal Figure 3: Real vs ideal single micro-iteration time as a function of number of GPUs for Zn-porphyrin using 655 basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' formance with 200 GPUs as the reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Figure 3 also shows the ideal theoretical scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' In this range, we observe a nearly ideal drop in the com- putation time for each micro-iteration, very close to the theoretically limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' The parallel efficiency remains high (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' higher than 94%) up to 400 GPUs, after which it drops to 83% for 500 GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Spectral Function Results The high quality of the RT-EOM-CC results has been previously demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='63,65 Thus here we focus on showcasing the new capabilities of the RT-EOM-CCSD TAMM implementation for more complex systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' For this purpose, we have simu- lated formaldehyde (H2CO) and ethyl trifluoroac- etate (ESCA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' For formaldehyde, we focus on the satellite region of the C and O core spectral func- 8 30 20 10 0 10 E - EQP (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='14 Spectral Function (1/eV) XPS (Expt) aug-cc-pVDZ aug-cc-pVTZ Sapporo-TZP H2CO 30 20 10 0 10 E - EQP (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='14 Spectral Function (1/eV) XPS (Expt) aug-cc-pVDZ aug-cc-pVTZ Sapporo-TZP H2CO Figure 4: Comparison of satellite regions of the C (top) and O (bottom) RT-EOM-CCSD core spectral functions to the XPS experiment95 for formaldehyde (H2CO) as a function of basis set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' The data has been shifted so that the quasiparticle peak is at 0 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' tions (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' We find that the theory reproduces the XPS semi-quantitatively, with the position of most of the satellite features relative to the quasi- particle peak in reasonable agreement with exper- iment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' The relative intensities of the peaks are not as well reproduced, probably due to i) the limita- tions of the local valence basis set used that misses the continuum background contribution, and ii) the limitations of the RT-EOM-CCSD to include all the relevant excitations in this energy range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' For the case of the ESCA molecule, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 5, we calculated the C core spectral function for each of the inequivalent C atoms in the sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' The agreement with experiment is excellent, | | | | | | | | 290 292 294 296 298 300 Binding Energy (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='2 Spectral Function (1/eV) C(H2OC) C(O2C) C(F3C) C(H3C) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='97 CH3-CH2-O-CO-CF3 XPS (Expt) RT-EOM-CCSD RT-EOM-CCSD (Shft) Figure 5: Comparison of the RT-EOM- CCSD/Sapporo-TZP C core spectral functions (red) to the experimental94 XPS (black dots) for the ethyl trifluoroacetate (ESCA) molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Each peak corresponds to an individual C core ionization and has been broadened to match the vibrational experimental broadening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Also shown are the same results shifted by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='89 eV (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' apart from a nearly-constant overall underestima- tion of the binding energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' When the underesti- mation shift is removed, the relative mean absolute error is only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='04 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' We also find that, unlike for the satellite peaks in H2CO, the relative intensities of the ESCA quasiparticle peaks are qualitatively reproduced by the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' By fitting the experi- ment to a skew Gaussian distribution (to account for the vibrational asymmetry), we find that the intensities of the C(H2OC), C(O2C) and C(F3C) peaks relative to the C(H3C) one are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='81, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='90 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='91, respectively, while for the theory the ratios are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='94, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='91 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' It was speculated94 that some of the intensity of the different C quasiparti- cle peaks might originate from underlying satellite peaks from lower energy cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' We find that for each of the individual spectral functions the first satellite peaks appear more than 10 eV above the quasiparticle, and thus all the intensity observed for the peaks between 291 and 299 eV can be as- signed exclusively to the quasiparticle transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 9 Conclusions We have successfully implemented the RT-EOM- CCSD method within the parallel TAMM infras- tructure using Cholesky decomposed two-body re- pulsion matrix elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' This implementation eliminates the memory bottleneck of the original approach associated with storing two-electron in- tegrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Unlike our earlier TCE-based RT-EOM- CCSD implementation, which relied only on real algebra, our new TAMM implementation supports explicit complex algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' This implementation is also flexible regarding the choice of the ref- erence function (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' where the hole state is lo- cated), and has checkpointing/restart capabilities at any stage of the workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' This is quite impor- tant since the propagation portion of the workflow can be very time-consuming and restarts are usu- ally needed in shared computing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' More- over, the TAMM RT-EOM-CCSD shows very good scalability, paving the way for simulations of larger, more realistic and chemically relevant systems, employing larger basis sets in conjunc- tion with reduced memory requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Illustra- tive calculations for the formaldehyde and ESCA molecules demonstrate that the predicted positions for the quasiparticle and satellite peaks are in good agreement with experimental values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' In particu- lar, the RT-EOM-CCSD reproduced the relative position of the different core ionizations in the ESCA molecule, highlighting its capabilities to study chemical speciation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' The method describes the positions of the satellite peaks without any cor- rections to the quasiparticle-satellite gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Finally, work is in progress on an atomic orbital- based implementation, coupled to more efficient solvers for the implicit Adams–Moulton propaga- tor that should reduce the computational time by at least an order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Other future method- ological developments include the implementation of spin-orbit coupling, and the multi-component coupled-cluster formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' These extensions will allow first-principles studies of multielectron dy- namics in previously unreachable large chemical systems, such as simulations involving multiple core holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Acknowledgement This work was supported by the Computational Chemical Sciences Program of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Department of Energy, Office of Sci- ence, BES, Chemical Sciences, Geosciences and Biosciences Division in the Center for Scalable and Predictive methods for Excitations and Cor- related phenomena (SPEC) at PNNL, with com- putational support from NERSC, a DOE Office of Science User Facility, under contract no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' DE- AC02-05CH11231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' also acknowledges sup- port from the Laboratory Directed Research and Development (LDRD) Program at PNNL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' References (1) Meirer, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Weckhuysen, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Spatial and temporal exploration of heterogeneous cat- alysts with synchrotron radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 2018, 3, 324–340.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (2) Qiao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' He, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Zhang, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Zhou, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Chen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Song, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Wei, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Synchrotron- radiation spectroscopic identification to- wards diverse local environments of single- atom catalysts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' A 2022, 10, 5771–5791.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (3) Kunin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Neumark, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Time-resolved radiation chemistry: femtosecond photo- electron spectroscopy of electron attach- ment and photodissociation dynamics in iodide–nucleobase clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 2019, 21, 7239–7255.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (4) Seah, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' The quantitative analysis of sur- faces by XPS: A review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Surf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Interface Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 1980, 2, 222–239.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (5) Bagus, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Ilton, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Nelin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' The interpretation of XPS spectra: Insights into materials properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Surf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 2013, 68, 273–304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (6) Shchukarev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' XPS at solid–aqueous so- lution interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Colloid Interface Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 2006, 122, 149–157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (7) Coester, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Bound states of a many-particle system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 1958, 7, 421–424.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (8) Coester, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' K¨ummel, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Short-range correla- tions in nuclear wave functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 1960, 17, 477–485.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 10 (9) ˇC´ıˇzek, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' On the Correlation Problem in Atomic and Molecular Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Calculation of Wavefunction Components in Ursell-Type Expansion Using Quantum-Field Theoretical Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 1966, 45, 4256– 4266.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (10) Paldus, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' ˇC´ıˇzek, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Shavitt, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Corre- lation Problems in Atomic and Molecular Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Extended Coupled-Pair Many- Electron Theory and Its Application to the BH3 Molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' A 1972, 5, 50–67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (11) K¨ummel, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' A biography of the coupled cluster method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' B 2003, 17, 5311–5325.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (12) Bartlett, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Musiał, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Coupled-cluster theory in quantum chemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 2007, 79, 291–352.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (13) Mukherjee, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Pal, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Use of cluster ex- pansion methods in the open-shell correla- tion problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Quantum Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 1989, 20, 291–373.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (14) Krylov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Equation-of-motion coupled- cluster methods for open-shell and electroni- cally excited species: the Hitchhiker’s guide to Fock space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Annu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 2008, 59, 433–462.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (15) Piecuch, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Kowalski, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Pimienta, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Mcguire, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Recent advances in electronic structure theory: Method of moments of coupled-cluster equations and renormalized coupled-cluster approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 2002, 21, 527–655.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (16) Crawford, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Schaefer, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' An introduc- tion to coupled cluster theory for computa- tional chemists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 2000, 14, 33–136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (17) Nooijen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Snijders, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Coupled cluster approach to the single-particle Green’s func- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Quantum Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 1992, 44, 55–83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (18) Nooijen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Snijders, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Coupled cluster Green’s function method: Working equations and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Quantum Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 1993, 48, 15–48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (19) Nooijen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Snijders, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Second or- der many-body perturbation approximations to the coupled cluster Green’s function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 1995, 102, 1681–1688.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (20) Meissner, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Bartlett, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Electron propa- gator theory with the ground state correlated by the coupled-cluster method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Quan- tum Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 1993, 48, 67–80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (21) Bhaskaran-Nair, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Kowalski, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Shel- ton, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Coupled cluster Green function: Model involving single and double excita- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 2016, 144, 144101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (22) Peng, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Kowalski, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Coupled-cluster Green’s function: Analysis of properties originating in the exponential parametriza- tion of the ground-state wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' A 2016, 94, 062512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (23) Peng, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Kowalski, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Properties of ad- vanced coupled-cluster Green’s function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 2018, 116, 561–569.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (24) Peng, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Bauman, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Gulania, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Kowal- ski, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Annu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Elsevier, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 17;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' pp 23–53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (25) Shee, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Zgid, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Coupled Cluster as an im- purity solver for Green’s function embedding methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Theory Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 2019, 15, 6010–6024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (26) Lange, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Berkelbach, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' On the re- lation between equation-of-motion coupled- cluster theory and the GW approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Theory Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 2018, 14, 4224– 4236.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (27) Hoodbhoy, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Negele, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Time-dependent coupled-cluster approximation to nuclear dy- namics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Application to a solvable model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' C 1978, 18, 2380.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (28) Hoodbhoy, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Negele, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Time-dependent coupled-cluster approximation to nuclear dy- namics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' General formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' C 1979, 19, 1971.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (29) Sch¨onhammer, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Gunnarsson, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Time- dependent approach to the calculation of 11 spectral functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' B 1978, 18, 6606.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (30) Kvaal, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Ab initio quantum dynamics using coupled-cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 2012, 136, 194109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (31) Arponen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Variational principles and linked-cluster exp S expansions for static and dynamic many-body problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 1983, 151, 311–382.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (32) Sato, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Pathak, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Orimo, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Ishikawa, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Communication: Time- dependent optimized coupled-cluster method for multielectron dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 2018, 148, 051101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (33) Pathak, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Sato, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Ishikawa, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Time- dependent optimized coupled-cluster method for multielectron dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' A coupled electron-pair approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 2020, 152, 124115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (34) Caillat, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Zanghellini, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Kitzler, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Koch, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Kreuzer, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Scrinzi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Corre- lated multielectron systems in strong laser fields: A multiconfiguration time-dependent Hartree-Fock approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' A 2005, 71, 012712.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (35) Kato, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Kono, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Time-dependent multi- configuration theory for electronic dynamics of molecules in an intense laser field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 2004, 392, 533–540.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (36) Sato, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Ishikawa, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Time-dependent complete-active-space self-consistent-field method for multielectron dynamics in in- tense laser fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' A 2013, 88, 023402.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (37) Ishikawa, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Sato, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' A review on ab ini- tio approaches for multielectron dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' IEEE J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Sel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Quantum Electron 2015, 21, 1–16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (38) Sato, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Pathak, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Orimo, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Ishikawa, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Time-dependent multi- configuration self-consistent-field and time- dependent optimized coupled-cluster meth- ods for intense laser-driven multielectron dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' arXiv preprint arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='10116 2022, (39) Pathak, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Sato, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Ishikawa, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Time- dependent optimized coupled-cluster method for multielectron dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' A second- order many-body perturbation approxima- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 2020, 153, 034110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (40) Pathak, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Sato, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Ishikawa, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Study of laser-driven multielectron dynamics of Ne atom using time-dependent optimised second-order many-body perturbation the- ory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 2020, 118, e1813910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (41) Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Govind, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Isborn, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' De- Prince III, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Lopata, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Real-time time- dependent electronic structure theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 2020, 120, 9951–9993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (42) Monkhorst, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Calculation of Properties with the Coupled-Cluster Method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Quantum Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 1977, 12, 421–432.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (43) Mukherjee, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Mukherjee, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' A response- function approach to the direct calculation of the transition-energy in a multiple-cluster ex- pansion formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 1979, 39, 325–335.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (44) Guha, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Mukherjee, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' A multireference coupled-cluster approach to quantum dy- namics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 1991, 186, 84–90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (45) Dalgaard, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Monkhorst, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Some aspects of the time-dependent coupled-cluster ap- proach to dynamic response functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' A 1983, 28, 1217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (46) Koch, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Jørgensen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Coupled cluster re- sponse functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 1990, 93, 3333–3344.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (47) Takahashi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Paldus, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Time-dependent coupled cluster approach: Excitation en- ergy calculation using an orthogonally spin- adapted formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 1986, 85, 1486–1501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (48) Prasad, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Time-dependent coupled clus- ter method: A new approach to the cal- culation of molecular absorption spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 1988, 88, 7005–7010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 12 (49) Sebastian, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Correlation effects in ion neu- tralization scattering with the use of a time- dependent coupled-cluster approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' B 1985, 31, 6976.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (50) Pigg, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Hagen, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Nam, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Papen- brock, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Time-dependent coupled-cluster method for atomic nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' C 2012, 86, 014308.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (51) Nascimento, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' DePrince III, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Linear absorption spectra from explicitly time-dependent equation-of-motion coupled- cluster theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Theory Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 2016, 12, 5834–5840.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (52) Nascimento, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' DePrince III, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Simulation of near-edge X-ray absorption fine structure with time-dependent equation- of-motion coupled-cluster theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 2017, 8, 2951–2957.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (53) Koulias, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Williams-Young, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Nascimento, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' DePrince III, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Relativistic real-time time-dependent equation-of-motion coupled-cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Theory Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 2019, 15, 6617– 6624.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (54) Nascimento, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' DePrince III, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' A general time-domain formulation of equation-of-motion coupled-cluster theory for linear spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 2019, 151, 204107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (55) Pathak, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Sato, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Ishikawa, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Time- dependent optimized coupled-cluster method for multielectron dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Approximate consideration of the triple excitation ampli- tudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 2021, 154, 234104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (56) Park, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Perera, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Bartlett, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Equa- tion of motion coupled-cluster for core ex- citation spectra: Two complementary ap- proaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 2019, 151, 164117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (57) Huber, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Klamroth, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Explicitly time- dependent coupled cluster singles doubles calculations of laser-driven many-electron dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 2011, 134, 054113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (58) Shushkov, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Miller III, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Real-time density-matrix coupled-cluster approach for closed and open systems at finite tempera- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 2019, 151, 134107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (59) White, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Chan, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' A Time- Dependent Formulation of Coupled-Cluster Theory for Many-Fermion Systems at Fi- nite Temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Theory Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 2018, 14, 5690–5700.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (60) Pedersen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Kvaal, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Symplectic inte- gration and physical interpretation of time- dependent coupled-cluster theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 2019, 150, 144106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (61) Rehr, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Vila, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Kas, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Hirsh- berg, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Kowalski, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Peng, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Equa- tion of motion coupled-cluster cumulant ap- proach for intrinsic losses in x-ray spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 2020, 152, 174113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (62) Vila, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Rehr, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Kas, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Kowal- ski, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Peng, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Real-time coupled-cluster approach for the cumulant Green’s function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Theory Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 2020, 16, 6983– 6992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (63) Vila, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Kas, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Rehr, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Kowalski, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Peng, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Equation-of-Motion Coupled-Cluster Cumulant Green’s Function for Excited States and X-Ray Spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 2021, 9, 734945.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (64) Vila, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Kowalski, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Peng, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Kas, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Rehr, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Real-Time Equation-of-Motion CCSD Cumulant Green’s Function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Theory Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 2022, 18, 1799–1807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (65) Vila, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Rehr, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Pathak, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Peng, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Pa- nyala, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Mutlu, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Bauman, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Kowal- ski, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Real-time equation-of-motion CC cu- mulant and CC Green’s function simulations of photoemission spectra of water and water dimer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 2022, 157, 044101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (66) Mutlu, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Panyala, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Kowalski, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Bauman, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Peng, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Brabec, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Kr- ishnamoorthy, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' TAMM: Tensor Algebra for Many-body Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' arXiv preprint arXiv:2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='01257 2022, 13 (67) Mutlu, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Kowalski, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Krishnamoor- thy, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Toward generalized tensor algebra for ab initio quantum chemistry methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Proceedings of the 6th ACM SIGPLAN International Workshop on Libraries, Lan- guages and Compilers for Array Program- ming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' pp 46–56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (68) Peng, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Kowalski, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Low-rank factoriza- tion of electron integral tensors and its appli- cation in electronic structure theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 2017, 672, 47–53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (69) Peng, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Kowalski, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Highly efficient and scalable compound decomposition of two- electron integral tensor and its application in coupled cluster calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Theory Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 2017, 13, 4179–4192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (70) Kas, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Rehr, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Reining, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Cumulant expansion of the retarded one-electron Green function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' B 2014, 90, 085112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (71) Kas, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Vila, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Rehr, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Cham- bers, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Real-time cumulant approach for charge-transfer satellites in x-ray photoemis- sion spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' B 2015, 91, 121112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (72) Kas, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Rehr, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Curtis, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Particle- hole cumulant approach for inelastic losses in x-ray spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' B 2016, 94, 035156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (73) Rehr, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Kas, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Strengths of plasmon satellites in XPS: Real-time cumulant ap- proach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Vac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' A 2021, 39, 060401.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (74) Quarteroni, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Sacco, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Saleri, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Numer- ical mathematics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Springer Science & Busi- ness Media, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (75) Nieplocha, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Palmer, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Tipparaju, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Krishnan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Trease, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Apr`a, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Ad- vances, Applications and Performance of the Global Arrays Shared Memory Program- ming Toolkit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' The International Journal of High Performance Computing Applications 2006, 20, 203–231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (76) Van Zee, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' van de Geijn, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' BLIS: A Framework for Rapidly Instantiating BLAS Functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' ACM Transactions on Mathe- matical Software 2015, 41, 14:1–14:33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (77) Springer, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Su, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Bientinesi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' HPTT: A High-Performance Tensor Transposition C++ Library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 2017, 56–62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (78) Lyakh, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' TAL-SH: Tensor Algebra Li- brary for Shared Memory Computers: Nodes equipped with multicore CPU, Nvidia GPU, and Intel Xeon Phi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' com/DmitryLyakh/TAL_SH, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (79) Valeev, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Libint - a library for the evalua- tion of molecular integrals of many-body op- erators over Gaussian functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' https:// github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='com/evaleev/libint, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (80) Kowalski, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Bair, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Bauman, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Boschen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Bylaska, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Daily, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' de Jong, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Dunning Jr, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Govind, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Harrison, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' From NWChem to NWChemEx: Evolving with the computa- tional chemistry landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 2021, 121, 4962–4998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (81) Gunnels, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Gustavson, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Henry, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Van De Geijn, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' FLAME: Formal linear algebra methods environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 2001, 27, 422–455.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (82) Solomonik, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Matthews, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Ham- mond, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Stanton, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Demmel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' A massively parallel tensor contraction framework for coupled-cluster computa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Parallel Distrib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 2014, 74, 3176–3190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (83) Epifanovsky, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Wormit, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Ku´s, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Landau, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Zuev, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Khistyaev, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Manohar, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Kaliman, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Dreuw, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Krylov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' New implementation of high- level correlated methods using a general block tensor library for high-performance electronic structure calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Wiley On- line Library 2013, (84) Ibrahim, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Williams, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Epi- fanovsky, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Krylov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Analysis and tuning of libtensor framework on multicore 14 architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 2014 21st International Con- ference on High Performance Computing (HiPC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' pp 1–10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (85) Røeggen, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Wisløff-Nilssen, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' On the Beebe-Linderberg two-electron integral ap- proximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 1986, 132, 154–160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (86) Koch, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' S´anchez de Mer´as, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Peder- sen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Reduced scaling in electronic structure calculations using Cholesky de- compositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 2003, 118, 9481–9484.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (87) Beebe, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Linderberg, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Simplifications in the generation and transformation of two- electron integrals in molecular calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Quantum Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 1977, 12, 683–705.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (88) Aquilante, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Pedersen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Lindh, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Low-cost evaluation of the exchange Fock matrix from Cholesky and density fitting rep- resentations of the electron repulsion inte- grals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 2007, 126, 194106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (89) Røeggen, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Johansen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Cholesky decom- position of the two-electron integral matrix in electronic structure calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 2008, 128, 194107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (90) https://cccbdb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='nist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content='gov/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (91) Kendall, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Dunning Jr, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Harri- son, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Electron affinities of the first-row atoms revisited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Systematic basis sets and wave functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 1992, 96, 6796–6806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (92) Noro, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Sekiya, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Koga, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Contracted polarization functions for the atoms helium through neon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 1997, 98, 25–32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (93) Noro, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Sekiya, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Koga, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Correlating basis sets for the H atom and the alkali- metal atoms from Li to Rb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 2003, 109, 85–90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (94) Travnikova, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Patanen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Soderstrom, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Lindblad, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Kas, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Vila, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' C´eolin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Marchenko, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Goldsztejn, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Guillemin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Energy-Dependent Rel- ative Cross Sections in Carbon 1s Pho- toionization: Separation of Direct Shake and Inelastic Scattering Effects in Single Molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' A 2019, 123, 7619–7636.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' (95) Kuramoto, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Ehara, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Nakatsuji, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Kitajima, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Tanaka, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' De Fanis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Tamenori, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Ueda, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' C 1s and O 1s photo- electron spectra of formaldehyde with satel- lites: theory and experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Spectros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Relat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' Phenomena 2005, 142, 253– 259.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} +page_content=' 15 Graphical TOC Entry For Table of Contents Only 16 C 1s Leadership-Class C 1S RT-EOM-CC + TAMM System XPS 0 1s O 1s RT-EOM-CC TAMM' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQf_gmP/content/2301.04249v1.pdf'} diff --git a/J9AzT4oBgHgl3EQfj_2L/vector_store/index.pkl b/J9AzT4oBgHgl3EQfj_2L/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..2fd724527288917eba18d91d8c5ea060d46370a7 --- /dev/null +++ b/J9AzT4oBgHgl3EQfj_2L/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:69628c7ebac7116f203e84354e06645d17b3f3499d73f9f6d2d0c9b393f97245 +size 183007 diff --git a/J9E4T4oBgHgl3EQf7g50/content/tmp_files/2301.05340v1.pdf.txt b/J9E4T4oBgHgl3EQf7g50/content/tmp_files/2301.05340v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..082f9d5bc641489b6577f805e2c369d54de432bb --- /dev/null +++ b/J9E4T4oBgHgl3EQf7g50/content/tmp_files/2301.05340v1.pdf.txt @@ -0,0 +1,1333 @@ +Topological superconductivity in helical crystals +Soma Yoshida1, Keiji Yada1, Yukio Tanaka1, Takehito Yokoyama2 +1 Department of Applied Physics, Nagoya University, Nagoya 464–8603, Japan +2 Department of Physics, Tokyo Institute if Technology, Tokyo 152–8551, Japan +We study superconductivity and surface Andreev bound states in helical crystals. We consider +the interlayer pairings along the helical hopping and investigate the surface local density of states +on the (001) and zigzag surfaces for all the possible irreducible representations. There are three +and four irreducible representations exhibiting the zero energy peaks in the local density of states +at the (001) and zigzag surfaces of helical lattices, respectively. By calculating the one dimensional +winging number, we show that these appearances of the zero energy peaks stem from the surface +Andreev bound states. +I. +INTRODUCTION +The symmetries of pair potentials are related to those +of the underlying crystals[1]. For example, in the sys- +tem with inversion symmetry, the symmetry of the pair +potential is classified into the even-parity spin-singlet or +odd-parity spin triplet states. In the transition of an un- +conventional superconductor (SC), one or more symme- +tries are broken in addition to U(1) symmetry breaking +in BCS SCs. Allowed pair potentials in the underlying +crystal lattice structure are classified by the irreducible +representations of the point group of the crystal lattice. +The symmetry of the pair potential has been extensively +studied in several SCs: cuprate, UTe2, and SrRuO4[2–7]. +Helical +crystals, +realized +in +materials +such +as +tellurium[8–14], have the right or left handedness. The +superconductivity in helical crystals has been found in, +e.g., NbRh2B2 and TaRh2B2[15–17]. As a result of the +helical crystal structures, current-induced orbital and +spin magnetizations in helical crystals have been the- +oretically proposed[18, 19], and chirality-induced spin +selectivity (CISS)[20–26] has been detected in helical +crystals[27–29]. +These effects inducing the magnetiza- +tion by the electric current are useful for the application +to spintronics. +The effect of helical molecules chemisorbed on the +conventional +SC +has +been +reported +in +the +recent +experiments[30–32]. Conductance spectra are observed +for the spin-singlet s-wave SC (Nb) through the helical +molecules by the STS and STM measurements. Interest- +ingly, they show zero bias conductance peaks. This result +is against the fact that a zero-bias conductance peak is +not exhibited on the surface of s-wave SCs because the +anisotropy of the gap function such as p-wave or d-wave +SCs is necessary to generate the zero energy bound states +on the surface[33–40]. The mechanism of this effect of +the helical molecules has not been established yet, while +this zero bias conductance peak structure suggests the +possibility of novel effect of helical structures. +It is known that the dispersionless Andreev bound +states (ABSs) are manifested as zero bias conductance +peaks on the surface of the unconventional SCs. +The +presence of the zero energy flat-band ABSs on the sur- +face is characterized by the topological number (winding +(a) Left-handed helix +(b) Right-handed helix +(c) 3D honeycomb lattice +A +B +K +M +K' +A +H' +H +L + Γ +(d) High symmetry points +FIG. 1. Helical lattice of the (a) left-handed helix and (b) +right-handed helix. (c) Three dimensional (3D) honeycomb +lattice is shown as a reference. (d) High symmetry points in +the Brillouin zone. A and B sites are marked by red and blue +balls, respectively. Red and blue bonds show the interlayer +hoppings between A and B sites, respectively. The hopping +amplitudes t2 and t3 are the interlayer ones along bonds in +the helical and 3D honeycomb lattices, respectively. +number) defined in the bulk system[41]. A SC with non- +trivial winding number is identified with the topological +SC[42–50], and the bound states protected by the wind- +ing number are robust against any perturbations as long +as the system remains the symmetry to define the topo- +logical number. Thus, it is interesting to investigate the +ABSs and winding number in the system with helical +structures to clarify the symmetry of the pairing in the +helical systems. +In the above experiments[30–32], the helical molecules +have been absorbed on the Nb substrate. In this paper, +we focus on the possibility that the pair potentials are +induced in the helical molecules by the superconducting +proximity effect. In this scenario, we have to clarify what +types of pairings are induced and how they generate the +bound states on the surface of helical lattices. For this +arXiv:2301.05340v1 [cond-mat.supr-con] 13 Jan 2023 + +2 +TABLE I. Irreducible representations (Irreps) and basis functions φIR +µ (k) of the pair potentials for interlayer pairing in the +helical and honeycomb lattices, where µ indicates the sub-lattice degree of freedom. D6h and D6 represent the point groups +(PG) in the helical and honeycomb lattices, respectively. The inter-site components are zero because we focus on the interlayer +pairings. Node structures are obtained at t2/t1 = 0.1 or t3/t1 = 0.1 and ∆0/t1 = 0.1. +PG Irrep +Spin +Node +φIR +A (k) +φIR +B (k) +D6h A1g singlet point +cos kz +φA(k) +A2u triplet +line +sin kz +φA(k) +B1u singlet +cos kz +−φA(k) +B2g +triplet +sin kz +−φA(k) +D6 +A1 +singlet point +cos(kx + kz) + cos(kx/2 − +√ +3ky/2 − kz) + cos(kx/2 + +√ +3ky/2 − kz) +φA(kx, ky, −kz) +A2 +triplet +line +sin(kx + kz) − sin(kx/2 − +√ +3ky/2 − kz) − sin(kx/2 + +√ +3ky/2 − kz) +−φA(kx, ky, −kz) +B1 +singlet +cos(kx + kz) + cos(kx/2 − +√ +3ky/2 − kz) + cos(kx/2 + +√ +3ky/2 − kz) +−φA(kx, ky, −kz) +B2 +triplet +sin(kx + kz) − sin(kx/2 − +√ +3ky/2 − kz) − sin(kx/2 + +√ +3ky/2 − kz) +φA(kx, ky, −kz) +E1 +singlet +line +2 cos(kx + kz) − cos(kx/2 − +√ +3ky/2 − kz) − cos(kx/2 + +√ +3ky/2 − kz) +−φA(kx, ky, −kz) +− cos(kx/2 − +√ +3ky/2 − kz) + cos(kx/2 + +√ +3ky/2 − kz) +−φA(kx, ky, −kz) +E1 +triplet +line +− sin(kx/2 − +√ +3ky/2 − kz) + sin(kx/2 + +√ +3ky/2 − kz) +φA(kx, ky, −kz) +2 sin(kx + kz) + sin(kx/2 − +√ +3ky/2 − kz) + sin(kx/2 + +√ +3ky/2 − kz) +φA(kx, ky, −kz) +E2 +singlet +line +2 cos(kx + kz) − cos(kx/2 − +√ +3ky/2 − kz) − cos(kx/2 + +√ +3ky/2 − kz) +φA(kx, ky, −kz) +− cos(kx/2 − +√ +3ky/2 − kz) + cos(kx/2 + +√ +3ky/2 − kz) +φA(kx, ky, −kz) +E2 +triplet +line +− sin(kx/2 − +√ +3ky/2 − kz) + sin(kx/2 + +√ +3ky/2 − kz) +−φA(kx, ky, −kz) +2 sin(kx + kz) + sin(kx/2 − +√ +3ky/2 − kz) + sin(kx/2 + +√ +3ky/2 − kz) +−φA(kx, ky, −kz) +(a) (001) surface +z +(b) zigzag surface +x +y +x +y +z +FIG. 2. +Semi-infinite model with (a) (001) and (b) zigzag +surface. +purpose, we adopt the model calculation of helical crys- +tals and investigate the surface bound states on the (001) +and zigzag surfaces of helical lattices for all the possible +nearest interlayer pairings. For A1 and E1 representa- +tions of spin-singlet and E2 representation of spin-triplet, +zero energy peaks in the surface density of states (SDOS) +are obtained on the (001) surface. For E1 and E2 repre- +sentations, zero energy peaks are obtained on the zigzag +surface. +In addition, we verify that the corresponding +winding numbers are non-trivial. +This paper is organized as follows: In Sec. II, we intro- +duce the tight-binding model for the helical lattices, the +recursive Green function method and one dimensional +(1D) winding number. In Sec. III A, we classify the pos- +sible pair potentials into the irreducible representations +of the point group. In Sec. III B, we show the numerical +results of the SDOS. In Sec. III C, we show the numeri- +cal results of the winding number and verify the consis- +tency between the appearance of zero energy peaks in the +SDOS and non-trivial winding number. We summarize +our results in Sec. IV. +II. +FORMULATION +In this paper, we consider a three dimensional (3D) he- +lical lattice with the D6 point groups as shown in Fig. 1. +We also consider the 3D honeycomb lattice with D6h as +a reference. The Hamiltonian ˆH of the helical and hon- +eycomb lattices are given by[18, 19] +ˆH = t1 +� +⟨ij⟩σ +ˆciσˆcjσ + t2 +� +[ij]σ +ˆciσˆcjσ + t3 +� +{ij}σ +ˆciσˆcjσ ++ +� +ijσ +[∆ijˆci↑ˆcj↓ + h.c.], +(1) +where ciσ (c† +iσ) is an annihilation (creation) operator for +an electron with the spin σ at the site i, t1, t2 and t3 are +hopping amplitudes, and ∆ij is the pair potential of the +superconductivity. In our paper, the chemical potential is +set to zero. The first term in Eq. (1) represents a nearest- +neighbor hopping in xy plane. +The second and third +terms represent nearest neighbor layer hoppings in the +helical and honeycomb lattices, respectively, as shown in +Figs. 1(b) and (c). We set t3 (t2) to zero when we consider +the helical (honeycomb) lattice. We consider the nearest +neighbor layer pairings depending on kz to investigate the +pair potentials generating the bound states on the (001) +surface as the blue plane in Fig. 2(a). In addition, we +investigate the bound states on the zigzag surface as the +red plane in Fig. 2(b). In the interlayer pairings, ∆ij only +has a finite value when the set of i and j belongs to the +same sub-lattice. Due to the spin-rotational symmetry, +it is sufficient to consider the anti-parallel spin pairings. +We calculate the SDOS on the (001) and zigzag sur- +faces of semi-infinite helical and honeycomb lattices. For +this purpose, we consider the clean system with the (001) + +3 +and zigzag surfaces as shown in Fig. 2. We assume the +translational invariance along the direction parallel to the +surface. +Thus, the momentum parallel to the surface +k∥ = (k1, k2) is conserved in our system. +The Green’s function at the sites i and i′, k∥ and the +complex frequency ω is defined as follows: +G(i, i′, k∥, ω) = +� +ω ˆI − ˆH +�−1 +, +(2) +where ˆI is a unit matrix. The SDOS is calculated from +the retarded Green’s function: +ρµ(E) = − 1 +2π +� +Im +� +G(i, i, k∥, E + iη) +� +dk∥, +(3) +ρ(E) = +� +µ=A,B +ρµ(E), +(4) +where E and η are the energy and smearing factor, re- +spectively, and µ show the sub-lattice of the site i at the +surface. To calculate the retarded Green’s function at the +surface, we apply the recursive Green function method +proposed by Umerski[51–53]. +The dispersionless ABSs generated on the surface of +anisotropic SCs are characterized by the non-trivial 1D +winding number defined in the bulk[41]. The BdG Hamil- +tonian in the bulk is written as +H(k) = 1 +2 +� +ˆε(k)αα′ +ˆ∆αα′ +ˆ∆† +αα′ +−ˆεT (−k)αα′ +� +, +(5) +where α and α′ are indices of the spin and sub-lattice +degrees of freedom. Having the time-reversal symmetry, +the BdG Hamiltonian satisfies +ΘH(k)Θ−1 = H∗(−k), +Θ = +� +iˆsyˆτ0 +0 +0 +iˆsyˆτ0 +� +, +(6) +where ˆsi and ˆτi (i = 0, x, y, z) are the Pauli matrices +in the spin and sub-lattice spaces, respectively. In addi- +tion, the BdG Hamiltonian has the particle-hole symme- +try written as: +CH(k)C−1 = −H∗(−k), +C = +� +0 +ˆs0ˆτ0 +ˆs0ˆτ0 +0 +� +. +(7) +In order to define the winding number, we introduce the +chiral operator as Γ = −iCΘ in the spin-singlet case and +Γ = SzCΘ in the spin-triplet case[41, 54], where Sz is the +z-component of the spin operator defined as: +Sz = +� +ˆszˆτ0 +0 +0 +−ˆszˆτ0 +� +. +(8) +Thus, the flat bands for the triplet pairs are unstable +against the spin-orbit interactions. +The 1D winding number manifesting the dispersionless +ABSs is defined with Γ for k∥ as: +w(k∥) = − 1 +4πi +� +dk⊥tr[ΓH−1(k)∂k⊥H(k)] +(9) +where k⊥ is a momentum perpendicular to the surface +and the integration is taken over the possible k⊥ on the +Brillouin zone. The winding number at k∥ is equal to the +integer value N+ − N−, where N± is the number of zero +energy states with an eigenvalue Γ = ±1 at k∥. +III. +RESULTS +(a) A2 +(b) A2u +(c) E1(singlet) +(d) E2(triplet) +FIG. 3. +Surface density of states at the (001) surface of the +helical and honeycomb lattices in the normal and supercon- +ducting states. The SDOS are normalized by ρN being the +zero energy SDOS of the normal state. The irreducible repre- +sentations are shown on top of each figure. The SDOS of A2u +is calculated in the honeycomb lattice, and the others are in +the helical lattice. We specify either spin singlet or triplet for +E1 and E2 representations. In the irreducible representations +that are not shown here, no zero energy peaks appear at the +(001) surface. +We take t1 as an energy unit and set other +hopping integrals as t2 = 0.1 or t3 = 0.1. The amplitudes of +pair potentials are set as ∆0 = 0.18 for A2 and A2u, ∆0 = 0.2 +for E1(singlet) and ∆0 = 0.4 for E2(triplet). +A. +Irreducible representations +We will investigate the possible pair potentials gener- +ating the bound states and the resulting surface bound +states in the helical lattice. For this purpose, we consider +the nearest layer pairings with the kz dependence. In this +case, the two electrons on the same sub-lattice constitute +the Cooper pair. Thus, ∆ij only has a finite value when +i and j belong to the same sub-lattice. +The possible order parameters are classified by the irre- +ducible representations of the point group symmetry[1]. + +250 +Normal +SC +200 +Nd/ +150 +(F) +100 +50 +0 +-1.0 +-0.5 +0.0 +0.5 +1.0 +E/△o20 +15 +Nd/( +10 +() +5 +0 +-1.0 +-0.5 +0.0 +0.5 +1.0 +E/△o25 +20 +Nd/(α)d +15 +10 +5 +0 +-1.0 +-0.5 +0.0 +0.5 +1.0 +E/△o15 +10 +() +5 +0 +-0.10 -0.05 +0.00 +0.05 +0.10 +E/△o4 +(a)E1(singlet) +(b)E1(triplet) +(c)E2(singlet) +(d)E2(triplet) +FIG. 4. +Surface density of states at the zigzag surface of +the helical lattice in the normal and superconducting states. +The figures are shown in the same manner as Fig. 3. In the +irreducible representations that are not shown here, no zero +energy peaks appear at the zigzag surface. The hopping pa- +rameters are set as the same values as in Fig. 3. The ampli- +tudes of the pair potential are set as ∆0 = 0.4 for E2(triplet) +and ∆0 = 0.2 for the other irreducible representations. +We decompose the pair potentials into the irreducible +representations and rewrite the superconducting parts of +the Hamiltonian as: +ˆH∆ = ∆0 +� +k,µ=A,B +[φIR +µ (k)ˆcµk↑ˆcµ−k↓ + h.c.], +(10) +where ∆0 is the amplitude of the pair potential, k and µ +are the momentum and index of sub-lattice, respectively, +and φIR +µ (k) is the basis function of the irreducible rep- +resentation of D6 or D6h. The basis functions φIR +µ (k) in +the helical lattice with D6 and honeycomb lattice with +D6h are shown in Table. I. There are two kinds of basis +functions distinguished by spin channels in E1 and E2 +representations. Hereafter, when necessary in E1 and E2 +representations, we append the spin channel to specify +the basis function; for example, we write E1 represen- +tation of the spin singlet as E1(singlet). There are two +basis functions in each E1 and E2 representation as seen +in Table. I. We will use the upper one in the model calcu- +lation. We have checked that similar results are obtained +for the lower basis function. +B. +Surface density of states +In this subsection, we show the numerical results of the +SDOS. We calculate the SDOS at the (001) and zigzag +surfaces for all the possible irreducible representations +shown in Table. I. We choose t1 as a unit of the energy +and set interlayer hoppings as t2/t1 = 0.1 or t3/t1 = 0.1. +In Figs. 3 and 4, we show the SDOS for the irreducible +representations exhibiting the zero energy peaks in the +SDOS. The SDOS of the irreducible representations be- +longing to D6 (D6h) point group are calculated at the +surface of the helical (honeycomb) lattice. The gap size +of E2(triplet) is accidentally much smaller than ∆0 in +our hopping parameters. Thus, we take ∆0 of E2(triplet) +larger than the ones for the other irreducible representa- +tions in Figs. 3 and 4. +The zero energy peaks appear at the (001) surface for +A2, A2u, E1(singlet), and E2(triplet) representations and +zigzag surface for E1 and E2 representations. +For the +other irreducible representations not shown in Figs. 3 +and 4, zero energy peaks are not obtained in the SDOS +(see Appendix A). In the helical lattice, there are three +representations, A2, E1(singlet) and E2(triplet) repre- +sentations, exhibiting the zero energy peak at the (001) +surface. On the other hand, A2u representation is the +only irreducible representation which shows zero energy +peak in the honeycomb lattice. At the zigzag surface, all +of the zero energy peaks in Fig. 4 are obtained in the he- +lical lattice. These appearance of the zero energy peaks +are characterized by 1D winding number in Eq. (9) as +discussed in the next subsection. +C. +One dimensional winding number +In this subsection, we calculate the 1D winding number +and investigate the correspondence between the presence +of the zero energy peaks and flat-band ABSs. In the nu- +merical calculation of the winding number, because of +the spin-rotational symmetry, we reduce the 8 × 8 BdG +Hamiltonian H(k) in Eq. (5) to a 4 × 4 matrix H4×4(k). +Thus, the winding numbers shown in this subsection take +half of the values defined in Eq. (9). +For all the ir- +reducible representations, we calculate the 1D winding +number in the Brillouin zone projected on the (001) and +zigzag surfaces. The hopping parameters are chosen as +t2/t1 = 0.1 or t3/t1 = 0.1 for all the irreducible rep- +resentations. For these parameters, the Fermi surfaces +are located around high symmetry points K, K′, H and +H′ shown in Fig.1(d). The winding numbers for the ir- +reducible representations considered in Figs. 3 and 4 are +shown in Figs. 5 and 6, respectively. The winding number +for the irreducible representations not shown in Figs. 5 +and 6 is zero over the surface Brillouin zone. +As shown in Fig. 5, the nodes of the gap function +for A2, A2u, E1(singlet) and E2(triplet) representations +make closed loops around K and K′ points in the Bril- +louin zone projected on the (001) surface. In addition, a +single nodal line goes through the K and K′ points for +E1(singlet) representation as shown in Fig.5(c), and two +nodal lines go through these points for E2(triplet) repre- +sentation as shown in Fig.5(d). The winding number has + +1.5 +1.0 +(E) +0.5 +0.0 +-0.10 -0.05 +0.00 +0.05 +0.10 +E/△o3.0 +Normal +2.5 +SC +2.0 +Nd/()d +1.5 +1.0 +0.5 +0.0 +-1.0 +-0.5 +0.0 +0.5 +1.0 +E/△o2.0 +1.5 +1.0 +0.5 +0.0 +-1.0 +-0.5 +0.0 +0.5 +1.0 +E/△o2.5 +2.0 +Nd/(α)d +1.5 +1.0 +0.5 +0.0 +-1.0 +-0.5 +0.0 +0.5 +1.0 +E/△o5 +(a)A2 +(b)A2u +(c)E1(singlet) +(d)E2(triplet) +FIG. 5. One dimensional winding number, Eq. (9), as a func- +tion of momentum k∥ parallel to the (001) surface. The irre- +ducible representations are shown on top of each figure. We +show the winding number for the irreducible representations +shown in Fig. 3, and those for the other irreducible represen- +tations are zero over the surface momentum k∥. The dashed +lines show the boundary of the Brillouin zone projected to +the (001) surface. For each irreducible representation, we set +the hopping parameters and pair potential as the same values +as in Fig.3. +Red and blue regions indicate w = ±2, while +white region represents w = 0. The black lines are the nodes +projected on the (001) surface. The nodal lines are drawn by +plotting the momenta at which +� +det[H4×4(k)] is less than +10−7t2 +1, where H4×4(k) is BdG Hamiltonian reduced to the +4 × 4 matrix. +the same value in a region surrounded by the nodal lines +and can change across the nodal line. The winding num- +ber takes w = +2 for A2 and A2u representations, and +w = ±2 for E1(singlet) and E2(triplet) representations. +For the irreducible representations shown in Fig. 6, +the nodal lines surround the K, K′, H and H′ points +projected on the zigzag surface. In particular, as shown +in Fig.6(a) and (d), there are two and three nodal lines +around these high symmetry points in E1(singlet) and +E2(triplet) representations, respectively. +The winding +number takes w = +1 in Fig. 6(b) and w = ±1 in the +other panels of Fig. 6. +The non-trivial values of the winding number obtained +in this subsections is consistent with the appearance of +the zero energy peaks in the SDOS shown in Figs. 3 and +4. Thus, the zero energy peaks shown in Figs. 3 and 4 +originate from the flat band ABSs protected by the topo- +logical number.[41] There are three irreducible represen- +tations, A1, E1(singlet) and E2 (triplet) representations, +generating the ABSs at the (001) surface of the helical +lattice, and four irreducible representations, E1 and E2 +(a)E1(singlet) +(b)E1(triplet) +(c)E2(singlet) +(d)E2(triplet) +FIG. 6. One dimensional winding number, Eq. (9), as a func- +tion of momentum k∥ parallel to the zigzag surface. +The +irreducible representations are shown on top of each figure. +We show the winding number for the irreducible represen- +tations shown in Fig. 4, and those for the other irreducible +representations are zero over the surface momentum k∥. The +dashed lines connect the high symmetry points projected on +the zigzag surface. For each irreducible representation, we set +the hopping parameters and pair potential as the same values +as in Fig.4. +Red and blue regions indicate w = ±1, while +white represents w = 0. The black lines are the line nodes +projected on the zigzag surface. The nodal lines are drawn +by plotting the momenta at which +� +det[H4×4(k)] is less than +10−7t2 +1, where H4×4(k) is BdG Hamiltonian reduced to the +4 × 4 matrix. +representations, generating the ABSs at the zigzag sur- +face of the helical lattice. +IV. +CONCLUSION +We have studied superconductivity in the helical lat- +tice with helical interlayer hopping and the 3D honey- +comb lattice as a reference. We have supposed the near- +est interlayer pairings under the mean field theory and +decomposed the pair potentials into all the irreducible +representations. +We have calculated the SDOS at the (001) and zigzag +surfaces for all the possible irreducible representations. +At the (001) surface of the helical lattice, the zero energy +peaks have appeared in the SDOS for A2, E1(singlet) +and E2(triplet) representations. At the zigzag surface, +the zero energy peaks have been obtained for E1 and E2 +representations. +Calculating the 1D winding number, +we have clarified the ABSs manifested as zero energy +peaks. +We have summarized the appearances of zero +energy peaks and values of winding number of all the + +1.5 +K' +1.0 +0.5 +K +=+2 +0.0 +0 +-0.5 +0 +0.4 +0.8 +1.2 +1.61.5 +1.0 +0.5 +w=十2 +0.0 +-0.5 +0 +0.4 +0.8 +1.2 +1.6 +kc1.5 +w=-l +1.0 +w=十l +0.5 +0.0 +0 +-0.5 +0 +0.4 +0.8 +1.2 +1.6 +kc1.5 +1.0 +w=十1 +0.5 +w=-l +0.0 +0 +-0.5 +3 +0 +0.4 +0.8 +1.2 +1.6 +kc1.5 +=十1 +1 +H +H' +0.5 +2 +K +K +K' +0 +3 +-0.5 +0.4 +0.8 +1.2 +1.61.5 +1 +w=十l +0.5 +2 +K +0 +0 +3 +-0.5 +0.4 +0.8 +1.2 +1.61.5 +T1二3 +1 +w=+l +0.5 +2 +K +0 +3 +0 +-0.5 +0.4 +0.8 +1.2 +1.61.5 +1 +w=十l +0.5 +2 +K +0 +1 +3 +-0.5 +0.4 +0.8 +1.2 +1.66 +TABLE II. Summary of the results in the helical lattice with +D6 and honeycomb lattice with D6h. +The basis functions +of each irreducible representation (Irrep) of each point group +(PG) are shown in Table. I. We clarify the spin channels of E1 +and E2 to distinguish the basis functions. Checks and crosses +indicate the presence and absence of the zero energy peak, +respectively. +The zero and finite numbers show the trivial +and non-trivial winding number. +PG +Irrep +Zero energy peak Winding number +Zigzag +(001) +Zigzag +(001) +surface +surface +surface +surface +D6h +A1g +× +× +0 +0 +A2u +× +✓ +0 +2 +B2g +× +× +0 +0 +B1u +× +× +0 +0 +D6 +A1 +× +× +0 +0 +A2 +× +✓ +0 +2 +B1 +× +× +0 +0 +B2 +× +× +0 +0 +E1(singlet) +✓ +✓ +±1 +±2 +✓ +✓ +±1 +±2 +E1(triplet) +✓ +× ++1 +0 +✓ +× +−1 +0 +E2(singlet) +✓ +× +±1 +0 +✓ +× +±1 +0 +E2(triplet) +✓ +✓ +±1 +±2 +✓ +✓ +±1 +±2 +possible irreducible representations in Table. II. +ACKNOWLEDGMENTS +S.Y. would like to take this opportunity to thank +the “Nagoya University Interdisciplinary Frontier Fel- +lowship” supported by Nagoya University and JST, +the +establishment +of +university +fellowships +towards +the creation of science technology innovation, Grant +Number JPMJFS2120. +T.Y. was supported by JSPS +KAKENHI Grant Number JP30578216 and the JSPS- +EPSRC Core-to-Core program ”Oxide Superspin”. Y.T. +was supported by Scientific Research (A) (KAKENHI +Grant No. +JP20H00131) and Scientific Research (B) +(KAKENHI No. +JP20H01857). +This work is sup- +ported by the JSPS Core-to-Core program (Grants No. +JPJSCCA20170002). +Appendix A: Numerical results of the surface +density of states without a zero energy peak +Here, we show the SDOS for the other representations +not exhibited in the main text. We show the SDOS on the +(001) and zigzag surfaces in Figs. 7 and 8, respectively. +The zero energy peaks do not appear for all the SDOS in +Figs.7 and 8. +[1] M. Sigrist and K. Ueda, Rev. Mod. Phys. 63, 239 (1991). +[2] C. C. Tsuei and J. R. Kirtley, Rev. Mod. Phys. 72, 969 +(2000). +[3] C. Kallin and J. Berlinsky, Rep. Prog. Phys. 79, 054502 +(2016). +[4] L. Jiao, S. Howard, S. Ran, Z. Wang, J. O. Rodriguez, +M. Sigrist, Z. Wang, N. P. Butch, +and V. Madhavan, +Nature 579, 523 (2020). +[5] P. J. Hirschfeld, M. M. Korshunov, and I. I. Mazin, Rep. +Prog. Phys. 74, 124508 (2011). +[6] S. Yoshida, K. Yada, and Y. Tanaka, Phys. Rev. B 104, +094506 (2021). +[7] D. Aoki, J.-P. Brison, J. Flouquet, K. Ishida, G. Knebel, +Y. Tokunaga, and Y. Yanase, J. Phys. Condens. Matter +34, 243002 (2022). +[8] J. R. Reitz, Phys. Rev. 105, 1233 (1957). +[9] R. S. Caldwell and H. Y. Fan, Phys. Rev. 114, 664 (1959). +[10] M. Hulin, J. Phys, Chem. of Solids 27, 441 (1966). +[11] L. D. Laude, B. Fitton, +and M. Anderegg, Phys. Rev. +Lett. 26, 637 (1971). +[12] K. Nakayama, M. Kuno, K. Yamauchi, S. Souma, K. Sug- +awara, T. Oguchi, T. Sato, and T. Takahashi, Phys. Rev. +B 95, 125204 (2017). +[13] S. S. Tsirkin, P. A. Puente, and I. Souza, Phys. Rev. B +97, 035158 (2018). +[14] M. Sakano, M. Hirayama, T. Takahashi, S. Akebi, +M. Nakayama, K. Kuroda, K. Taguchi, T. Yoshikawa, +K. Miyamoto, T. Okuda, K. Ono, H. Kumigashira, +T. Ideue, Y. Iwasa, N. Mitsuishi, K. Ishizaka, S. Shin, +T. Miyake, S. Murakami, T. Sasagawa, and T. Kondo, +Phys. Rev. Lett. 124, 136404 (2020). +[15] E. +M. +Carnicom, +W. +Xie, +T. +Klimczuk, +J. +Lin, +K. G´ornicka, Z. Sobczak, N. P. Ong, +and R. J. Cava, +Science Advances 4, eaar7969 (2018). +[16] D. A. Mayoh, A. D. Hillier, K. G¨otze, D. M. Paul, G. Bal- +akrishnan, +and M. R. Lees, Phys. Rev. B 98, 014502 +(2018). +[17] K. Matano, R. Ogura, M. Fountaine, H. O. Jeschke, +S. Kawasaki, and G.-q. Zheng, Phys. Rev. B 104, 224508 +(2021). +[18] T. Yoda, T. Yokoyama, and S. Murakami, Sci. Rep. 5, +12024 (2015). +[19] T. Yoda, T. Yokoyama, +and S. Murakami, Nano Lett. +18, 916 (2018). +[20] K. Ray, S. P. Ananthavel, D. H. Waldeck, and R. Naa- +man, Science 283, 814 (1999). +[21] B. G¨ohler, V. Hamelbeck, T. Z. Markus, M. Kettner, +G. F. Hanne, Z. Vager, R. Naaman, +and H. Zacharias, +Science 331, 894 (2011). +[22] R. Naaman and D. H. Waldeck, Annu. Rev. Phys. Chem +66, 263 (2015). +[23] K. Michaeli, N. Kantor-Uriel, R. Naaman, +and D. H. +Waldeck, Chem. Soc. Rev. 45, 6478 (2016). +[24] R. Naaman, Y. Paltiel, +and D. H. Waldeck, J. Phys. +Chem. Lett. 11, 3660 (2020). +[25] D. H. Waldeck, R. Naaman, and Y. Paltiel, APL Mate- +rials 9, 040902 (2021). + +7 +(a) A1g +(b) B1u +(c) B2g +(d) A1 +(e) B1 +(f) B2 +(g) E1(triplet) +(h) E2(singlet) +FIG. 7. Surface density of states on the (001) surface. The +figures are shown in the same manner as Fig. 3. The hop- +ping parameters are set as the same values as in Fig. 3. The +amplitudes of the pair potential are set as ∆0 = 0.2. +[26] F. Evers, A. Aharony, N. Bar-Gill, O. Entin-Wohlman, +P. Hedeg˚ard, +O. Hod, +P. Jelinek, +G. Kamieniarz, +M. Lemeshko, K. Michaeli, V. Mujica, R. Naaman, +Y. Paltiel, S. Refaely-Abramson, O. Tal, J. Thijssen, +M. Thoss, J. M. van Ruitenbeek, L. Venkataraman, D. H. +Waldeck, B. Yan, +and L. Kronik, Adv. Mater. 34, +2106629 (2022). +[27] A. Inui, R. Aoki, Y. Nishiue, K. Shiota, Y. Kousaka, +H. Shishido, D. Hirobe, M. Suda, J.-i. Ohe, J.-i. Kishine, +H. M. Yamamoto, and Y. Togawa, Phys. Rev. Lett. 124, +166602 (2020). +(a) A1g +(b) A2u +(c) B1u +(d) B2g +(e) A1 +(f) A2 +(g) B1 +(h) B2 +FIG. 8. Surface density of states on the zigzag surface. The +figures are shown in the same manner as Fig. 3. The hopping +parameters and amplitudes of the pair potential are set as the +same values as in Fig. 7. +[28] K. Shiota, A. Inui, Y. Hosaka, R. Amano, Y. ¯Onuki, +M. Hedo, T. Nakama, D. Hirobe, J.-i. Ohe, J.-i. Kishine, +H. M. Yamamoto, H. Shishido, +and Y. Togawa, Phys. +Rev. Lett. 127, 126602 (2021). +[29] H. Shishido, R. Sakai, Y. Hosaka, and Y. Togawa, Appl. +Phys. Lett. 119, 182403 (2021). +[30] H. Alpern, E. Katzir, S. Yochelis, N. Katz, Y. Paltiel, +and O. Millo, New J. Phys. 18, 113048 (2016). +[31] H. Alpern, K. Yavilberg, T. Dvir, N. Sukenik, M. Klang, +S. Yochelis, +H. Cohen, +E. Grosfeld, +H. Steinberg, +Y. Paltiel, and O. Millo, Nano Lett. 19, 5167 (2019). + +2.5 +Normal +SC +2.0 +Nd/(α)d +1.5 +1.0 +0.5 +0.0 +-1.0 +-0.5 +0.0 +0.5 +1.0 +E/△o2.0 +1.5 +Nd/()d +1.0 +0.5 +0.0 +-1.0 +-0.5 +0.0 +0.5 +1.0 +E/△o2.0 +1.5 +Nd/()d +1.0 +0.5 +0.0 +-1.0 +-0.5 +0.0 +0.5 +1.0 +E/△o3.0 +Nd/(α)d +2.0 +1.0 +0.0 +-1.0 +-0.5 +0.0 +0.5 +1.0 +E/△o2.0 +1.5 +Nd/(α)d +1.0 +0.5 +0.0 +-1.0 +-0.5 +0.0 +0.5 +1.0 +E/△o2.0 +1.5 +Nd/(α)d +1.0 +0.5 +0.0 +-1.0 +-0.5 +0.0 +0.5 +1.0 +E/△o2.0 +1.5 +Nd/()d +1.0 +0.5 +0.0 +-1.0 +-0.5 +0.0 +0.5 +1.0 +E/△o2.0 +1.5 +Nd/(α)d +1.0 +0.5 +0.0 +-1.0 +-0.5 +0.0 +0.5 +1.0 +E/△oNormal +4.0 +SC +3.0 +Nd/()d +2.0 +1.0 +0.0 +-1.0 +-0.5 +0.0 +0.5 +1.0 +E/△o40 +30 +Nd/(α)d +20 +10 +0 +-1.0 +-0.5 +0.0 +0.5 +1.0 +E/△o4.0 +3.0 +Nd/()d +2.0 +1.0 +0.0 +-1.0 +-0.5 +0.0 +0.5 +1.0 +E/△o40 +30 +Nd/(α)d +20 +10 +0 +-1.0 +-0.5 +0.0 +0.5 +1.0 +E/△o2.0 +1.5 +Nd/(α)d +1.0 +0.5 +0.0 +-1.0 +-0.5 +0.0 +0.5 +1.0 +E/△o3.0 +Nd/()d +2.0 +1.0 +0.0 +-1.0 +-0.5 +0.0 +0.5 +1.0 +E/△o2.0 +1.5 +Nd/(α)d +1.0 +0.5 +0.0 +-1.0 +-0.5 +0.0 +0.5 +1.0 +E/△o5.0 +4.0 +Nd/()d +3.0 +2.0 +1.0 +0.0 +-1.0 +-0.5 +0.0 +0.5 +1.0 +E/△o8 +[32] H. Alpern, M. Amundsen, R. Hartmann, N. Sukenik, +A. Spuri, S. Yochelis, T. Prokscha, V. Gutkin, Y. Ana- +hory, E. Scheer, J. Linder, Z. Salman, O. Millo, Y. Paltiel, +and A. Di Bernardo, Phys. Rev. Mater. 5, 114801 (2021). +[33] A. Andreev, Sov Phys JETP 19, 1228 (1964). +[34] L. J. Buchholtz and G. Zwicknagl, Phys. Rev. B 23, 5788 +(1981). +[35] J. Hara and K. Nagai, Prog. Theor. Phys. 76, 1237 +(1986). +[36] C.-R. Hu, Phys. Rev. Lett. 72, 1526 (1994). +[37] Y. Tanaka and S. Kashiwaya, Phys. Rev. Lett. 74, 3451 +(1995). +[38] S. Kashiwaya and Y. Tanaka, Rep. Prog. Phys. 63, 1641 +(2000). +[39] T. L¨ofwander, V. S. Shumeiko, and G. Wendin, Super- +cond. Sci. Technol. 14, R53 (2001). +[40] Y. Asano, Y. Tanaka, and S. Kashiwaya, Phys. Rev. B +69, 134501 (2004). +[41] M. Sato, Y. Tanaka, K. Yada, and T. Yokoyama, Phys. +Rev. B , 224511 (2011). +[42] X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 +(2011). +[43] M. Leijnse and K. Flensberg, Semicond. Sci. Technol. 27, +124003 (2012). +[44] C. Beenakker, Annu. Rev. Condens. Matter Phys. 4, 113 +(2013). +[45] A. P. Schnyder and P. M. R. Brydon, J. Phy. Condens. +Matter 27, 243201 (2015). +[46] Y. Tanaka, M. Sato, and N. Nagaosa, J. Phys. Soc. Jpn. +81, 011013 (2012). +[47] C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu, +Rev. Mod. Phys. 88, 035005 (2016). +[48] M. Sato and S. Fujimoto, J. Phys. Soc. Jpn. 85, 072001 +(2016). +[49] M. Sato and Y. Ando, Rep. Prog. Phys. 80, 076501 +(2017). +[50] P. Marra, J. Appl. Phys. 132, 231101 (2022). +[51] A. Umerski, Phys. Rev. B 55, 5266 (1997). +[52] K. Yada, A. A. Golubov, Y. Tanaka, and S. Kashiwaya, +J. Phys. Soc. Jpn. 83, 074706 (2014). +[53] D. Takagi, S. Tamura, +and Y. Tanaka, Phys. Rev. B +101, 024509 (2020). +[54] S. Kobayashi, Y. Tanaka, +and M. Sato, Phys. Rev. B +92, 214514 (2015). + diff --git a/J9E4T4oBgHgl3EQf7g50/content/tmp_files/load_file.txt b/J9E4T4oBgHgl3EQf7g50/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..55204ce12d8619b1aeadcfdfe6d89e787e353650 --- /dev/null +++ b/J9E4T4oBgHgl3EQf7g50/content/tmp_files/load_file.txt @@ -0,0 +1,1062 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf,len=1061 +page_content='Topological superconductivity in helical crystals Soma Yoshida1, Keiji Yada1, Yukio Tanaka1, Takehito Yokoyama2 1 Department of Applied Physics, Nagoya University, Nagoya 464–8603, Japan 2 Department of Physics, Tokyo Institute if Technology, Tokyo 152–8551, Japan We study superconductivity and surface Andreev bound states in helical crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' We consider the interlayer pairings along the helical hopping and investigate the surface local density of states on the (001) and zigzag surfaces for all the possible irreducible representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' There are three and four irreducible representations exhibiting the zero energy peaks in the local density of states at the (001) and zigzag surfaces of helical lattices, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' By calculating the one dimensional winging number, we show that these appearances of the zero energy peaks stem from the surface Andreev bound states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' INTRODUCTION The symmetries of pair potentials are related to those of the underlying crystals[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' For example, in the sys- tem with inversion symmetry, the symmetry of the pair potential is classified into the even-parity spin-singlet or odd-parity spin triplet states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' In the transition of an un- conventional superconductor (SC), one or more symme- tries are broken in addition to U(1) symmetry breaking in BCS SCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Allowed pair potentials in the underlying crystal lattice structure are classified by the irreducible representations of the point group of the crystal lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' The symmetry of the pair potential has been extensively studied in several SCs: cuprate, UTe2, and SrRuO4[2–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Helical crystals, realized in materials such as tellurium[8–14], have the right or left handedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' The superconductivity in helical crystals has been found in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=', NbRh2B2 and TaRh2B2[15–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' As a result of the helical crystal structures, current-induced orbital and spin magnetizations in helical crystals have been the- oretically proposed[18, 19], and chirality-induced spin selectivity (CISS)[20–26] has been detected in helical crystals[27–29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' These effects inducing the magnetiza- tion by the electric current are useful for the application to spintronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' The effect of helical molecules chemisorbed on the conventional SC has been reported in the recent experiments[30–32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Conductance spectra are observed for the spin-singlet s-wave SC (Nb) through the helical molecules by the STS and STM measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Interest- ingly, they show zero bias conductance peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' This result is against the fact that a zero-bias conductance peak is not exhibited on the surface of s-wave SCs because the anisotropy of the gap function such as p-wave or d-wave SCs is necessary to generate the zero energy bound states on the surface[33–40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' The mechanism of this effect of the helical molecules has not been established yet, while this zero bias conductance peak structure suggests the possibility of novel effect of helical structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' It is known that the dispersionless Andreev bound states (ABSs) are manifested as zero bias conductance peaks on the surface of the unconventional SCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=" The presence of the zero energy flat-band ABSs on the sur- face is characterized by the topological number (winding (a) Left-handed helix (b) Right-handed helix (c) 3D honeycomb lattice A B K M K' A H' H L Γ (d) High symmetry points FIG." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Helical lattice of the (a) left-handed helix and (b) right-handed helix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' (c) Three dimensional (3D) honeycomb lattice is shown as a reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' (d) High symmetry points in the Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' A and B sites are marked by red and blue balls, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Red and blue bonds show the interlayer hoppings between A and B sites, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' The hopping amplitudes t2 and t3 are the interlayer ones along bonds in the helical and 3D honeycomb lattices, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' number) defined in the bulk system[41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' A SC with non- trivial winding number is identified with the topological SC[42–50], and the bound states protected by the wind- ing number are robust against any perturbations as long as the system remains the symmetry to define the topo- logical number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Thus, it is interesting to investigate the ABSs and winding number in the system with helical structures to clarify the symmetry of the pairing in the helical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' In the above experiments[30–32], the helical molecules have been absorbed on the Nb substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' In this paper, we focus on the possibility that the pair potentials are induced in the helical molecules by the superconducting proximity effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' In this scenario, we have to clarify what types of pairings are induced and how they generate the bound states on the surface of helical lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' For this arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='05340v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='supr-con] 13 Jan 2023 2 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Irreducible representations (Irreps) and basis functions φIR µ (k) of the pair potentials for interlayer pairing in the helical and honeycomb lattices, where µ indicates the sub-lattice degree of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' D6h and D6 represent the point groups (PG) in the helical and honeycomb lattices, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' The inter-site components are zero because we focus on the interlayer pairings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Node structures are obtained at t2/t1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='1 or t3/t1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='1 and ∆0/t1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' PG Irrep Spin Node φIR A (k) φIR B (k) D6h A1g singlet point cos kz φA(k) A2u triplet line sin kz φA(k) B1u singlet cos kz −φA(k) B2g triplet sin kz −φA(k) D6 A1 singlet point cos(kx + kz) + cos(kx/2 − √ 3ky/2 − kz) + cos(kx/2 + √ 3ky/2 − kz) φA(kx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' ky,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' −kz) A2 triplet line sin(kx + kz) − sin(kx/2 − √ 3ky/2 − kz) − sin(kx/2 + √ 3ky/2 − kz) −φA(kx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' ky,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' −kz) B1 singlet cos(kx + kz) + cos(kx/2 − √ 3ky/2 − kz) + cos(kx/2 + √ 3ky/2 − kz) −φA(kx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' ky,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' −kz) B2 triplet sin(kx + kz) − sin(kx/2 − √ 3ky/2 − kz) − sin(kx/2 + √ 3ky/2 − kz) φA(kx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' ky,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' −kz) E1 singlet line 2 cos(kx + kz) − cos(kx/2 − √ 3ky/2 − kz) − cos(kx/2 + √ 3ky/2 − kz) −φA(kx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' ky,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' −kz) − cos(kx/2 − √ 3ky/2 − kz) + cos(kx/2 + √ 3ky/2 − kz) −φA(kx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' ky,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' −kz) E1 triplet line − sin(kx/2 − √ 3ky/2 − kz) + sin(kx/2 + √ 3ky/2 − kz) φA(kx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' ky,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' −kz) 2 sin(kx + kz) + sin(kx/2 − √ 3ky/2 − kz) + sin(kx/2 + √ 3ky/2 − kz) φA(kx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' ky,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' −kz) E2 singlet line 2 cos(kx + kz) − cos(kx/2 − √ 3ky/2 − kz) − cos(kx/2 + √ 3ky/2 − kz) φA(kx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' ky,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' −kz) − cos(kx/2 − √ 3ky/2 − kz) + cos(kx/2 + √ 3ky/2 − kz) φA(kx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' ky,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' −kz) E2 triplet line − sin(kx/2 − √ 3ky/2 − kz) + sin(kx/2 + √ 3ky/2 − kz) −φA(kx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' ky,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' −kz) 2 sin(kx + kz) + sin(kx/2 − √ 3ky/2 − kz) + sin(kx/2 + √ 3ky/2 − kz) −φA(kx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' ky,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' −kz) (a) (001) surface z (b) zigzag surface x y x y z FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Semi-infinite model with (a) (001) and (b) zigzag surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' purpose, we adopt the model calculation of helical crys- tals and investigate the surface bound states on the (001) and zigzag surfaces of helical lattices for all the possible nearest interlayer pairings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' For A1 and E1 representa- tions of spin-singlet and E2 representation of spin-triplet, zero energy peaks in the surface density of states (SDOS) are obtained on the (001) surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' For E1 and E2 repre- sentations, zero energy peaks are obtained on the zigzag surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' In addition, we verify that the corresponding winding numbers are non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' This paper is organized as follows: In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' II, we intro- duce the tight-binding model for the helical lattices, the recursive Green function method and one dimensional (1D) winding number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' III A, we classify the pos- sible pair potentials into the irreducible representations of the point group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' III B, we show the numerical results of the SDOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' III C, we show the numeri- cal results of the winding number and verify the consis- tency between the appearance of zero energy peaks in the SDOS and non-trivial winding number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' We summarize our results in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' FORMULATION In this paper, we consider a three dimensional (3D) he- lical lattice with the D6 point groups as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' We also consider the 3D honeycomb lattice with D6h as a reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' The Hamiltonian ˆH of the helical and hon- eycomb lattices are given by[18, 19] ˆH = t1 � ⟨ij⟩σ ˆciσˆcjσ + t2 � [ij]σ ˆciσˆcjσ + t3 � {ij}σ ˆciσˆcjσ + � ijσ [∆ijˆci↑ˆcj↓ + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='], (1) where ciσ (c† iσ) is an annihilation (creation) operator for an electron with the spin σ at the site i, t1, t2 and t3 are hopping amplitudes, and ∆ij is the pair potential of the superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' In our paper, the chemical potential is set to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' The first term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' (1) represents a nearest- neighbor hopping in xy plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' The second and third terms represent nearest neighbor layer hoppings in the helical and honeycomb lattices, respectively, as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 1(b) and (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' We set t3 (t2) to zero when we consider the helical (honeycomb) lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' We consider the nearest neighbor layer pairings depending on kz to investigate the pair potentials generating the bound states on the (001) surface as the blue plane in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' In addition, we investigate the bound states on the zigzag surface as the red plane in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' In the interlayer pairings, ∆ij only has a finite value when the set of i and j belongs to the same sub-lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Due to the spin-rotational symmetry, it is sufficient to consider the anti-parallel spin pairings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' We calculate the SDOS on the (001) and zigzag sur- faces of semi-infinite helical and honeycomb lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' For this purpose, we consider the clean system with the (001) 3 and zigzag surfaces as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' We assume the translational invariance along the direction parallel to the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Thus, the momentum parallel to the surface k∥ = (k1, k2) is conserved in our system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' The Green’s function at the sites i and i′, k∥ and the complex frequency ω is defined as follows: G(i, i′, k∥, ω) = � ω ˆI − ˆH �−1 , (2) where ˆI is a unit matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' The SDOS is calculated from the retarded Green’s function: ρµ(E) = − 1 2π � Im � G(i, i, k∥, E + iη) � dk∥, (3) ρ(E) = � µ=A,B ρµ(E), (4) where E and η are the energy and smearing factor, re- spectively, and µ show the sub-lattice of the site i at the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' To calculate the retarded Green’s function at the surface, we apply the recursive Green function method proposed by Umerski[51–53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' The dispersionless ABSs generated on the surface of anisotropic SCs are characterized by the non-trivial 1D winding number defined in the bulk[41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' The BdG Hamil- tonian in the bulk is written as H(k) = 1 2 � ˆε(k)αα′ ˆ∆αα′ ˆ∆† αα′ −ˆεT (−k)αα′ � , (5) where α and α′ are indices of the spin and sub-lattice degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Having the time-reversal symmetry, the BdG Hamiltonian satisfies ΘH(k)Θ−1 = H∗(−k), Θ = � iˆsyˆτ0 0 0 iˆsyˆτ0 � , (6) where ˆsi and ˆτi (i = 0, x, y, z) are the Pauli matrices in the spin and sub-lattice spaces, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' In addi- tion, the BdG Hamiltonian has the particle-hole symme- try written as: CH(k)C−1 = −H∗(−k), C = � 0 ˆs0ˆτ0 ˆs0ˆτ0 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' (7) In order to define the winding number, we introduce the chiral operator as Γ = −iCΘ in the spin-singlet case and Γ = SzCΘ in the spin-triplet case[41, 54], where Sz is the z-component of the spin operator defined as: Sz = � ˆszˆτ0 0 0 −ˆszˆτ0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' (8) Thus, the flat bands for the triplet pairs are unstable against the spin-orbit interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' The 1D winding number manifesting the dispersionless ABSs is defined with Γ for k∥ as: w(k∥) = − 1 4πi � dk⊥tr[ΓH−1(k)∂k⊥H(k)] (9) where k⊥ is a momentum perpendicular to the surface and the integration is taken over the possible k⊥ on the Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' The winding number at k∥ is equal to the integer value N+ − N−, where N± is the number of zero energy states with an eigenvalue Γ = ±1 at k∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' RESULTS (a) A2 (b) A2u (c) E1(singlet) (d) E2(triplet) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Surface density of states at the (001) surface of the helical and honeycomb lattices in the normal and supercon- ducting states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' The SDOS are normalized by ρN being the zero energy SDOS of the normal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' The irreducible repre- sentations are shown on top of each figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' The SDOS of A2u is calculated in the honeycomb lattice, and the others are in the helical lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' We specify either spin singlet or triplet for E1 and E2 representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' In the irreducible representations that are not shown here, no zero energy peaks appear at the (001) surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' We take t1 as an energy unit and set other hopping integrals as t2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='1 or t3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' The amplitudes of pair potentials are set as ∆0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='18 for A2 and A2u, ∆0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='2 for E1(singlet) and ∆0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='4 for E2(triplet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Irreducible representations We will investigate the possible pair potentials gener- ating the bound states and the resulting surface bound states in the helical lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' For this purpose, we consider the nearest layer pairings with the kz dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' In this case, the two electrons on the same sub-lattice constitute the Cooper pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Thus, ∆ij only has a finite value when i and j belong to the same sub-lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' The possible order parameters are classified by the irre- ducible representations of the point group symmetry[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 250 Normal SC 200 Nd/ 150 (F) 100 50 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='0 E/△o20 15 Nd/( 10 () 5 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='0 E/△o25 20 Nd/(α)d 15 10 5 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='0 E/△o15 10 () 5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='10 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='10 E/△o4 (a)E1(singlet) (b)E1(triplet) (c)E2(singlet) (d)E2(triplet) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Surface density of states at the zigzag surface of the helical lattice in the normal and superconducting states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' The figures are shown in the same manner as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' In the irreducible representations that are not shown here, no zero energy peaks appear at the zigzag surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' The hopping pa- rameters are set as the same values as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' The ampli- tudes of the pair potential are set as ∆0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='4 for E2(triplet) and ∆0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='2 for the other irreducible representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' We decompose the pair potentials into the irreducible representations and rewrite the superconducting parts of the Hamiltonian as: ˆH∆ = ∆0 � k,µ=A,B [φIR µ (k)ˆcµk↑ˆcµ−k↓ + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='], (10) where ∆0 is the amplitude of the pair potential, k and µ are the momentum and index of sub-lattice, respectively, and φIR µ (k) is the basis function of the irreducible rep- resentation of D6 or D6h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' The basis functions φIR µ (k) in the helical lattice with D6 and honeycomb lattice with D6h are shown in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' There are two kinds of basis functions distinguished by spin channels in E1 and E2 representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Hereafter, when necessary in E1 and E2 representations, we append the spin channel to specify the basis function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' for example, we write E1 represen- tation of the spin singlet as E1(singlet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' There are two basis functions in each E1 and E2 representation as seen in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' We will use the upper one in the model calcu- lation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' We have checked that similar results are obtained for the lower basis function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Surface density of states In this subsection, we show the numerical results of the SDOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' We calculate the SDOS at the (001) and zigzag surfaces for all the possible irreducible representations shown in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' We choose t1 as a unit of the energy and set interlayer hoppings as t2/t1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='1 or t3/t1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 3 and 4, we show the SDOS for the irreducible representations exhibiting the zero energy peaks in the SDOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' The SDOS of the irreducible representations be- longing to D6 (D6h) point group are calculated at the surface of the helical (honeycomb) lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' The gap size of E2(triplet) is accidentally much smaller than ∆0 in our hopping parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Thus, we take ∆0 of E2(triplet) larger than the ones for the other irreducible representa- tions in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' The zero energy peaks appear at the (001) surface for A2, A2u, E1(singlet), and E2(triplet) representations and zigzag surface for E1 and E2 representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' For the other irreducible representations not shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 3 and 4, zero energy peaks are not obtained in the SDOS (see Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' In the helical lattice, there are three representations, A2, E1(singlet) and E2(triplet) repre- sentations, exhibiting the zero energy peak at the (001) surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' On the other hand, A2u representation is the only irreducible representation which shows zero energy peak in the honeycomb lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' At the zigzag surface, all of the zero energy peaks in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 4 are obtained in the he- lical lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' These appearance of the zero energy peaks are characterized by 1D winding number in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' (9) as discussed in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' One dimensional winding number In this subsection, we calculate the 1D winding number and investigate the correspondence between the presence of the zero energy peaks and flat-band ABSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' In the nu- merical calculation of the winding number, because of the spin-rotational symmetry, we reduce the 8 × 8 BdG Hamiltonian H(k) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' (5) to a 4 × 4 matrix H4×4(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Thus, the winding numbers shown in this subsection take half of the values defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' For all the ir- reducible representations, we calculate the 1D winding number in the Brillouin zone projected on the (001) and zigzag surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' The hopping parameters are chosen as t2/t1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='1 or t3/t1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='1 for all the irreducible rep- resentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' For these parameters, the Fermi surfaces are located around high symmetry points K, K′, H and H′ shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='1(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' The winding numbers for the ir- reducible representations considered in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 3 and 4 are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 5 and 6, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' The winding number for the irreducible representations not shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 5 and 6 is zero over the surface Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 5, the nodes of the gap function for A2, A2u, E1(singlet) and E2(triplet) representations make closed loops around K and K′ points in the Bril- louin zone projected on the (001) surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' In addition, a single nodal line goes through the K and K′ points for E1(singlet) representation as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='5(c), and two nodal lines go through these points for E2(triplet) repre- sentation as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='5(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' The winding number has 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='0 (E) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='10 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='10 E/△o3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='0 Normal 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='5 SC 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='0 Nd/()d 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='0 E/△o2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='0 E/△o2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='0 Nd/(α)d 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='0 E/△o5 (a)A2 (b)A2u (c)E1(singlet) (d)E2(triplet) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' One dimensional winding number, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' (9), as a func- tion of momentum k∥ parallel to the (001) surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' The irre- ducible representations are shown on top of each figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' We show the winding number for the irreducible representations shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 3, and those for the other irreducible represen- tations are zero over the surface momentum k∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' The dashed lines show the boundary of the Brillouin zone projected to the (001) surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' For each irreducible representation, we set the hopping parameters and pair potential as the same values as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Red and blue regions indicate w = ±2, while white region represents w = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' The black lines are the nodes projected on the (001) surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' The nodal lines are drawn by plotting the momenta at which � det[H4×4(k)] is less than 10−7t2 1, where H4×4(k) is BdG Hamiltonian reduced to the 4 × 4 matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' the same value in a region surrounded by the nodal lines and can change across the nodal line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' The winding num- ber takes w = +2 for A2 and A2u representations, and w = ±2 for E1(singlet) and E2(triplet) representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' For the irreducible representations shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 6, the nodal lines surround the K, K′, H and H′ points projected on the zigzag surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' In particular, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='6(a) and (d), there are two and three nodal lines around these high symmetry points in E1(singlet) and E2(triplet) representations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' The winding number takes w = +1 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 6(b) and w = ±1 in the other panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' The non-trivial values of the winding number obtained in this subsections is consistent with the appearance of the zero energy peaks in the SDOS shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Thus, the zero energy peaks shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 3 and 4 originate from the flat band ABSs protected by the topo- logical number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' [41] There are three irreducible represen- tations, A1, E1(singlet) and E2 (triplet) representations, generating the ABSs at the (001) surface of the helical lattice, and four irreducible representations, E1 and E2 (a)E1(singlet) (b)E1(triplet) (c)E2(singlet) (d)E2(triplet) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' One dimensional winding number, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' (9), as a func- tion of momentum k∥ parallel to the zigzag surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' The irreducible representations are shown on top of each figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' We show the winding number for the irreducible represen- tations shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 4, and those for the other irreducible representations are zero over the surface momentum k∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' The dashed lines connect the high symmetry points projected on the zigzag surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' For each irreducible representation, we set the hopping parameters and pair potential as the same values as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Red and blue regions indicate w = ±1, while white represents w = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' The black lines are the line nodes projected on the zigzag surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' The nodal lines are drawn by plotting the momenta at which � det[H4×4(k)] is less than 10−7t2 1, where H4×4(k) is BdG Hamiltonian reduced to the 4 × 4 matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' representations, generating the ABSs at the zigzag sur- face of the helical lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' CONCLUSION We have studied superconductivity in the helical lat- tice with helical interlayer hopping and the 3D honey- comb lattice as a reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' We have supposed the near- est interlayer pairings under the mean field theory and decomposed the pair potentials into all the irreducible representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' We have calculated the SDOS at the (001) and zigzag surfaces for all the possible irreducible representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' At the (001) surface of the helical lattice, the zero energy peaks have appeared in the SDOS for A2, E1(singlet) and E2(triplet) representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' At the zigzag surface, the zero energy peaks have been obtained for E1 and E2 representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Calculating the 1D winding number, we have clarified the ABSs manifested as zero energy peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' We have summarized the appearances of zero energy peaks and values of winding number of all the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content="5 K' 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='5 K =+2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='5 0 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='66 TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Summary of the results in the helical lattice with D6 and honeycomb lattice with D6h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' The basis functions of each irreducible representation (Irrep) of each point group (PG) are shown in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' We clarify the spin channels of E1 and E2 to distinguish the basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Checks and crosses indicate the presence and absence of the zero energy peak, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' The zero and finite numbers show the trivial and non-trivial winding number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' PG Irrep Zero energy peak Winding number Zigzag (001) Zigzag (001) surface surface surface surface D6h A1g × × 0 0 A2u × ✓ 0 2 B2g × × 0 0 B1u × × 0 0 D6 A1 × × 0 0 A2 × ✓ 0 2 B1 × × 0 0 B2 × × 0 0 E1(singlet) ✓ ✓ ±1 ±2 ✓ ✓ ±1 ±2 E1(triplet) ✓ × +1 0 ✓ × −1 0 E2(singlet) ✓ × ±1 0 ✓ × ±1 0 E2(triplet) ✓ ✓ ±1 ±2 ✓ ✓ ±1 ±2 possible irreducible representations in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' ACKNOWLEDGMENTS S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' would like to take this opportunity to thank the “Nagoya University Interdisciplinary Frontier Fel- lowship” supported by Nagoya University and JST, the establishment of university fellowships towards the creation of science technology innovation, Grant Number JPMJFS2120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' was supported by JSPS KAKENHI Grant Number JP30578216 and the JSPS- EPSRC Core-to-Core program ”Oxide Superspin”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' was supported by Scientific Research (A) (KAKENHI Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' JP20H00131) and Scientific Research (B) (KAKENHI No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' JP20H01857).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' This work is sup- ported by the JSPS Core-to-Core program (Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' JPJSCCA20170002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Appendix A: Numerical results of the surface density of states without a zero energy peak Here, we show the SDOS for the other representations not exhibited in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' We show the SDOS on the (001) and zigzag surfaces in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 7 and 8, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' The zero energy peaks do not appear for all the SDOS in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='7 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Sigrist and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Ueda, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 63, 239 (1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' [2] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Tsuei and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Kirtley, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 72, 969 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' [3] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Kallin and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Berlinsky, Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 79, 054502 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' [4] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Jiao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Howard, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Ran, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Rodriguez, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Sigrist, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Wang, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Butch, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Madhavan, Nature 579, 523 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' [5] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Hirschfeld, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Korshunov, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Mazin, Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 74, 124508 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' [6] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Yoshida, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Yada, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Tanaka, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' B 104, 094506 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' [7] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Aoki, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Brison, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Flouquet, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Ishida, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Knebel, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Tokunaga, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Yanase, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Matter 34, 243002 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' [8] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Reitz, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 105, 1233 (1957).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' [9] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Caldwell and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Fan, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 114, 664 (1959).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' [10] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Hulin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Phys, Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' of Solids 27, 441 (1966).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' [11] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Laude, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Fitton, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Anderegg, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 26, 637 (1971).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' [12] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Nakayama, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Kuno, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Yamauchi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Souma, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Sug- awara, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Oguchi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Sato, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Takahashi, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' B 95, 125204 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' [13] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Tsirkin, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Puente, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Souza, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' B 97, 035158 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' [14] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Sakano, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Hirayama, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Takahashi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Akebi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Nakayama, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Kuroda, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Taguchi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Yoshikawa, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Miyamoto, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Okuda, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Ono, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Kumigashira, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Ideue, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Iwasa, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Mitsuishi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Ishizaka, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Shin, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Miyake, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Murakami, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Sasagawa, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Kondo, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 124, 136404 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' [15] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Carnicom, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Xie, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Klimczuk, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Lin, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' G´ornicka, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Sobczak, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Ong, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Cava, Science Advances 4, eaar7969 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' [16] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Mayoh, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Hillier, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' G¨otze, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Paul, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Bal- akrishnan, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Lees, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' B 98, 014502 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' [17] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Matano, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Ogura, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Fountaine, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Jeschke, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Kawasaki, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='-q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Zheng, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' B 104, 224508 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' [18] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Yoda, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Yokoyama, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Murakami, Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 5, 12024 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' [19] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Yoda, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Yokoyama, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Murakami, Nano Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 18, 916 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' [20] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Ray, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Ananthavel, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Waldeck, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Naa- man, Science 283, 814 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' [21] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' G¨ohler, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Hamelbeck, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Markus, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Kettner, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Hanne, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Vager, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Naaman, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Zacharias, Science 331, 894 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' [22] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Naaman and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Waldeck, Annu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Chem 66, 263 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' [23] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Michaeli, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Kantor-Uriel, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Naaman, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Waldeck, Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 45, 6478 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' [24] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Naaman, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Paltiel, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Waldeck, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 11, 3660 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' [25] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Waldeck, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Naaman, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Paltiel, APL Mate- rials 9, 040902 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 7 (a) A1g (b) B1u (c) B2g (d) A1 (e) B1 (f) B2 (g) E1(triplet) (h) E2(singlet) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Surface density of states on the (001) surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' The figures are shown in the same manner as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' The hop- ping parameters are set as the same values as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' The amplitudes of the pair potential are set as ∆0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' [26] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Evers, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Aharony, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Bar-Gill, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Entin-Wohlman, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Hedeg˚ard, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Hod, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Jelinek, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Kamieniarz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Lemeshko, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Michaeli, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Mujica, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Naaman, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Paltiel, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Refaely-Abramson, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Tal, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Thijssen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Thoss, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' van Ruitenbeek, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Venkataraman, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Waldeck, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Yan, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Kronik, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 34, 2106629 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' [27] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Inui, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Aoki, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Nishiue, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Shiota, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Kousaka, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Shishido, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Hirobe, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Suda, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Ohe, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Kishine, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Yamamoto, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Togawa, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 124, 166602 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' (a) A1g (b) A2u (c) B1u (d) B2g (e) A1 (f) A2 (g) B1 (h) B2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Surface density of states on the zigzag surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' The figures are shown in the same manner as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' The hopping parameters and amplitudes of the pair potential are set as the same values as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' [28] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Shiota, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Inui, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Hosaka, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Amano, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' ¯Onuki, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Hedo, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Nakama, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Hirobe, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Ohe, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Kishine, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Yamamoto, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Shishido, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Togawa, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 127, 126602 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' [29] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Shishido, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Sakai, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Hosaka, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Togawa, Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 119, 182403 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' [30] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Alpern, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Katzir, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Yochelis, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Katz, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Paltiel, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Millo, New J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 18, 113048 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' [31] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Alpern, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Yavilberg, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Dvir, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Sukenik, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Klang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Yochelis, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Cohen, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Grosfeld, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Steinberg, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Paltiel, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Millo, Nano Lett.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Hartmann, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Sukenik, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Spuri, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Yochelis, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Prokscha, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Gutkin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Ana- hory, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Scheer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Linder, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Salman, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Millo, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Paltiel, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Di Bernardo, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 5, 114801 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' [33] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Andreev, Sov Phys JETP 19, 1228 (1964).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' [34] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Buchholtz and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Zwicknagl, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' B 23, 5788 (1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' [35] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Hara and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Nagai, Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 76, 1237 (1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' [36] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Hu, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 72, 1526 (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' [37] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Tanaka and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Kashiwaya, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 74, 3451 (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' [38] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Kashiwaya and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Tanaka, Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 63, 1641 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' [39] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' L¨ofwander, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Shumeiko, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Wendin, Super- cond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 14, R53 (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' [40] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Asano, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Tanaka, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Kashiwaya, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' B 69, 134501 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' [41] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Sato, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Tanaka, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Yada, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Yokoyama, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' B , 224511 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' [42] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Qi and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Zhang, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 83, 1057 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' [43] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Leijnse and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Flensberg, Semicond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 27, 124003 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' [44] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Beenakker, Annu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Matter Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 4, 113 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' [45] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Schnyder and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Brydon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Phy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Matter 27, 243201 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' [46] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Tanaka, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Sato, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Nagaosa, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Jpn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 81, 011013 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' [47] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Chiu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Teo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Schnyder, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Ryu, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 88, 035005 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' [48] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Sato and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Fujimoto, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Jpn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 85, 072001 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' [49] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Sato and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Ando, Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 80, 076501 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' [50] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Marra, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 132, 231101 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' [51] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Umerski, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' B 55, 5266 (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' [52] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Yada, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Golubov, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Tanaka, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Kashiwaya, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Jpn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' 83, 074706 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' [53] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Takagi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Tamura, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Tanaka, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' B 101, 024509 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' [54] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Kobayashi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Tanaka, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Sato, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} +page_content=' B 92, 214514 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E4T4oBgHgl3EQf7g50/content/2301.05340v1.pdf'} diff --git a/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf b/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..306c86b982140aec4101b4b262653eb05daac451 Binary files /dev/null and b/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf differ diff --git a/JNE2T4oBgHgl3EQfpAjJ/content/tmp_files/2301.04025v1.pdf.txt b/JNE2T4oBgHgl3EQfpAjJ/content/tmp_files/2301.04025v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..bd0a13bda986a629d38a0adbbf6f31f8be76389a --- /dev/null +++ b/JNE2T4oBgHgl3EQfpAjJ/content/tmp_files/2301.04025v1.pdf.txt @@ -0,0 +1,228 @@ +arXiv:2301.04025v1 [math.PR] 10 Jan 2023 +A NOTE ON THE LIMIT LAW OF ONE-SIDED TREE DESTRUCTION +MARKUS KUBA AND ALOIS PANHOLZER +ABSTRACT. This short note serves an addendum to the article ”Destruction of very sim- +ple trees” by Fill, Kapur and Panholzer (2004). Therein, the limit law of one-sided tree +destruction was determined by its moment sequence. We add an identification of the limit +law, using recent results of Bertoin (2022), in terms of the local time of a noise reinforced +Bessel process. +1. ADDENDUM: IDENTIFICATION OF THE LIMIT LAW +In Theorem 5.1 of [4] the limit law of the cost of root isolation Yn in the class of very +simple trees, a subclass of simply generated trees, was obtained. The random variable Yn +itself satisfies the distributional equation +Yn +L= YKn + tn +n ≥ 2, +for some random variable Kn [4], with initial value Y1 = t1 and toll function tn = na, +with a ≥ 0. It was shown that for a parameter σ > 0 and a′ = a + 1 +2 it holds +Yn +σna′ +L→ Ya′, +where Yα′ is uniquely determined by its moment sequence ms = E(Y s +a′), for positive +integers s ≥ 1: +ms = +s! +2s/2 +s +� +k=1 +Γ(ka′) +Γ(ka′ + 1 +2). +Recent results of Bertoin [2] on the noise reinforced Bessel process and its local time +allows us to observe the following Corollary to Theorem 5.1 of [4]. +Corollary 1. The limit law Ya′ has the same distribution as a moment-shifted (also called +size biased or moment-tilted) ˆLt of a local time of a noise reinforced Bessel process, scaled +by +1 +√ +2: +Ya′ L= 1 +√ +2 · tilt1 +� ˆLt/κ(p, t) +� +, +with factor κ(p, t) given in (1), dimension d = 1 and reinforcement parameter p = 1 +2 − 1 +4a′ . +Additionally, Ya′ L= +1 +κ(p,1) +√ +2 · ˆI, where the scaled exponential functional ˆI is given by +ˆI = +� ∞ +0 +exp(−1 +2 +ˆξt)dt, +with ˆξt denoting the subordinator with Laplace-Exponent ˆΦ, determined by +ˆΦ(r) = 2− 1 +2 � 1 +4a′ +� 1 +2 +Γ( 1 +2) +1 +2B( 1 +2, 4a′r). +2000 Mathematics Subject Classification. 05C05, 05A15, 05A19. +Key words and phrases. Tree destruction, cutting down, local time, limit law. +1 + +2 +M. KUBA AND A. PANHOLZER +Remark 1 (Moments of Gamma type). For a′ = 1 +2 one obtains a Rayleigh distribution, as +noted before [4]. For m ∈ N and m · a′ = 1 +2, i.e. a′ = 1 +4, we observe cancellations and +obtain interesting moments of Gamma type [5]. For example, for a′ = 1 +4 we get +ms = +1 +2s/2 +Γ( 1 +4)Γ( 1 +2)Γ(s + 1) +Γ( s+1 +4 )Γ( s+2 +4 ) +, +s ≥ 1. +Density functions for such moment sequences can be obtained using inverse Mellin com- +putations; compare with [1, Proof of Theorem 4.1] or see [5, Theorem 5.4]. +Proof. The identification is directly done using moment sequences appearing in [2]. There, +a noise reinforced Bessel process of dimension d > 0 and with reinforcement parameter +p ∈ (−∞, 1/2) was considered. Set +α = 1 − d/2 ∈ (0, 1) and β = +α +1 − 2p, +or equivalently +p = 1 +2 − α +2β , with β > 0. +The local time ˆLt of the noise reinforced Bessel process has power moments (see [2, The- +orem 1.2]), which can be written as +E( ˆLt +s) = κ(p, t)s · +1 − 2p +Γ(1 + α) · Γ(s) · +s−1 +� +j=1 +Γ(jβ) +Γ(α + jβ). +with scale factor given by +κ(p, t) = +(2t)α · Γ(1 + α) +(1 − 2p)α · Γ(1 − α). +(1) +This implies that the scaled random variable L = ˆLt/κ(p, t) has moment sequence µs = +E(Ls), given by +µs = +1 − 2p +Γ(1 + α) · Γ(s) · +s−1 +� +j=1 +Γ(jβ) +Γ(α + jβ), +s ≥ 1. +Using standard results on moment shifts (see for example [1, Lemma 3.4]), also called size +biased distributions or tilted distributions, we observe that the random variable T = tilt(L) +has moment sequence E(T s) = µs+1 +µ1 , or more explicitly +E(T s) = Γ(s + 1) · +s +� +j=1 +Γ(jβ) +Γ(α + jβ). +Since Γ(s + 1) = s!, we finally set β = a′ and α = 1 +2. This leads to the desired moment +sequence. Finally, we note that the choices lead to a dimension d = 1 and p = 1 +2 − +1 +4a′ , +such that 1 − 2p = +1 +4a′ . +Concerning the second statement we use the definition of ˆI and its Laplace exponent [2, +Equations 4.3 and 4.4]. We use Corollary 4.3 of [2], which relates ˆI and ˆL1. Note that +there is a small misprint [3] in the stated equation, as the prefactor +c = (1/2 − p)−α 1 − 2p +Γ(1 − α), +which can be rewritten into +�1 − 2p +2 +�−α 1 − 2p +Γ(1 − α) = 2α(1 − 2p)1−α +Γ(1 − α) += E(ˆL1), + +LIMIT LAW OF ONE-SIDED TREE DESTRUCTION +3 +should be replaced by its reciprocal. We state here the corrected version: +E +� +f(ˆI) +� += +1 +E(ˆL1) +· E +�ˆL1f(ˆL1) +� +. +Choosing f as the power functions, we observe a tilt of the moments [1, Lemma 3.4], +leading to the moment sequence of ˆI as the shifted moments of ˆL1. +□ +Remark 2 (More moments of Gamma type). As already pointed out [2], for p = 0, such +that α = β, the random variable ˆLt simplifies to a Mittag-Leffler distribution: +E( ˆLt +s) = κ(0, t)s · +1 +Γ(1 + α) · Γ(s) · Γ(α) +Γ(sα). = κ(0, t)s Γ(s + 1) +Γ(sα + 1). +The random variable T also leads to moments of Gamma types for small β, proportional +to α: β = α/m, m ∈ N, leading to moments +E(T s) = Γ(s + 1) · +m +� +j=1 +Γ( jα +m ) +Γ( (s+j)α +m +) +, +s ≥ 1. +Again, density functions can be obtained for such moments. +2. OUTLOOK AND ACKNOWLEDGMENTS +The authors are currently investigating into other occurrences of such limit laws in com- +binatorial probability. +The authors warmly thank Jean Bertoin for feedback on his work and explanations. +REFERENCES +[1] Cyril Banderier, Markus Kuba and Michael Wallner. Phase transistions of composition schemes: Mittag-Leffler and Mixed Poisson distributions. +Submitted, 2021. +[2] Jean Bertoin. On the local times of noise reinforced Bessel processes. Annales Henri Lebesgue, +vol- +ume 5, pages 1277–1294, 2022. +[3] Jean Bertoin. Personal communication. 2022. +[4] James Allen Fill, Nevin Kapur and Alois Panholzer. Destruction of very simple trees. Algorithmica, vol- +ume 46, pages 345—366, 2006. +[5] Svante Janson. Moments of Gamma type and the Brownian supremum process area. Probability Surveys, 7, +1–52, 2010. +MARKUS KUBA, DEPARTMENT APPLIED MATHEMATICS AND PHYSICS, UNIVERSITY OF APPLIED SCI- +ENCES - TECHNIKUM WIEN, H ¨OCHST ¨ADTPLATZ 5, 1200 WIEN +Email address: kuba@technikum-wien.at +ALOIS PANHOLZER, INSTITUT F ¨UR DISKRETE MATHEMATIK UND GEOMETRIE, TECHNISCHE UNIVER- +SIT ¨AT WIEN, WIEDNER HAUPTSTR. 8-10/104, 1040 WIEN, AUSTRIA +Email address: Alois.Panholzer@tuwien.ac.at + diff --git a/JNE2T4oBgHgl3EQfpAjJ/content/tmp_files/load_file.txt b/JNE2T4oBgHgl3EQfpAjJ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..15f56820e4c58ffad78279321c31861f2eb2e2af --- /dev/null +++ b/JNE2T4oBgHgl3EQfpAjJ/content/tmp_files/load_file.txt @@ -0,0 +1,86 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf,len=85 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content='04025v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content='PR] 10 Jan 2023 A NOTE ON THE LIMIT LAW OF ONE-SIDED TREE DESTRUCTION MARKUS KUBA AND ALOIS PANHOLZER ABSTRACT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' This short note serves an addendum to the article ”Destruction of very sim- ple trees” by Fill, Kapur and Panholzer (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' Therein, the limit law of one-sided tree destruction was determined by its moment sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' We add an identification of the limit law, using recent results of Bertoin (2022), in terms of the local time of a noise reinforced Bessel process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' ADDENDUM: IDENTIFICATION OF THE LIMIT LAW In Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content='1 of [4] the limit law of the cost of root isolation Yn in the class of very simple trees, a subclass of simply generated trees, was obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' The random variable Yn itself satisfies the distributional equation Yn L= YKn + tn n ≥ 2, for some random variable Kn [4], with initial value Y1 = t1 and toll function tn = na, with a ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' It was shown that for a parameter σ > 0 and a′ = a + 1 2 it holds Yn σna′ L→ Ya′, where Yα′ is uniquely determined by its moment sequence ms = E(Y s a′), for positive integers s ≥ 1: ms = s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' 2s/2 s � k=1 Γ(ka′) Γ(ka′ + 1 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' Recent results of Bertoin [2] on the noise reinforced Bessel process and its local time allows us to observe the following Corollary to Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content='1 of [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' The limit law Ya′ has the same distribution as a moment-shifted (also called size biased or moment-tilted) ˆLt of a local time of a noise reinforced Bessel process, scaled by 1 √ 2: Ya′ L= 1 √ 2 · tilt1 � ˆLt/κ(p, t) � , with factor κ(p, t) given in (1), dimension d = 1 and reinforcement parameter p = 1 2 − 1 4a′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' Additionally, Ya′ L= 1 κ(p,1) √ 2 · ˆI, where the scaled exponential functional ˆI is given by ˆI = � ∞ 0 exp(−1 2 ˆξt)dt, with ˆξt denoting the subordinator with Laplace-Exponent ˆΦ, determined by ˆΦ(r) = 2− 1 2 � 1 4a′ � 1 2 Γ( 1 2) 1 2B( 1 2, 4a′r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' 2000 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' 05C05, 05A15, 05A19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' Tree destruction, cutting down, local time, limit law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' 1 2 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' KUBA AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' PANHOLZER Remark 1 (Moments of Gamma type).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' For a′ = 1 2 one obtains a Rayleigh distribution, as noted before [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' For m ∈ N and m · a′ = 1 2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' a′ = 1 4, we observe cancellations and obtain interesting moments of Gamma type [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' For example, for a′ = 1 4 we get ms = 1 2s/2 Γ( 1 4)Γ( 1 2)Γ(s + 1) Γ( s+1 4 )Γ( s+2 4 ) , s ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' Density functions for such moment sequences can be obtained using inverse Mellin com- putations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' compare with [1, Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content='1] or see [5, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' The identification is directly done using moment sequences appearing in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' There, a noise reinforced Bessel process of dimension d > 0 and with reinforcement parameter p ∈ (−∞, 1/2) was considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' Set α = 1 − d/2 ∈ (0, 1) and β = α 1 − 2p, or equivalently p = 1 2 − α 2β , with β > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' The local time ˆLt of the noise reinforced Bessel process has power moments (see [2, The- orem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content='2]), which can be written as E( ˆLt s) = κ(p, t)s · 1 − 2p Γ(1 + α) · Γ(s) · s−1 � j=1 Γ(jβ) Γ(α + jβ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' with scale factor given by κ(p, t) = (2t)α · Γ(1 + α) (1 − 2p)α · Γ(1 − α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' (1) This implies that the scaled random variable L = ˆLt/κ(p, t) has moment sequence µs = E(Ls), given by µs = 1 − 2p Γ(1 + α) · Γ(s) · s−1 � j=1 Γ(jβ) Γ(α + jβ), s ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' Using standard results on moment shifts (see for example [1, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content='4]), also called size biased distributions or tilted distributions, we observe that the random variable T = tilt(L) has moment sequence E(T s) = µs+1 µ1 , or more explicitly E(T s) = Γ(s + 1) · s � j=1 Γ(jβ) Γ(α + jβ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' Since Γ(s + 1) = s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=', we finally set β = a′ and α = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' This leads to the desired moment sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' Finally, we note that the choices lead to a dimension d = 1 and p = 1 2 − 1 4a′ , such that 1 − 2p = 1 4a′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' Concerning the second statement we use the definition of ˆI and its Laplace exponent [2, Equations 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' We use Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content='3 of [2], which relates ˆI and ˆL1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' Note that there is a small misprint [3] in the stated equation, as the prefactor c = (1/2 − p)−α 1 − 2p Γ(1 − α), which can be rewritten into �1 − 2p 2 �−α 1 − 2p Γ(1 − α) = 2α(1 − 2p)1−α Γ(1 − α) = E(ˆL1), LIMIT LAW OF ONE-SIDED TREE DESTRUCTION 3 should be replaced by its reciprocal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' We state here the corrected version: E � f(ˆI) � = 1 E(ˆL1) E �ˆL1f(ˆL1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' Choosing f as the power functions, we observe a tilt of the moments [1, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content='4], leading to the moment sequence of ˆI as the shifted moments of ˆL1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' □ Remark 2 (More moments of Gamma type).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' As already pointed out [2], for p = 0, such that α = β, the random variable ˆLt simplifies to a Mittag-Leffler distribution: E( ˆLt s) = κ(0, t)s · 1 Γ(1 + α) · Γ(s) · Γ(α) Γ(sα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' = κ(0, t)s Γ(s + 1) Γ(sα + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' The random variable T also leads to moments of Gamma types for small β, proportional to α: β = α/m, m ∈ N, leading to moments E(T s) = Γ(s + 1) · m � j=1 Γ( jα m ) Γ( (s+j)α m ) , s ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' Again, density functions can be obtained for such moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' OUTLOOK AND ACKNOWLEDGMENTS The authors are currently investigating into other occurrences of such limit laws in com- binatorial probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' The authors warmly thank Jean Bertoin for feedback on his work and explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' REFERENCES [1] Cyril Banderier, Markus Kuba and Michael Wallner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' Phase transistions of composition schemes: Mittag-Leffler and Mixed Poisson distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' Submitted, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' [2] Jean Bertoin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' On the local times of noise reinforced Bessel processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' Annales Henri Lebesgue, vol- ume 5, pages 1277–1294, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' [3] Jean Bertoin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' Personal communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' [4] James Allen Fill, Nevin Kapur and Alois Panholzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' Destruction of very simple trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' Algorithmica, vol- ume 46, pages 345—366, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' [5] Svante Janson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' Moments of Gamma type and the Brownian supremum process area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' Probability Surveys, 7, 1–52, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' MARKUS KUBA, DEPARTMENT APPLIED MATHEMATICS AND PHYSICS, UNIVERSITY OF APPLIED SCI- ENCES - TECHNIKUM WIEN, H ¨OCHST ¨ADTPLATZ 5, 1200 WIEN Email address: kuba@technikum-wien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content='at ALOIS PANHOLZER, INSTITUT F ¨UR DISKRETE MATHEMATIK UND GEOMETRIE, TECHNISCHE UNIVER- SIT ¨AT WIEN, WIEDNER HAUPTSTR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content=' 8-10/104, 1040 WIEN, AUSTRIA Email address: Alois.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content='Panholzer@tuwien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} +page_content='at' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfpAjJ/content/2301.04025v1.pdf'} diff --git a/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf b/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf new file mode 100644 index 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b/KdE5T4oBgHgl3EQfYA9D/content/tmp_files/2301.05570v1.pdf.txt @@ -0,0 +1,578 @@ +JOHN CLARK’S LATIN VERSE MACHINE: 19TH CENTURY +COMPUTATIONAL CREATIVITY +Mike Sharples +Emeritus Professor of Educational Technology +The Open University +UK +mike.sharples@open.ac.uk +ABSTRACT +John Clark was inventor of the Eureka machine to generate hexameter Latin verse. He labored for 13 +years from 1832 to implement the device that could compose at random over 26 million different lines +of well-formed verse. This article considers Clark as an early cognitive scientist. Clark described +his machine as an illustration of a theory of “kaleidoscopic evolution” whereby the Latin verse is +“conceived in the mind of the machine” then mechanically produced and displayed. We describe the +background to automated generation of verse, the design and mechanics of Eureka, its reception in +London in 1845 and its place in the history of language generation by machine. The article interprets +Clark’s theory of kaleidoscopic evolution in terms of modern cognitive science. It suggests that Clark +has not been given the recognition he deserves as a pioneer of computational creativity. +Keywords History of computing · Cognitive simulation · Natural language genereation · Computational creativity · +hexameter Latin verse · 19th century cognitive science · Kaleidoscopic evolution · Charles Babbage · Thinking Machine +1 +Introduction +In April 1845, handbills were posted in London advertising “Egyptian Hall. The Eureka, a machine for making Latin +Verses, exhibited daily, from 12 to 5 & 7 to 9; with illustrative lectures. Admission 1s."1 +A visitor to the Egyptian Hall in Piccadilly would have seen a wooden cabinet on legs, about the size of a writing bureau. +In the course of a lecture about the device, its operator pulled a small rope attached to a lever, and the machine sprang +into motion. The interior whirred with clockwork, it played the National Anthem, and through tiny, glazed apertures at +the front of the machine letters appeared, producing a line of well-formed Latin, such as: +IMPIA VERBA DOMI CONJUNGUNT CRIMINA MALA +which roughly translates as “wicked words at home connect evil crimes.” At each pull of the lever, the machine formed +a new line. It was capable of generating over 26 million different lines of hexameter Latin verse. +The inventor of this strange machine was John Clark. Clark spent 13 years designing and perfecting the Eureka as a +demonstration of what he termed the principle of “kaleidoscopic evolution”. This paper proposes that Clark should +be regarded as an early cognitive scientist. As a contemporary of Charles Babbage, he should be given comparable +recognition in the annals of computing history. +1The handbill and other documents relating to John Clark are preserved in the archive of the Alfred Gillett Trust, The Grange, +Farm Road, Street, Somerset, BA16 0BQ, UK. +arXiv:2301.05570v1 [cs.CY] 13 Jan 2023 + +John Clark’s Latin Verse Machine +2 +Background +The idea of automated production of Latin verse can be traced back at least to the 17th century. In 1677, a mathematician +named John Peter published a booklet with the title Artificial Versifying, or the School-boy’s Recreation: A New Way to +Make Latin Verses [20]. The front cover claimed: +Whereby any one of ordinary Capacity, that only knows the A.B.C. and can Count 9 (though he +understands not one word of Latin, or what a Verse means) may be plainly taught, (and in as little a +time as this is Reading over,) how to make Hundreds of Hexameter Verses, which shall be True Latin, +True Verse, and good Sense. +The booklet depicts six grids, each containing letters. By choosing any six-digit number, the reader could interrogate +the tables and generate a line of six Latin words in hexameter rhythm or meter such as “Perfida dicta mihi confirmant +somnia multa.” +Peter’s tables are an elaborate game, similar to modern Word Search puzzles where words are hidden within a grid of +letters. Each table hides nine words that can be found by starting from a letter on the top row (indexed by the chosen +digit) and counting forward nine squares to get the next letter and so on. John Peter could have just listed the nine words +for each table, but that would have ruined the mystery. +The tables are ordered according to the strictures of Latin grammar, so the words always produce grammatically correct +sentences: Adjective Noun Adverb Verb Noun Adjective. Lastly, Peter carefully chose the words for each table to +conform to the hexameter rhythm of classical Latin poetry. The Artificial Versifying booklet proved popular and led to +imitations of the tables over the next century. +For many centuries, scientists had been devising machines to mechanize arithmetic, calculate tides and determine +the positions of planets. Music boxes that played a single musical tune were produced from the late 18th century. +The German inventor Johann Nepomuk Maelzel manufactured music-playing automatons including, in 1805, the +Panharmonicon which could imitate orchestral instruments [8]. However, nobody until John Clark had demonstrated a +mechanical device to automate the creative process of versification. +3 +John Clark +John Clark (1785-1853) was born in the village of Greinton near Glastonbury at the southwest of England [2]. He was a +first cousin to Cyrus and James Clark, founders of the C&J Clark shoe company. After leaving school, he joined his +uncle’s woollen stocking business, then worked in his family home as a grocer, before setting up a printing firm in +Bridgwater, Somerset. He gained a local reputation as a philosopher, author and poet, including writing a continuation +of Lord Byron’s Don Juan. Catherine Impey, a local historian, describes John as being “a remarkable character, full of +strange idiosyncrasies...He used to wear a sailor blue jacket with brass buttons because it was prettier than a coat, & +was quite regardless of fashion in other ways, wearing his shirt open all down the front.”2 +Clark’s abiding passion was scientific invention. In 1813 he gained a patent for a new method of rubberizing fabric +for blow-up air beds which he sold in 1825 to Charles Macintosh who applied this process to waterproofing material +and ran a successful raincoat business. Clark was also a skilled clock repairer. For 13 years, from 1832 to 1845, he +transferred his knowledge of clockwork to the design, construction and refinement of his Eureka machine for producing +hexameter Latin verses [1]. +4 +The mechanics of Eureka +In its exhibition version, the Eureka machine was housed in a cabinet on legs, the size of a modern washing machine, +painted blue-green with a glossy varnish, with a gilded front panel where a row of small windows displayed the line of +verse (Figure 1). +Opening the hinged back of the machine reveals3 a clockwork mechanism (Figure 2). A large weight (left of the picture) +provides power and a flywheel (top right) governs speed. At the rear of the picture are wooden bars, or staves, that drop +onto wires or differing lengths to compose a line on Latin verse. At the bottom is a cylinder, as in a music box, that +played the National Anthem to accompany the composing. +2Cited in notes on John Clark by Roger Clark, a descendent of the Clark family, writing in 1950. The notes are held in the archive +of John Clark at the Arthur Gillett Trust. +3The present tense is used to describe the construction since, as we discuss later, the Eureka machine still exists in working order. +2 + +John Clark’s Latin Verse Machine +Figure 1: The Eureka Latin verse machine. The verse is displayed on a row of six small windows in the lower strip. +Figure 2: Eureka with the rear door open, showing the weight on the left, driving mechanism on the right, row of +wooden staves (shown from behind) and, at the bottom, a large music cylinder that played the National Anthem. +Below the main drive mechanism is a line of six wooden cylinders, or drums, from which project long rigid metal wires +like spines on a porcupine (Figure 3). A line of wires, of differing lengths, forms the letters of a single Latin word. +Each wire is bent to a right angle at the end. The drums and wires form the “composing” section of the machine. The +drums rotate independently to produce a new line of verse. Behind the clockwork in Figure 2 can be seen a line of 47 +long vertical wooden staves. On the front of each stave is written a vertical line of letters in alphabetic order: +A Æ B C D E F G H I J K L M N O Œ P Q R S T U V X Y Z +These form the “interpreting” section. When the machine is set in motion the staves slowly descend to rest on the wires, +each displaying a letter of the new verse in the viewing slot. The length of a wire produces a single letter. The longer +the wire, the shorter distance a stave has to fall, so the earlier the letter in the alphabet. Conversely, a short wire allows +the stave to fall further, producing a letter at the end of the alphabet, such as V or X (Figure 4). In modern terms, the +machine was “programmed” by Clark installing rigid wires of appropriate lengths to form words – the Latin words +were literally hard-wired onto the drums. The machine is wound up by inserting and turning a key on the right of the +machine which raises the weight. Moving a lever at the left of the machine sets the clockwork in motion. John Clark +describes the mechanism thus [7]: +3 + +John Clark’s Latin Verse Machine +Figure 3: Three of the six drums from the “composing” section of the Eureka machine, showing the rigid wires of +differing lengths. +Figure 4: A diagram showing one drum of the Eureka interpreting mechanism, with staves descending onto the wires to +display the word DENSA in the viewing window. +4 + +M +N +H +Z +M +X +G +Y +K +F +L +X +1 +K +U +E +v +H +1 +T +D +U +G +H +C +T +F +G +R +B +E +F +Q +A +P +D +E +N +A +C +D +M +N +B +c +L +M +AE +B +K +1 +A +AE +一 +K +A +H +一 +G +H +F +G +E +F +D +E +C +D +B +C +A +B +A +AE +A +DrumJohn Clark’s Latin Verse Machine +The entire Machine contains about eighty-six wheels, giving motion to cylinders, cranks, spirals, +pullies, levers, springs, ratchets, quadrants, tractors, snails, worm and fly, heart-wheels, eccentric +wheels, and star wheels;—all of which are in essential and effective motion, with varying degrees of +velocity, each performing its part in proper time and place. +The machine goes through cycles of generating and resetting, as follows4: +1. The clutch lifts, the control cam moves, and the flywheel begins to revolve. +2. Brake pressure is released, and the machine starts. +3. The flywheel gathers speed, the control frame moves up, and the board that resets the letter staves moves down. +4. The 47 staves independently move slowly down, under gravity. The changing letters can be seen through the +viewing windows on the front panel. +5. Each stave comes to rest on a “composing” wire. The front panel shows the appropriate letter. +6. The control cam completes a half cycle and the internal bell rings. +7. The clutch is applied. +8. A complete Latin verse can be read through the apertures on the front panel. +9. The cycle continues, and the letter staves move back up to their start positions. +10. When all the staves are up and in line, a “drum kick” occurs. Each drum rotates a different amount, bringing +new rows of wires to the top. +11. The machine can either be stopped after each cycle or left to continue until the driving weight fully drops. The +machine can go through five complete cycles each lasting about a minute before needing to be rewound. +It should be noted that the machine is designed for creativity not calculation. There is no requirement for its movements +to be replicable but there is a need to impart arbitrary motion so that each new line of verse is different and unpredictable. +Clark included a mechanism to set each drum rotating independently by giving it a rotational kick from a weight-driven +pawl that engages a ratchet on the drum to spin it. Then a spring-loaded roller moves to connect with a star wheel on +each drum to stop it in a position where the staves can descend onto its line of wires to form a Latin word. The spin +mechanism is similar to that seen on mechanical slot machines that were introduced some 40 years later in the late 19th +century. It is not pseudo-random since the spin of each drum is affected by physical conditions such as temperature and +friction. +Clark carefully chose the words coded on each drum. The ordering of the words on the six drums is ADJECTIVE +NOUN ADVERB VERB NOUN ADJECTIVE. The machine is designed to always produce a line of dactylic hexameter +verse, though it may not always make much sense. He generally chose gloomy words to give a solemn feel to the +production, for example: +M ¯ART˘I ˘A C ¯ASTR ˘A F ˇOR¯IS PR ¯ÆN ¯ARR ¯ANT PR ¯OEL˘I ˘A M ¯ULT˘A5 +which Clark himself translated as “martial encampments foreshow many oppositions abroad.” Table 1 shows the +complete set of words for each drum. The physical width of a drum is determined by the longest word coded on it, +thus drum 4 is wider than drum 3. Some words, and the general morbid tone, were copied from John Peter but Clark +extended the vocabulary of previous versifying tables while retaining grammar and meter. His choice of Latin words has +been criticized for being repetitive and in a few instances incorrect. However, “Faults and shortcomings there may be, +but it still stands as a monument to the patience and ingenuity of John Clark” [2]. It was also a practical demonstration +Clark’s theory of linguistic creativity. +5 +19th Century Cognitive Science +The Stanford Encyclopedia of Philosophy [30] characterizes Cognitive Science as “the interdisciplinary study of mind +and intelligence, embracing philosophy, psychology, artificial intelligence, neuroscience, linguistics, and anthropology. +Its intellectual origins are in the mid-1950s when researchers in several fields began to develop theories of mind based +on complex representations and computational procedures.” +4Adapted from a typewritten “Description of the Mechanism of the Latin Verse Machine” by the conservator Peter Jealous, +September 1970. Held in the Clark archive at the Alfred Gillett Trust. +5The diacritic marks added here indicate the hexameter rhythm: dee dum dum, dee dum, dum dee, dee dee dee, dee dum dum, +dee dum. +5 + +John Clark’s Latin Verse Machine +Table 1: Lists of Latin words for each drum on Eureka. +Here, we show how John Clark, working 100 years before the “intellectual origins” of cognitive science, explored a +theory of poetic composition that was based on a representation of mental conception realized through a computational +mechanism. Although the representation was less complex than those founded on digital computers, it had the hallmarks +of a symbolic process theory of mind. Clark named this theory “kaleidoscopic evolution”. +To put it in historic context, by the early 19th century there was discussion among mathematical scientists as to whether +computation machines such as Leibniz’s calculator and Babbage’s Difference Engine could be said to emulate mental +processes. In June 1833, Lady Byron (mother of Ada Byron) visited Babbage’s house to view what she called the +“thinking machine”. Babbage himself was careful never to ascribe mental powers to his machines [9]. Clark had no +such qualms. +In 1837 (five years before the first published description of Babbage’s Analytical Engine) John Clark published a +22-page booklet, which he printed himself, with the title “The General History and Description of a Machine for +Composing Hexameter Latin Verses” [5]. A revised version was published in 1848, printed by Frederick Wood of +Bridgwater, Somerset [7]. The new version omits a lengthy discussion of poetic forms. It also includes some revealing +alterations to wording in the description of the machine and its foundations in a theory of mechanical composition. The +account below will refer to the 1848 printing unless stated otherwise. Reference will also be made to a letter Clark +wrote to The Athenaeum magazine in July 1845, in response to a correspondent from that magazine [6]. In that letter, +referring to the exhibition of Eureka in the Egyptian Hall, Clark states: +The machine is neither more nor less than a practical illustration of the law of evolution. The process +of composition is not by words already formed, but from separate letters. This fact is perfectly +obvious, although some spectators may probably have mistaken the effect for the cause—the result +for the principle—which is that of kaleidoscopic evolution; and as an illustration of this principle +it is that the machine is interesting—a principle affording a far greater scope of extension than has +hitherto been attempted. +6 + +John Clark’s Latin Verse Machine +The machine contains letters in alphabetical arrangement. Out of these, through the medium of +numbers, rendered tangible by being expressed by indentures on wheel work, the instrument selects +such as are requisite to form the verse conceived; the components of words suited to form hexameters +being alone previously calculated, the harmonious combination of which will be found to be practically +interminable. +Clark did not use the term “evolution” in its modern definition of evolution of species but the earlier sense of “unfolding” +or “coming into being”, applied to a foetus, or in this case a line of verse. Why did he apply the modifier “kaleidoscopic” +to this process of composition? +A distinctive feature of cognitive science is the use of technology as a mechanism to understand mental structures and +processes. In modern times this is the digital computer; in the mid-19th century, Clark called upon a contemporary +scientific instrument, the kaleidoscope. +The kaleidoscope was invented in 1815 by Sir David Brewster as an instrument to create regular patterns from irregular +or random arrangements of objects [3]. Clark applied this as a metaphor and mechanism for the creative process of +versification. He describes the interior of his machine thus: +And in the interior is a large Kaleidoscope, which regularly constructs a geometric figure, the form +whereof is ascertained by the falling of numerous probes into its indentures, and thus the Latin verse +about to be composed is determined. This action is performed at the commencement of the operation, +and is the precise time when the Line of Verse is conceived, previous to its mechanical composition. +Clark was referring to the kaleidoscopic pattern of wires radiating from the drums that determined how the lettered +staves would fall and thus which words would form. +The booklet then refers to “the mind of the machine” to distinguish the “moment of conception” from the mechanical +production of the verse (his emphasis below): +There is a certain point of time, which may be called the Moment of Conception, at which instant of +time the identical Latin verse, which is about to be produced, is conceived in the mind of the machine, +(if the expression be allowable,) and that identical verse, which is then and there conceived, will be +mechanically produced and displayed. +This visible display of the Line conceived, is effected by the mechanical movements of the Automaton. +But the conception of the Line is not mechanical, nor can it possibly be rendered a visible or tangible +thing, in being an imagination only, partaking somewhat of the nature of an arithmetic series. In +like manner we cannot see or feel human imagination, but we can render it audible or visible by the +mechanical instruments of the tongue or pen. +The Hexameter Automaton bears some affinity to an animated being. It possesses a material and +an immaterial part, a corporeal and an incorporeal power. Yet it is a thing far inferior to an animal, +inasmuch that it possesses no volition or intention, nor any consciousness of its own existence. +What follows is an attempt to explain Clark’s conception of kaleidoscopic evolution, with extracts from Clark’s original +wording in quotes (his emphasis). +Mechanical reproduction of verse requires two elements: a method of representing the verse (“to produce the effect by +mechanical means”) and a mechanism to output combinations of well-formed words (“to arrange the machinery in a +convenient and eligible manner”). +Language can be described as formal sequences of numbers to mechanize composition (“the powers and properties +of infinite numbers, applied to poetical composition, etc.”). The language of Latin hexameter verse may be harder +for humans to write than everyday English prose (“which act as fetters of confinement to the writers of verses, much +increasing their difficulties”), but its formal constraints make it easier to implement on a computational device (“have +an opposite effect when applied to a machine”). +By devising a schema for a line of verse – of the form ADJECTIVE NOUN ADVERB VERB NOUN ADJECTIVE +– and selecting words for each category with the appropriate meter, Clark produced a tabular representation (“the +foregoing Tabular principle”) where words from each column in sequence could be combined arbitrarily to create a +well-formed verse (Table 1). The process is analogous to a kaleidoscope which, through its careful alignment of mirrors, +creates beautiful patterns from arbitrary shapes (“Forming an indefinite number of regular Geometrical Figures by a +Machine”). Just as a kaleidoscope necessarily creates regular geometric patterns, so the Eureka machine is designed +7 + +John Clark’s Latin Verse Machine +to produce only well-formed verses (“the machine now proposed, cannot possibly form other than Hexameter Latin +Verses”). +The verses must not only be grammatically well-formed but also be original (“the Automaton has never repeated any +line of verse which it has previously made”). The machine must display creative intelligence, producing verses that +make sense (“It is requisite that each Line or Verse shall be correct, not only in measure and in accent, but it is also +essential that the verse be fraught with idea or intelligence”). +Clark coded the letters for each word as numbers and implemented these on the Eureka machine as filaments of wire in +the six drums, with the length of each wire corresponding to a letter of the Latin alphabet (“Out of these, through the +medium of numbers, rendered tangible by being expressed by indentures on wheel work, the instrument selects such as +are requisite to form the verse conceived”). +The Eureka machine simulates human creative composition (“the Hexameter Automaton bears some affinity to an +animate being”), whereby a Latin scholar who has learned the patterns of hexameter verse can combine these mentally +(“an incorporeal power”). The machine mimics this creative process of versification (“at which instant of time the +identical Latin verse, which is about to be produced, is conceived in the mind of the machine”) and then displays the +verse (“that identical verse, which is then and there conceived, will be mechanically produced and displayed”). At the +“moment of conception” its drums randomly align to produce a novel combination of words (“thus the Latin verse about +to be composed, is determined”). The staves then drop down to display the series of letters that form a line of verse. +The creative mental combination and its mechanical realization are separate processes. The creativity is hidden from +view (“we cannot see or feel human imagination, but we can render is audible or visible by the mechanical instruments +of tongue or pen”). Many great ideas have been conceived but not expressed in words (“It is also possible that this +supposed idea with thousands of others, equally beautiful, and necessarily existing in embryo from eternity to eternity, +may possibly never be disclosed”). +The machine can continue to simulate creativity without human assistance (“thousands of Verses may be theoretically +conceived, and also mechanically composed and decomposed during the night, or in the intentional absence of all +Intelligent Beings, or Spectators: provided that the weight, or power, which actuates the machine, be continued”). Clark +did not see the Eureka machine as a useless mechanical curiosity but as a working demonstration of his theory of +kaleidoscopic evolution, a theory with applications in science and technology. In a “Questions and Answers” section of +his booklet Clark asks, “To what uses may the Hexameter Machine be put?” He answers with: +It may also be asked of what use is an Acorn? Not much in its present state: but if planted and +suffered to grow, it may possibly produce an Oak... Thus one invention brings forward another... +Every new thing is an intellectual accession, and every accession may, possibly, be of important use. +In his letter to The Athenaeum [6], Clark writes “as an illustration of this principle [of kaleidoscopic evolution] it is that +this machine is interesting—a principle affording far greater scope of extension than has hitherto been attempted”. In +his booklet he envisages automated speech output: +A Speaking Automaton6, has lately been completed: it is actuated by a performer, but if it were +combined with the Hexameter Machine, it would produce its sentences spontaneously. +Clark ends the booklet with a brief account of the history of “Androides and Automatic Figures” concluding with: +The attention of the present age is deservedly directed to the admirable Calculating Machine, designed +by Mr. Babbage. This is now in a considerable degree of progress towards completion. +6 +Reception +By the time he exhibited Eureka, Clark was aged 60. His dress and manners must have been out of place among the +London intelligentsia. In his memoirs, William Ballantyne Hodgson, a Scottish educational reformer and political +scientist describes a visit to the Egyptian Hall to see the Eureka machine [16]: +We walked together as far as Piccadilly to the Egyptian Hall, where I saw the Eureka, an instrument +for making Latin verse, of which I enclose you a brief account. Had not heard of it before. Barham is +exhibiting it just now for the inventor, Clark, whom I also saw. Though I cannot say much for the +6Presumably a reference to the Euphonia machine, a talking head able to mimic human speech, which was demonstrated in the +Egyptian Hall in 1846. +8 + +John Clark’s Latin Verse Machine +sense of the verses, there are occasional and recurring errors in quantity, and I suspect that the range +of the machine is much more limited than is alleged. The inventor spent fifteen years upon it—five +more years than are needed to make a boy into a verse-making machine, and still less perfect. Clarke +is a strange, simple-looking old man. Babbage said the other day that he was as great a curiosity as +his machine. +The contrast is striking between the suave Cambridge-educated raconteur Babbage and the simply dressed, provincial +Clark. By 1845, Babbage had won a Gold Medal from the Astronomical Society and gained over £17,000 of Government +grants for his Difference Engine engine to calculate mathematical and astronomical tables, despite never completing a +full working machine [28]. Meanwhile Clark was demonstrating his Eureka machine daily in an exhibition hall to a +bemused public. +An article in The Illustrated London News of July 19, 1845 [32] shows a line drawing of the Eureka cabinet with a +factual description of the machine which “has lately been brought to the metropolis, to contribute to the ‘sights of the +season’.” Most the content of that newspaper article is taken from verbatim from Clark’s 1837 booklet. +A letter from “P.A. Nuttall” to The Athenaeum magazine dated June 28, 1845 is less reserved [17]. The Eureka, Nuttall +opines, is “little better than a mere puzzle, which any school-boy might perform by a simpler process” since it merely +produces six words of a regular Latin grammatical pattern (“the first word is uniformly a dactyl, and an adjective of the +neuter plural, the second word a trochee . .. ”). All a school-boy has do it is write Latin words of each type on slips of +paper, put them into six piles, then draw out words at random from each pile in sequence to form a Latin verse. “It may +be a very curious and instructive amusement,—but nothing more.” +Clark knew that he could have simplified his machine by printing complete words on the drums then rotating each drum +to display a line of words through windows, like a 20th-century slot machine. But that would have been less general +and creative. As Clark attempted to explain in his response to Nuttall, published in The Athenaeum on July 2, 1845 [6], +the method he had adopted for Eureka of having the staves fall individually onto the wires is not an effect to entertain +the audience but an intrinsic part of its design. Working with letters not words has “far greater scope of extension than +has hitherto been attempted.” By representing words as lengths of wire, he could not only re-program the machine by +adjusting the lengths, but also envision a machine that would automatically interpret the lengths to form new verses. +A glimpse of Clarke’s vision for extension to his theory of kaleidoscopic evolution can be seen in a letter he wrote to +his sister in August 1845 7: +I hope to be home soon with a long story to tell. A most astonishing discovery has been made with +the Hexameter Machine. It would make so many millions of verses that to produce all will not be +done for a century or more, but we have discovered that if we take 100 or so of its productions that +these will produce thousands & millions of other verses, by another machine, & so on. It has opened +a new field of scientific speculation. +It appears from this fragment that Clark had been speculating about a sequence of machines, each of which takes +outputs from the previous machine and combines these in new ways to generate longer and more varied results. +The exhibition of the Eureka machine in London coincided with what has been called “hexameter mania” [22]. This +was a lively argument about the teaching of Classics in schools. In 19th century England, boys in elite schools were +required not only to speak and write Latin, but also to compose Latin verse in a manner that supposedly emulated the +great Roman poets. By the mid 19th century, this practice had descended into rote learning, with pupils looking up +textbooks of verse composition to grind out pastiches of Ovid and Virgil. A machine that could automate this process at +the rate of 10,000 verses a week, like a demented schoolboy, added fuel to those commentators who derided the useless +rituals of elite schools and were calling for a modern curriculum that embraced the Victorian advances in science and +technology. Thus, Clark became embroiled in contemporary debates about the value to the individual and society of +writing hexameter Latin verse. +The satirical magazine Punch [23] wrote, tongue in cheek: +That notable invention, the Eureka, or Latin verse-grinder, was tried yesterday before a committee of +young gentlemen from the public schools, who are anxious to have their Latin exercises done with +the least possible trouble... Several double-barrelled Eurekas were ordered for Eton, Harrow, and +Rugby. +7Letter by John Clark to his sister Sarah (Clark) Metford, August 1845. The letter is held in the Clark archive at the Alfred Gillett +Trust. +9 + +John Clark’s Latin Verse Machine +The debate over the value, if any, of students churning out assignments that could just as well be generated by machine +resonates with current concerns about students employing AI language systems to write their essays and assignments +[25]. +The Eureka machine was soon consigned to a footnote in the history of Latin versification. An article for Chambers’s +Edinburgh Journal in 1850 confuses the Eureka with the Euphonia (a machine to mimic the human voice), stating that +“by its aid the most illiterate person could produce thousands of Latin verses” [4]. +7 +Legacy +In the 20th century, the Oulipo movement, originating among writers and mathematicians in France, experimented with +automated production of novels and poems but made no reference that we can find to Clark and his verse machine [25]. +The best-known mention in literature of a machine to generate verse comes in George Orwell’s Nineteen Eighty-Four +[19, p. 43]. Orwell depicts the Records Department of the Ministry of Truth where rubbishy newspapers and sentimental +songs were “composed entirely by mechanical means on a special kind of kaleidoscope known as a versificator.” Of all +the words Orwell might have used to describe a machine for churning out prose and songs he chose “kaleidoscope” and +“versificator.” Was that a nod to Clark’s “Kaleidoscopic Evolution” by a “machine for making Latin verses?” +With the advent of digital computers from the 1950s onwards, many academics and students of computer science +(including the author of this paper) experimented with programs to generate poems but without acknowledging the +pioneering work of John Clark. In recent years Clark’s work has been re-discovered. In 1963, D.W. Blandford wrote a +factual account of Clark and his machine which included a table of words on each drum (Table 1) [2]. Jason David Hall, +Professor of Modern Literature and Culture at the University of Exeter, has written academic papers [10], [11] and a +book, Nineteenth-Century Verse and Technology: Machines of Meter [12] examining the relation between machine +culture and poetic meter in the 19th century. These publications explore the Eureka machine as a producer of metrical +verse within the mechanization of science, education, culture, travel and work in Victorian England. The book also +examines how 19th-century writers on psychology and physiology saw the human production of meter as an inherently +automatic process. +Clark has been recognized by a few researchers in computational creativity as a pioneer of mechanized composition. +Douglas Summers Stay, in his book Machinamenta has commented on the connection between the Eureka and the +kaleidoscope as a machine to produce patterns by random combination. The problem with both, he notes, is how to +make a machine that is not limited by the inventor’s initial choice of settings, but can grow in ability over time [27, p.5]. +Other mentions of Eureka in the literature on computational creativity include [12], [33] and [25]. +As for the Eureka machine itself, it is not clear where it went immediately after Clark’s death in 1853 8. Around +1856 it was housed in the Counting House of the Clarks Shoe Company in Street, Somerset. Then, in 1889 or 1890 it +was moved to the geology Museum at Crispin Hall in Street, to the concern of Alfred Gillett, donor of the geological +collection, who thought it let down his exhibits. The last person capable of operating the machine during that period +was John Aubrey Clark, son of the founder of the shoe company. When the Geology Museum dispersed in 1948, the +Eureka was moved back to the Clarks factory. It was restored to working order in 1950 by Leslie W.M. Husbands, a +local clockmaker, and Frederick Berry, the typewriter and sewing machine engineer at Clarks. Then it was moved to the +company Records Office. By 1963 it was in the Clarks company museum. In 1970-71 it was renovated for a second +time by the Clarks Special Projects Engineer Peter Jealous, in collaboration with Husbands, and in 1979 it was housed +in the company’s Shoe Museum. By 1996, no longer working, it was put into storage. It stayed there until 2015 when it +went to Devon for extensive restoration by accredited conservators Richard Jaeschke who conserved the casework and +exposed the gilding and verse on the front of the machine, and Neil Bollen who conserved the mechanical parts and +restored the machine to a gentle working order. The restoration was part of a collaboration between the University of +Exeter and the Alfred Gillett Trust, funded by the Arts and Humanities Research Council, to understand, document +and restore Eureka. Eureka now rests on a plinth at the Alfred Gillett Trust, Street, Somerset as a centerpiece of its +collection of artefacts connected with the Clarks shoe company 9. +8 +Conclusion +John Clark does not fit comfortably into the narrative of pre-electronic computing, where urbane and sometimes irascible +mathematicians based in the great European cities of Paris and London developed machines of intricate engineering to +8The history of the Eureka machine is drawn from [1]. +9Eureka can be accessed at the Alfred Gillett Trust in Street, Somerset, by appointment. Researchers can also view archives and +collections related to Clarks shoe company, the family, and Street whilst a new museum is being developed. +10 + +John Clark’s Latin Verse Machine +solve problems in accounting, astronomy and navigation. Clark was an outsider. He never studied at university, he +was a self-taught poet and philosopher, he lived most of his life in the market town of Bridgwater on the Southwest +corner of England. He began work on the Eureka machine in 1830, when he was forty-five. When, at the age of 60, he +arrived in London it was to exhibit his machine at the Egyptian Hall which by that time was noted more for its popular +entertainment than display of scientific discovery. +Clark himself, in the final section of his booklet [7], places his device among “The construction of different species of +Androides and Automata [that have] occasionally engaged the attention of Mechanists, of all nations” including “a fine +automaton, representing a Bengal Tiger. The deep sounds of its roaring are admirably produced by the organ pipes +of its internal structure.” It is hardly surprising given his background, his reception in London (“as great a curiosity +as his machine”), and the criticism of Eureka for producing gloomy and occasionally ill-formed 10 Latin verses, that +John Clark has not been included in histories of computing and artificial intelligence (e.g., [21], [29], [18]). The +comprehensive book on The Origins of Digital Computers [24] mentions Clark only in the bibliography. +However, the enterprise in which Clark was engaged, to understand and simulate the creative production of language, +has been a recurring interest of computer scientists from Christopher Strachey in the 1950s [26], through the extensive +work of Sheldon Klein and colleagues at the University of Wisconsin to automate the writing of novels [14] and recent +work on computational poetry (see [15] for a review), to the investment by companies including Microsoft, Google, +Meta and Baidu in pre-trained AI models for language generation. +This author prompted the GPT-3 Transformer language model from OpenAI to generate original Latin verse. It +responded in seconds with well-formed Latin but it failed in rhyme and meter. Clark would have been intrigued and +amused. +9 +Acknowledgements +My sincere thanks to Karina Virahsawmy, Assistant Curator, Alfred Gillett Trust, for allowing me access to the John +Clark archive and giving a personal demonstration of the machine, to Neil Bollen, conservator, for his patient and +detailed responses to my queries about how the machine functions, and to Rafael Pérez y Pérez, co-author of Story +Machines: How Computers Have Become Creative Writers, for setting me on the journey to Eureka. +10 +References +[1] C. Berry, “LVM Latin Verse Machine c 1830 – c1845: Collection Description compiled by Charlotte Berry”, The +Alfred Gillett Trust, Street, Somerset, United Kingdom, 2014. +[2] D. W. Blandford, “The Eureka,” Greece & Rome, vol. 10, no .1, pp. 71–78, 1963. +[3] D. Brewster, The Kaleidoscope: Its History, Theory, and Construction with its Application to the Fine and Useful +Arts, 2nd ed. London, UK: John Murray, 1858. +[4] Chambers’s Edinburgh Journal, “Latin versification for the million,” Chambers’s Edinburgh Journal, no. 13, p. +205, 1850. +[5] J. Clark, The General History and Description of a Machine for Composing Hexameter Latin Verses. Bridgwater, +Somerset, UK: John Clark, 1837. +[6] J. Clark, “Letter to The Athenaeum”, The Athenaeum, no. 923, Jul. 5, 1845, p.p. 669–670. +[7] J. Clark, The General History and Description of a Machine for Composing Hexameter Latin Verses. Bridgwater, +Somerset, UK: Frederick Wood, 1848. +[8] A. Engberg-Pedersen, “The sense of tact: Hoffmann, Maelzel, and mechanical music,” The Germanic Review: +Literature, Culture, Theory, vol. 93 no. 4, pp. 351–372, 2018. +[9] C. D. Green, “Was Babbages’s Analytical Engine intended to be a mechanical model of the mind?,” History of +Psychology, vol. 8, no. 1, pp. 35–45. +[10] J. D. Hall, “Popular prosody: Spectacle and the politics of Victorian versification,” Nineteenth-Century Literature, +vol. 62, no. 2, pp. 222–249, 2007. +[11] J. D. Hall, “Mechanized metrics: From verse science to laboratory prosody, 1880–1918,” Configurations: A +Journal of Literature, Science and Technology, vol. 3, no. 17, pp. 285–308, 2009. +10“From the metrical point of view the inventor would have done better to have produced lines of five words each, arranged +perhaps in the order of the so-called Golden Line with nouns and adjectives carefully balanced” [2, p.77]. +11 + +John Clark’s Latin Verse Machine +[12] J. D. Hall, Nineteenth-Century Verse and Technology: Machines of Meter, Basingstoke, UK: Palgrave Macmillan, +2017. +[13] M. Hrešková and K. Machová, “Haiku poetry generation using interactive evolution vs. poem models,” Acta +Electronica et Informatica, vol. 17, no. 1, pp. 10–16, 2017. +[14] S. Klein et al., “Automatic novel writing: A status report,” The University of Wisconsin, Technical Report #186, +Computer Sciences Department, The University of Wisconsin, 1973. +[15] C. Linardaki, “Poetry at the first steps of Artificial Intelligence,” Humanist Studies & the Digital Age, Vol. 7, No. +1, 2022, doi: 10.5399/uo/hsda/7.1.6. +[16] J. M. D. Meiklejohn, Ed., Life and Letters of William Ballantyne Hodgson, Edinburgh, UK: David Douglas, 1883. +[17] P. A. Nuttall, “Letter to The Athenaeum,” The Athenaeum, no. 922, Jun. 28, 1845, p. 638. +[18] G. O’Regan, Introduction to the History of Computing: A Computer History Primer. Springer International +Publishing, 2016. +[19] G. Orwell, Nineteen Eighty-Four (Penguin Classics edition, 2000), Penguin Random House, 1949. +[20] J. Peter, Artificial Versifying, or the School-boy’s Recreation. London, UK: John Sims, 1677. +[21] V. Pratt, Thinking Machines: The Evolution of Artificial Intelligence. Oxford: Basil Blackwell, 1987. +[22] Y. Prins, “Metrical translation: nineteenth-century Homers and the hexameter mania,” in Nation, Language, and +the Ethics of Translation, M. Wood and S. Bermann, Eds., Princeton, USA: Princeton University Press, 2005, +p. 229–256. +[23] Punch, vol. 9, July to December 1845, London: Punch, p. 20. +[24] B. Randell, Ed., The Origins of Digital Computers: Selected Papers, 3rd ed., Berlin: Springer-Verlag, 1982. +[25] M. Sharples and R. Pérez y Pérez, Story Machines: How Computers Have Become Creative Writers, Routledge, +2022. +[26] C. Strachey, “The ‘thinking’ machine,” Encounter, Oct. 1954, pp. 25-31, 1954. +[27] D. Summers Stay, Machinamenta: The Thousand Year Quest to Build a Creative Machine, Machinamenta +Publishing, 2011. +[28] D. Swade, The Cogwheel Brain, London, UK: Abacus, 2000. +[29] D. Swade, “Pre-electronic computing,” in Dependable and Historic Computing, C.B. Jones and J.L. Lloyd, Eds., +Berlin, Germany: Springer, 2011, pp. 58–83. +[30] P. Thagard. “Cognitive Science.” Stanford Enyclopedia of Philosophy, https://plato.stanford.edu/entries/cognitive- +science/ (accessed Oct. 5, 2022). +[31] The Athenaeum, “Announcement,” The Athenaeum, no. 922, Jun. 28, 1845, p. 635. +[32] The Illustrated London News, “The Eureka,” The Illustrated London News, Jul. 19, 1845, p. 37. +[33] V. Todorovic, and D. Grba, “Wandering machines: narrativity in generative art,” J. Sci. and Technol. of the Arts, +vol. 11, no2, pp. 50-58, 2019. +12 + diff --git a/KdE5T4oBgHgl3EQfYA9D/content/tmp_files/load_file.txt b/KdE5T4oBgHgl3EQfYA9D/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3ba8cc85567d008f4a7c3137020a681558bdea8d --- /dev/null +++ b/KdE5T4oBgHgl3EQfYA9D/content/tmp_files/load_file.txt @@ -0,0 +1,453 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf,len=452 +page_content='JOHN CLARK’S LATIN VERSE MACHINE: 19TH CENTURY COMPUTATIONAL CREATIVITY Mike Sharples Emeritus Professor of Educational Technology The Open University UK mike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content='sharples@open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content='uk ABSTRACT John Clark was inventor of the Eureka machine to generate hexameter Latin verse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' He labored for 13 years from 1832 to implement the device that could compose at random over 26 million different lines of well-formed verse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' This article considers Clark as an early cognitive scientist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Clark described his machine as an illustration of a theory of “kaleidoscopic evolution” whereby the Latin verse is “conceived in the mind of the machine” then mechanically produced and displayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' We describe the background to automated generation of verse, the design and mechanics of Eureka, its reception in London in 1845 and its place in the history of language generation by machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The article interprets Clark’s theory of kaleidoscopic evolution in terms of modern cognitive science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' It suggests that Clark has not been given the recognition he deserves as a pioneer of computational creativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Keywords History of computing · Cognitive simulation · Natural language genereation · Computational creativity · hexameter Latin verse · 19th century cognitive science · Kaleidoscopic evolution · Charles Babbage · Thinking Machine 1 Introduction In April 1845, handbills were posted in London advertising “Egyptian Hall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The Eureka, a machine for making Latin Verses, exhibited daily, from 12 to 5 & 7 to 9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' with illustrative lectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Admission 1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' "1 A visitor to the Egyptian Hall in Piccadilly would have seen a wooden cabinet on legs, about the size of a writing bureau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' In the course of a lecture about the device, its operator pulled a small rope attached to a lever, and the machine sprang into motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The interior whirred with clockwork, it played the National Anthem, and through tiny, glazed apertures at the front of the machine letters appeared, producing a line of well-formed Latin, such as: IMPIA VERBA DOMI CONJUNGUNT CRIMINA MALA which roughly translates as “wicked words at home connect evil crimes.” At each pull of the lever, the machine formed a new line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' It was capable of generating over 26 million different lines of hexameter Latin verse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The inventor of this strange machine was John Clark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Clark spent 13 years designing and perfecting the Eureka as a demonstration of what he termed the principle of “kaleidoscopic evolution”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' This paper proposes that Clark should be regarded as an early cognitive scientist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' As a contemporary of Charles Babbage, he should be given comparable recognition in the annals of computing history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 1The handbill and other documents relating to John Clark are preserved in the archive of the Alfred Gillett Trust, The Grange, Farm Road, Street, Somerset, BA16 0BQ, UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content='05570v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content='CY] 13 Jan 2023 John Clark’s Latin Verse Machine 2 Background The idea of automated production of Latin verse can be traced back at least to the 17th century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' In 1677, a mathematician named John Peter published a booklet with the title Artificial Versifying, or the School-boy’s Recreation: A New Way to Make Latin Verses [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The front cover claimed: Whereby any one of ordinary Capacity, that only knows the A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' and can Count 9 (though he understands not one word of Latin, or what a Verse means) may be plainly taught, (and in as little a time as this is Reading over,) how to make Hundreds of Hexameter Verses, which shall be True Latin, True Verse, and good Sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The booklet depicts six grids, each containing letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' By choosing any six-digit number, the reader could interrogate the tables and generate a line of six Latin words in hexameter rhythm or meter such as “Perfida dicta mihi confirmant somnia multa.” Peter’s tables are an elaborate game, similar to modern Word Search puzzles where words are hidden within a grid of letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Each table hides nine words that can be found by starting from a letter on the top row (indexed by the chosen digit) and counting forward nine squares to get the next letter and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' John Peter could have just listed the nine words for each table, but that would have ruined the mystery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The tables are ordered according to the strictures of Latin grammar, so the words always produce grammatically correct sentences: Adjective Noun Adverb Verb Noun Adjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Lastly, Peter carefully chose the words for each table to conform to the hexameter rhythm of classical Latin poetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The Artificial Versifying booklet proved popular and led to imitations of the tables over the next century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' For many centuries, scientists had been devising machines to mechanize arithmetic, calculate tides and determine the positions of planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Music boxes that played a single musical tune were produced from the late 18th century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The German inventor Johann Nepomuk Maelzel manufactured music-playing automatons including, in 1805, the Panharmonicon which could imitate orchestral instruments [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' However, nobody until John Clark had demonstrated a mechanical device to automate the creative process of versification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 3 John Clark John Clark (1785-1853) was born in the village of Greinton near Glastonbury at the southwest of England [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' He was a first cousin to Cyrus and James Clark, founders of the C&J Clark shoe company.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' After leaving school, he joined his uncle’s woollen stocking business, then worked in his family home as a grocer, before setting up a printing firm in Bridgwater, Somerset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' He gained a local reputation as a philosopher, author and poet, including writing a continuation of Lord Byron’s Don Juan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Catherine Impey, a local historian, describes John as being “a remarkable character, full of strange idiosyncrasies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content='He used to wear a sailor blue jacket with brass buttons because it was prettier than a coat, & was quite regardless of fashion in other ways, wearing his shirt open all down the front.”2 Clark’s abiding passion was scientific invention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' In 1813 he gained a patent for a new method of rubberizing fabric for blow-up air beds which he sold in 1825 to Charles Macintosh who applied this process to waterproofing material and ran a successful raincoat business.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Clark was also a skilled clock repairer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' For 13 years, from 1832 to 1845, he transferred his knowledge of clockwork to the design, construction and refinement of his Eureka machine for producing hexameter Latin verses [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 4 The mechanics of Eureka In its exhibition version, the Eureka machine was housed in a cabinet on legs, the size of a modern washing machine, painted blue-green with a glossy varnish, with a gilded front panel where a row of small windows displayed the line of verse (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Opening the hinged back of the machine reveals3 a clockwork mechanism (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' A large weight (left of the picture) provides power and a flywheel (top right) governs speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' At the rear of the picture are wooden bars, or staves, that drop onto wires or differing lengths to compose a line on Latin verse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' At the bottom is a cylinder, as in a music box, that played the National Anthem to accompany the composing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 2Cited in notes on John Clark by Roger Clark, a descendent of the Clark family, writing in 1950.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The notes are held in the archive of John Clark at the Arthur Gillett Trust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 3The present tense is used to describe the construction since, as we discuss later, the Eureka machine still exists in working order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 2 John Clark’s Latin Verse Machine Figure 1: The Eureka Latin verse machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The verse is displayed on a row of six small windows in the lower strip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Figure 2: Eureka with the rear door open, showing the weight on the left, driving mechanism on the right, row of wooden staves (shown from behind) and, at the bottom, a large music cylinder that played the National Anthem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Below the main drive mechanism is a line of six wooden cylinders, or drums, from which project long rigid metal wires like spines on a porcupine (Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' A line of wires, of differing lengths, forms the letters of a single Latin word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Each wire is bent to a right angle at the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The drums and wires form the “composing” section of the machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The drums rotate independently to produce a new line of verse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Behind the clockwork in Figure 2 can be seen a line of 47 long vertical wooden staves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' On the front of each stave is written a vertical line of letters in alphabetic order: A Æ B C D E F G H I J K L M N O Œ P Q R S T U V X Y Z These form the “interpreting” section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' When the machine is set in motion the staves slowly descend to rest on the wires, each displaying a letter of the new verse in the viewing slot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The length of a wire produces a single letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The longer the wire, the shorter distance a stave has to fall, so the earlier the letter in the alphabet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Conversely, a short wire allows the stave to fall further, producing a letter at the end of the alphabet, such as V or X (Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' In modern terms, the machine was “programmed” by Clark installing rigid wires of appropriate lengths to form words – the Latin words were literally hard-wired onto the drums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The machine is wound up by inserting and turning a key on the right of the machine which raises the weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Moving a lever at the left of the machine sets the clockwork in motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' John Clark describes the mechanism thus [7]: 3 John Clark’s Latin Verse Machine Figure 3: Three of the six drums from the “composing” section of the Eureka machine, showing the rigid wires of differing lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Figure 4: A diagram showing one drum of the Eureka interpreting mechanism, with staves descending onto the wires to display the word DENSA in the viewing window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 4 M N H Z M X G Y K F L X 1 K U E v H 1 T D U G H C T F G R B E F Q A P D E N A C D M N B c L M AE B K 1 A AE 一 K A H 一 G H F G E F D E C D B C A B A AE A DrumJohn Clark’s Latin Verse Machine The entire Machine contains about eighty-six wheels, giving motion to cylinders, cranks, spirals, pullies, levers, springs, ratchets, quadrants, tractors, snails, worm and fly, heart-wheels, eccentric wheels, and star wheels;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content='—all of which are in essential and effective motion, with varying degrees of velocity, each performing its part in proper time and place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The machine goes through cycles of generating and resetting, as follows4: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The clutch lifts, the control cam moves, and the flywheel begins to revolve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Brake pressure is released, and the machine starts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The flywheel gathers speed, the control frame moves up, and the board that resets the letter staves moves down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The 47 staves independently move slowly down, under gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The changing letters can be seen through the viewing windows on the front panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Each stave comes to rest on a “composing” wire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The front panel shows the appropriate letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The control cam completes a half cycle and the internal bell rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The clutch is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' A complete Latin verse can be read through the apertures on the front panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The cycle continues, and the letter staves move back up to their start positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' When all the staves are up and in line, a “drum kick” occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Each drum rotates a different amount, bringing new rows of wires to the top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The machine can either be stopped after each cycle or left to continue until the driving weight fully drops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The machine can go through five complete cycles each lasting about a minute before needing to be rewound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' It should be noted that the machine is designed for creativity not calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' There is no requirement for its movements to be replicable but there is a need to impart arbitrary motion so that each new line of verse is different and unpredictable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Clark included a mechanism to set each drum rotating independently by giving it a rotational kick from a weight-driven pawl that engages a ratchet on the drum to spin it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Then a spring-loaded roller moves to connect with a star wheel on each drum to stop it in a position where the staves can descend onto its line of wires to form a Latin word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The spin mechanism is similar to that seen on mechanical slot machines that were introduced some 40 years later in the late 19th century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' It is not pseudo-random since the spin of each drum is affected by physical conditions such as temperature and friction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Clark carefully chose the words coded on each drum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The ordering of the words on the six drums is ADJECTIVE NOUN ADVERB VERB NOUN ADJECTIVE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The machine is designed to always produce a line of dactylic hexameter verse, though it may not always make much sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' He generally chose gloomy words to give a solemn feel to the production, for example: M ¯ART˘I ˘A C ¯ASTR ˘A F ˇOR¯IS PR ¯ÆN ¯ARR ¯ANT PR ¯OEL˘I ˘A M ¯ULT˘A5 which Clark himself translated as “martial encampments foreshow many oppositions abroad.” Table 1 shows the complete set of words for each drum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The physical width of a drum is determined by the longest word coded on it, thus drum 4 is wider than drum 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Some words, and the general morbid tone, were copied from John Peter but Clark extended the vocabulary of previous versifying tables while retaining grammar and meter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' His choice of Latin words has been criticized for being repetitive and in a few instances incorrect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' However, “Faults and shortcomings there may be, but it still stands as a monument to the patience and ingenuity of John Clark” [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' It was also a practical demonstration Clark’s theory of linguistic creativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 5 19th Century Cognitive Science The Stanford Encyclopedia of Philosophy [30] characterizes Cognitive Science as “the interdisciplinary study of mind and intelligence, embracing philosophy, psychology, artificial intelligence, neuroscience, linguistics, and anthropology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Its intellectual origins are in the mid-1950s when researchers in several fields began to develop theories of mind based on complex representations and computational procedures.” 4Adapted from a typewritten “Description of the Mechanism of the Latin Verse Machine” by the conservator Peter Jealous, September 1970.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Held in the Clark archive at the Alfred Gillett Trust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 5The diacritic marks added here indicate the hexameter rhythm: dee dum dum, dee dum, dum dee, dee dee dee, dee dum dum, dee dum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 5 John Clark’s Latin Verse Machine Table 1: Lists of Latin words for each drum on Eureka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Here, we show how John Clark, working 100 years before the “intellectual origins” of cognitive science, explored a theory of poetic composition that was based on a representation of mental conception realized through a computational mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Although the representation was less complex than those founded on digital computers, it had the hallmarks of a symbolic process theory of mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Clark named this theory “kaleidoscopic evolution”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' To put it in historic context, by the early 19th century there was discussion among mathematical scientists as to whether computation machines such as Leibniz’s calculator and Babbage’s Difference Engine could be said to emulate mental processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' In June 1833, Lady Byron (mother of Ada Byron) visited Babbage’s house to view what she called the “thinking machine”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Babbage himself was careful never to ascribe mental powers to his machines [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Clark had no such qualms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' In 1837 (five years before the first published description of Babbage’s Analytical Engine) John Clark published a 22-page booklet, which he printed himself, with the title “The General History and Description of a Machine for Composing Hexameter Latin Verses” [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' A revised version was published in 1848, printed by Frederick Wood of Bridgwater, Somerset [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The new version omits a lengthy discussion of poetic forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' It also includes some revealing alterations to wording in the description of the machine and its foundations in a theory of mechanical composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The account below will refer to the 1848 printing unless stated otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Reference will also be made to a letter Clark wrote to The Athenaeum magazine in July 1845, in response to a correspondent from that magazine [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' In that letter, referring to the exhibition of Eureka in the Egyptian Hall, Clark states: The machine is neither more nor less than a practical illustration of the law of evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The process of composition is not by words already formed, but from separate letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' This fact is perfectly obvious, although some spectators may probably have mistaken the effect for the cause—the result for the principle—which is that of kaleidoscopic evolution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' and as an illustration of this principle it is that the machine is interesting—a principle affording a far greater scope of extension than has hitherto been attempted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 6 John Clark’s Latin Verse Machine The machine contains letters in alphabetical arrangement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Out of these, through the medium of numbers, rendered tangible by being expressed by indentures on wheel work, the instrument selects such as are requisite to form the verse conceived;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' the components of words suited to form hexameters being alone previously calculated, the harmonious combination of which will be found to be practically interminable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Clark did not use the term “evolution” in its modern definition of evolution of species but the earlier sense of “unfolding” or “coming into being”, applied to a foetus, or in this case a line of verse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Why did he apply the modifier “kaleidoscopic” to this process of composition?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' A distinctive feature of cognitive science is the use of technology as a mechanism to understand mental structures and processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' In modern times this is the digital computer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' in the mid-19th century, Clark called upon a contemporary scientific instrument, the kaleidoscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The kaleidoscope was invented in 1815 by Sir David Brewster as an instrument to create regular patterns from irregular or random arrangements of objects [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Clark applied this as a metaphor and mechanism for the creative process of versification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' He describes the interior of his machine thus: And in the interior is a large Kaleidoscope, which regularly constructs a geometric figure, the form whereof is ascertained by the falling of numerous probes into its indentures, and thus the Latin verse about to be composed is determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' This action is performed at the commencement of the operation, and is the precise time when the Line of Verse is conceived, previous to its mechanical composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Clark was referring to the kaleidoscopic pattern of wires radiating from the drums that determined how the lettered staves would fall and thus which words would form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The booklet then refers to “the mind of the machine” to distinguish the “moment of conception” from the mechanical production of the verse (his emphasis below): There is a certain point of time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' which may be called the Moment of Conception,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' at which instant of time the identical Latin verse,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' which is about to be produced,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' is conceived in the mind of the machine,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' (if the expression be allowable,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=') and that identical verse,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' which is then and there conceived,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' will be mechanically produced and displayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' This visible display of the Line conceived, is effected by the mechanical movements of the Automaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' But the conception of the Line is not mechanical, nor can it possibly be rendered a visible or tangible thing, in being an imagination only, partaking somewhat of the nature of an arithmetic series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' In like manner we cannot see or feel human imagination, but we can render it audible or visible by the mechanical instruments of the tongue or pen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The Hexameter Automaton bears some affinity to an animated being.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' It possesses a material and an immaterial part, a corporeal and an incorporeal power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Yet it is a thing far inferior to an animal, inasmuch that it possesses no volition or intention, nor any consciousness of its own existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' What follows is an attempt to explain Clark’s conception of kaleidoscopic evolution, with extracts from Clark’s original wording in quotes (his emphasis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Mechanical reproduction of verse requires two elements: a method of representing the verse (“to produce the effect by mechanical means”) and a mechanism to output combinations of well-formed words (“to arrange the machinery in a convenient and eligible manner”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Language can be described as formal sequences of numbers to mechanize composition (“the powers and properties of infinite numbers, applied to poetical composition, etc.”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The language of Latin hexameter verse may be harder for humans to write than everyday English prose (“which act as fetters of confinement to the writers of verses, much increasing their difficulties”), but its formal constraints make it easier to implement on a computational device (“have an opposite effect when applied to a machine”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' By devising a schema for a line of verse – of the form ADJECTIVE NOUN ADVERB VERB NOUN ADJECTIVE – and selecting words for each category with the appropriate meter, Clark produced a tabular representation (“the foregoing Tabular principle”) where words from each column in sequence could be combined arbitrarily to create a well-formed verse (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The process is analogous to a kaleidoscope which, through its careful alignment of mirrors, creates beautiful patterns from arbitrary shapes (“Forming an indefinite number of regular Geometrical Figures by a Machine”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Just as a kaleidoscope necessarily creates regular geometric patterns, so the Eureka machine is designed 7 John Clark’s Latin Verse Machine to produce only well-formed verses (“the machine now proposed, cannot possibly form other than Hexameter Latin Verses”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The verses must not only be grammatically well-formed but also be original (“the Automaton has never repeated any line of verse which it has previously made”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The machine must display creative intelligence, producing verses that make sense (“It is requisite that each Line or Verse shall be correct, not only in measure and in accent, but it is also essential that the verse be fraught with idea or intelligence”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Clark coded the letters for each word as numbers and implemented these on the Eureka machine as filaments of wire in the six drums, with the length of each wire corresponding to a letter of the Latin alphabet (“Out of these, through the medium of numbers, rendered tangible by being expressed by indentures on wheel work, the instrument selects such as are requisite to form the verse conceived”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The Eureka machine simulates human creative composition (“the Hexameter Automaton bears some affinity to an animate being”), whereby a Latin scholar who has learned the patterns of hexameter verse can combine these mentally (“an incorporeal power”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The machine mimics this creative process of versification (“at which instant of time the identical Latin verse, which is about to be produced, is conceived in the mind of the machine”) and then displays the verse (“that identical verse, which is then and there conceived, will be mechanically produced and displayed”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' At the “moment of conception” its drums randomly align to produce a novel combination of words (“thus the Latin verse about to be composed, is determined”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The staves then drop down to display the series of letters that form a line of verse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The creative mental combination and its mechanical realization are separate processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The creativity is hidden from view (“we cannot see or feel human imagination, but we can render is audible or visible by the mechanical instruments of tongue or pen”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Many great ideas have been conceived but not expressed in words (“It is also possible that this supposed idea with thousands of others, equally beautiful, and necessarily existing in embryo from eternity to eternity, may possibly never be disclosed”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The machine can continue to simulate creativity without human assistance (“thousands of Verses may be theoretically conceived, and also mechanically composed and decomposed during the night, or in the intentional absence of all Intelligent Beings, or Spectators: provided that the weight, or power, which actuates the machine, be continued”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Clark did not see the Eureka machine as a useless mechanical curiosity but as a working demonstration of his theory of kaleidoscopic evolution, a theory with applications in science and technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' In a “Questions and Answers” section of his booklet Clark asks, “To what uses may the Hexameter Machine be put?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' He answers with: It may also be asked of what use is an Acorn?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Not much in its present state: but if planted and suffered to grow, it may possibly produce an Oak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Thus one invention brings forward another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Every new thing is an intellectual accession, and every accession may, possibly, be of important use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' In his letter to The Athenaeum [6], Clark writes “as an illustration of this principle [of kaleidoscopic evolution] it is that this machine is interesting—a principle affording far greater scope of extension than has hitherto been attempted”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' In his booklet he envisages automated speech output: A Speaking Automaton6, has lately been completed: it is actuated by a performer, but if it were combined with the Hexameter Machine, it would produce its sentences spontaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Clark ends the booklet with a brief account of the history of “Androides and Automatic Figures” concluding with: The attention of the present age is deservedly directed to the admirable Calculating Machine, designed by Mr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Babbage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' This is now in a considerable degree of progress towards completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 6 Reception By the time he exhibited Eureka, Clark was aged 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' His dress and manners must have been out of place among the London intelligentsia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' In his memoirs, William Ballantyne Hodgson, a Scottish educational reformer and political scientist describes a visit to the Egyptian Hall to see the Eureka machine [16]: We walked together as far as Piccadilly to the Egyptian Hall, where I saw the Eureka, an instrument for making Latin verse, of which I enclose you a brief account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Had not heard of it before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Barham is exhibiting it just now for the inventor, Clark, whom I also saw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Though I cannot say much for the 6Presumably a reference to the Euphonia machine, a talking head able to mimic human speech, which was demonstrated in the Egyptian Hall in 1846.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 8 John Clark’s Latin Verse Machine sense of the verses, there are occasional and recurring errors in quantity, and I suspect that the range of the machine is much more limited than is alleged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The inventor spent fifteen years upon it—five more years than are needed to make a boy into a verse-making machine, and still less perfect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Clarke is a strange, simple-looking old man.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Babbage said the other day that he was as great a curiosity as his machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The contrast is striking between the suave Cambridge-educated raconteur Babbage and the simply dressed, provincial Clark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' By 1845, Babbage had won a Gold Medal from the Astronomical Society and gained over £17,000 of Government grants for his Difference Engine engine to calculate mathematical and astronomical tables, despite never completing a full working machine [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Meanwhile Clark was demonstrating his Eureka machine daily in an exhibition hall to a bemused public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' An article in The Illustrated London News of July 19, 1845 [32] shows a line drawing of the Eureka cabinet with a factual description of the machine which “has lately been brought to the metropolis, to contribute to the ‘sights of the season’.” Most the content of that newspaper article is taken from verbatim from Clark’s 1837 booklet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' A letter from “P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Nuttall” to The Athenaeum magazine dated June 28, 1845 is less reserved [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The Eureka, Nuttall opines, is “little better than a mere puzzle, which any school-boy might perform by a simpler process” since it merely produces six words of a regular Latin grammatical pattern (“the first word is uniformly a dactyl, and an adjective of the neuter plural, the second word a trochee .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content='. ”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' All a school-boy has do it is write Latin words of each type on slips of paper, put them into six piles, then draw out words at random from each pile in sequence to form a Latin verse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' “It may be a very curious and instructive amusement,—but nothing more.” Clark knew that he could have simplified his machine by printing complete words on the drums then rotating each drum to display a line of words through windows, like a 20th-century slot machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' But that would have been less general and creative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' As Clark attempted to explain in his response to Nuttall, published in The Athenaeum on July 2, 1845 [6], the method he had adopted for Eureka of having the staves fall individually onto the wires is not an effect to entertain the audience but an intrinsic part of its design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Working with letters not words has “far greater scope of extension than has hitherto been attempted.” By representing words as lengths of wire, he could not only re-program the machine by adjusting the lengths, but also envision a machine that would automatically interpret the lengths to form new verses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' A glimpse of Clarke’s vision for extension to his theory of kaleidoscopic evolution can be seen in a letter he wrote to his sister in August 1845 7: I hope to be home soon with a long story to tell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' A most astonishing discovery has been made with the Hexameter Machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' It would make so many millions of verses that to produce all will not be done for a century or more, but we have discovered that if we take 100 or so of its productions that these will produce thousands & millions of other verses, by another machine, & so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' It has opened a new field of scientific speculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' It appears from this fragment that Clark had been speculating about a sequence of machines, each of which takes outputs from the previous machine and combines these in new ways to generate longer and more varied results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The exhibition of the Eureka machine in London coincided with what has been called “hexameter mania” [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' This was a lively argument about the teaching of Classics in schools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' In 19th century England, boys in elite schools were required not only to speak and write Latin, but also to compose Latin verse in a manner that supposedly emulated the great Roman poets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' By the mid 19th century, this practice had descended into rote learning, with pupils looking up textbooks of verse composition to grind out pastiches of Ovid and Virgil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' A machine that could automate this process at the rate of 10,000 verses a week, like a demented schoolboy, added fuel to those commentators who derided the useless rituals of elite schools and were calling for a modern curriculum that embraced the Victorian advances in science and technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Thus, Clark became embroiled in contemporary debates about the value to the individual and society of writing hexameter Latin verse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The satirical magazine Punch [23] wrote, tongue in cheek: That notable invention, the Eureka, or Latin verse-grinder, was tried yesterday before a committee of young gentlemen from the public schools, who are anxious to have their Latin exercises done with the least possible trouble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Several double-barrelled Eurekas were ordered for Eton, Harrow, and Rugby.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 7Letter by John Clark to his sister Sarah (Clark) Metford, August 1845.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The letter is held in the Clark archive at the Alfred Gillett Trust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 9 John Clark’s Latin Verse Machine The debate over the value, if any, of students churning out assignments that could just as well be generated by machine resonates with current concerns about students employing AI language systems to write their essays and assignments [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The Eureka machine was soon consigned to a footnote in the history of Latin versification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' An article for Chambers’s Edinburgh Journal in 1850 confuses the Eureka with the Euphonia (a machine to mimic the human voice), stating that “by its aid the most illiterate person could produce thousands of Latin verses” [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 7 Legacy In the 20th century, the Oulipo movement, originating among writers and mathematicians in France, experimented with automated production of novels and poems but made no reference that we can find to Clark and his verse machine [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The best-known mention in literature of a machine to generate verse comes in George Orwell’s Nineteen Eighty-Four [19, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Orwell depicts the Records Department of the Ministry of Truth where rubbishy newspapers and sentimental songs were “composed entirely by mechanical means on a special kind of kaleidoscope known as a versificator.” Of all the words Orwell might have used to describe a machine for churning out prose and songs he chose “kaleidoscope” and “versificator.” Was that a nod to Clark’s “Kaleidoscopic Evolution” by a “machine for making Latin verses?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' With the advent of digital computers from the 1950s onwards, many academics and students of computer science (including the author of this paper) experimented with programs to generate poems but without acknowledging the pioneering work of John Clark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' In recent years Clark’s work has been re-discovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' In 1963, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Blandford wrote a factual account of Clark and his machine which included a table of words on each drum (Table 1) [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Jason David Hall, Professor of Modern Literature and Culture at the University of Exeter, has written academic papers [10], [11] and a book, Nineteenth-Century Verse and Technology: Machines of Meter [12] examining the relation between machine culture and poetic meter in the 19th century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' These publications explore the Eureka machine as a producer of metrical verse within the mechanization of science, education, culture, travel and work in Victorian England.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The book also examines how 19th-century writers on psychology and physiology saw the human production of meter as an inherently automatic process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Clark has been recognized by a few researchers in computational creativity as a pioneer of mechanized composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Douglas Summers Stay, in his book Machinamenta has commented on the connection between the Eureka and the kaleidoscope as a machine to produce patterns by random combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The problem with both, he notes, is how to make a machine that is not limited by the inventor’s initial choice of settings, but can grow in ability over time [27, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Other mentions of Eureka in the literature on computational creativity include [12], [33] and [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' As for the Eureka machine itself, it is not clear where it went immediately after Clark’s death in 1853 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Around 1856 it was housed in the Counting House of the Clarks Shoe Company in Street, Somerset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Then, in 1889 or 1890 it was moved to the geology Museum at Crispin Hall in Street, to the concern of Alfred Gillett, donor of the geological collection, who thought it let down his exhibits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The last person capable of operating the machine during that period was John Aubrey Clark, son of the founder of the shoe company.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' When the Geology Museum dispersed in 1948, the Eureka was moved back to the Clarks factory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' It was restored to working order in 1950 by Leslie W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Husbands, a local clockmaker, and Frederick Berry, the typewriter and sewing machine engineer at Clarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Then it was moved to the company Records Office.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' By 1963 it was in the Clarks company museum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' In 1970-71 it was renovated for a second time by the Clarks Special Projects Engineer Peter Jealous, in collaboration with Husbands, and in 1979 it was housed in the company’s Shoe Museum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' By 1996, no longer working, it was put into storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' It stayed there until 2015 when it went to Devon for extensive restoration by accredited conservators Richard Jaeschke who conserved the casework and exposed the gilding and verse on the front of the machine, and Neil Bollen who conserved the mechanical parts and restored the machine to a gentle working order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The restoration was part of a collaboration between the University of Exeter and the Alfred Gillett Trust, funded by the Arts and Humanities Research Council, to understand, document and restore Eureka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Eureka now rests on a plinth at the Alfred Gillett Trust, Street, Somerset as a centerpiece of its collection of artefacts connected with the Clarks shoe company 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 8 Conclusion John Clark does not fit comfortably into the narrative of pre-electronic computing, where urbane and sometimes irascible mathematicians based in the great European cities of Paris and London developed machines of intricate engineering to 8The history of the Eureka machine is drawn from [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 9Eureka can be accessed at the Alfred Gillett Trust in Street, Somerset, by appointment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Researchers can also view archives and collections related to Clarks shoe company, the family, and Street whilst a new museum is being developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 10 John Clark’s Latin Verse Machine solve problems in accounting, astronomy and navigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Clark was an outsider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' He never studied at university, he was a self-taught poet and philosopher, he lived most of his life in the market town of Bridgwater on the Southwest corner of England.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' He began work on the Eureka machine in 1830, when he was forty-five.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' When, at the age of 60, he arrived in London it was to exhibit his machine at the Egyptian Hall which by that time was noted more for its popular entertainment than display of scientific discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Clark himself, in the final section of his booklet [7], places his device among “The construction of different species of Androides and Automata [that have] occasionally engaged the attention of Mechanists, of all nations” including “a fine automaton, representing a Bengal Tiger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The deep sounds of its roaring are admirably produced by the organ pipes of its internal structure.” It is hardly surprising given his background, his reception in London (“as great a curiosity as his machine”), and the criticism of Eureka for producing gloomy and occasionally ill-formed 10 Latin verses, that John Clark has not been included in histories of computing and artificial intelligence (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=', [21], [29], [18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' The comprehensive book on The Origins of Digital Computers [24] mentions Clark only in the bibliography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' However,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' the enterprise in which Clark was engaged,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' to understand and simulate the creative production of language,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' has been a recurring interest of computer scientists from Christopher Strachey in the 1950s [26],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' through the extensive work of Sheldon Klein and colleagues at the University of Wisconsin to automate the writing of novels [14] and recent work on computational poetry (see [15] for a review),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' to the investment by companies including Microsoft,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Google,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Meta and Baidu in pre-trained AI models for language generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' This author prompted the GPT-3 Transformer language model from OpenAI to generate original Latin verse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' It responded in seconds with well-formed Latin but it failed in rhyme and meter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Clark would have been intrigued and amused.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 9 Acknowledgements My sincere thanks to Karina Virahsawmy, Assistant Curator, Alfred Gillett Trust, for allowing me access to the John Clark archive and giving a personal demonstration of the machine, to Neil Bollen, conservator, for his patient and detailed responses to my queries about how the machine functions, and to Rafael Pérez y Pérez, co-author of Story Machines: How Computers Have Become Creative Writers, for setting me on the journey to Eureka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 10 References [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Berry, “LVM Latin Verse Machine c 1830 – c1845: Collection Description compiled by Charlotte Berry”, The Alfred Gillett Trust, Street, Somerset, United Kingdom, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' [2] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Blandford, “The Eureka,” Greece & Rome, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 10, no .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content='1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 71–78, 1963.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' [3] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Brewster, The Kaleidoscope: Its History, Theory, and Construction with its Application to the Fine and Useful Arts, 2nd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' London, UK: John Murray, 1858.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' [4] Chambers’s Edinburgh Journal, “Latin versification for the million,” Chambers’s Edinburgh Journal, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 13, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 205, 1850.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' [5] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Clark, The General History and Description of a Machine for Composing Hexameter Latin Verses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Bridgwater, Somerset, UK: John Clark, 1837.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' [6] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Clark, “Letter to The Athenaeum”, The Athenaeum, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 923, Jul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 5, 1845, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 669–670.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' [7] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Clark, The General History and Description of a Machine for Composing Hexameter Latin Verses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Bridgwater, Somerset, UK: Frederick Wood, 1848.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' [8] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Engberg-Pedersen, “The sense of tact: Hoffmann, Maelzel, and mechanical music,” The Germanic Review: Literature, Culture, Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 93 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 351–372, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' [9] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Green, “Was Babbages’s Analytical Engine intended to be a mechanical model of the mind?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=',” History of Psychology, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 8, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 35–45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' [10] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Hall, “Popular prosody: Spectacle and the politics of Victorian versification,” Nineteenth-Century Literature, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 62, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 222–249, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' [11] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Hall, “Mechanized metrics: From verse science to laboratory prosody, 1880–1918,” Configurations: A Journal of Literature, Science and Technology, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 3, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 17, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 285–308, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 10“From the metrical point of view the inventor would have done better to have produced lines of five words each, arranged perhaps in the order of the so-called Golden Line with nouns and adjectives carefully balanced” [2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content='77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 11 John Clark’s Latin Verse Machine [12] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Hall, Nineteenth-Century Verse and Technology: Machines of Meter, Basingstoke, UK: Palgrave Macmillan, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' [13] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Hrešková and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Machová, “Haiku poetry generation using interactive evolution vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' poem models,” Acta Electronica et Informatica, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 17, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 10–16, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' [14] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Klein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=', “Automatic novel writing: A status report,” The University of Wisconsin, Technical Report #186, Computer Sciences Department, The University of Wisconsin, 1973.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' [15] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Linardaki, “Poetry at the first steps of Artificial Intelligence,” Humanist Studies & the Digital Age, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 7, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 1, 2022, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content='5399/uo/hsda/7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' [16] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Meiklejohn, Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=', Life and Letters of William Ballantyne Hodgson, Edinburgh, UK: David Douglas, 1883.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' [17] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Nuttall, “Letter to The Athenaeum,” The Athenaeum, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 922, Jun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 28, 1845, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 638.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' [18] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' O’Regan, Introduction to the History of Computing: A Computer History Primer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Springer International Publishing, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' [19] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Orwell, Nineteen Eighty-Four (Penguin Classics edition, 2000), Penguin Random House, 1949.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' [20] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Peter, Artificial Versifying, or the School-boy’s Recreation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' London, UK: John Sims, 1677.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' [21] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Pratt, Thinking Machines: The Evolution of Artificial Intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Oxford: Basil Blackwell, 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' [22] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Prins, “Metrical translation: nineteenth-century Homers and the hexameter mania,” in Nation, Language, and the Ethics of Translation, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Wood and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Bermann, Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=', Princeton, USA: Princeton University Press, 2005, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 229–256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' [23] Punch, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 9, July to December 1845, London: Punch, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' [24] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Randell, Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=', The Origins of Digital Computers: Selected Papers, 3rd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=', Berlin: Springer-Verlag, 1982.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' [25] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Sharples and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Pérez y Pérez, Story Machines: How Computers Have Become Creative Writers, Routledge, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' [26] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Strachey, “The ‘thinking’ machine,” Encounter, Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 1954, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 25-31, 1954.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' [27] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Summers Stay, Machinamenta: The Thousand Year Quest to Build a Creative Machine, Machinamenta Publishing, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' [28] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Swade, The Cogwheel Brain, London, UK: Abacus, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' [29] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Swade, “Pre-electronic computing,” in Dependable and Historic Computing, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Jones and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Lloyd, Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=', Berlin, Germany: Springer, 2011, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 58–83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' [30] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Thagard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' “Cognitive Science.” Stanford Enyclopedia of Philosophy, https://plato.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content='stanford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content='edu/entries/cognitive- science/ (accessed Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 5, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' [31] The Athenaeum, “Announcement,” The Athenaeum, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 922, Jun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 28, 1845, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 635.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' [32] The Illustrated London News, “The Eureka,” The Illustrated London News, Jul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 19, 1845, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' [33] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Todorovic, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Grba, “Wandering machines: narrativity in generative art,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' and Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' of the Arts, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 11, no2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 50-58, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} +page_content=' 12' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE5T4oBgHgl3EQfYA9D/content/2301.05570v1.pdf'} diff --git 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generalization of the thermofield double state +Arash Azizia +aThe Institute for Quantum Science and Engineering, Texas A&M University, College Station, TX +77843, USA +E-mail: sazizi@tamu.edu +Abstract: We elaborate more on κ-mode, a mode that was found by a combination of the +opposite sign norm Rindler modes in the right and left Rindler wedges. Especially, we show +how the thermofield double state can be extended to a generalized non-thermofield double +state by considering a relation between κ-vacua, similar to the Minkowski-Rindler vacua +relation. A general κ ̸= 1 vacuum, in contrast to the well-known case of the Minkowski +vacuum, is no longer thermal when reduced to a specific Rindler wedge. +arXiv:2301.13672v1 [hep-th] 31 Jan 2023 + +Contents +1 +Introduction +1 +2 +The Klein-Gordon inner product +3 +2.1 +The definition and properties of the inner product +3 +2.2 +The inner product and commutation relations +5 +2.3 +Some examples: Minkowski plane wave and Rindler +6 +2.4 +Combining positive and negative norm modes +7 +3 +Kappa mode: combining opposite sign norm Rindler modes +8 +3.1 +Constructing Kappa modes +8 +3.2 +Rindler and Unruh-Minkowski as special cases +10 +3.3 +Bogoliubov transformation between κ-modes +11 +3.4 +Bogoliubov transformation between a κ-mode and the Minkowski plane wave +12 +3.5 +Commutation relations for different κ +13 +4 +Combining same sign norm Rindler modes +14 +4.1 +Bogoliubov transformation between different modes +16 +4.2 +Bogoliubov transformation between the same sign norm and the opposite +sign norm modes +17 +5 +Generalized non-thermofield double states +17 +5.1 +Relating different κ-vacua +18 +5.2 +κ-vacuum in terms of Rindler +18 +5.3 +κ-vacuum in terms of Minkowski +18 +6 +Conclusion +19 +A Proof of the lemma (2.12) +20 +1 +Introduction +Quantum field theory (QFT) in curved spacetime1 gives us intriguing results such as Hawk- +ing radiation [2] and the Unruh effect [3]. These phenomena have not only far-reaching +consequences, where any consistent theory of quantum gravity should reproduce them at +appropriate limits, but also appear in diverse research fields such as theoretical condensed +matter [4, 5], quantum optics [6, 7], and quantum information [8, 9], in addition to their +1For a recent review on mathematical background see [1]. +– 1 – + +original research program, high energy theory. The latter is especially reflected in the re- +cent developments on resolving the Hawking information paradox [10] which have been +conducted by two groups in [11–14]. +The quintessence of Hawking radiation and the Unruh effect is the fact that generally +there is not a unique vacuum in QFT. Fulling [15] was one of the first scholars who realized +this point. The canonical way of describing QFT is as follows: +• Consider a field theory of desired spin in a given manifold. The curved spacetime is +treated classically (no back reaction from quantum fields on the background). The so- +lutions of the equation of motion, i.e., field modes, can be derived from the Lagrangian +of the theory in the curved background. +• The positive norm modes, with respect to the appropriate inner product, can be +computed. The negative norm modes, hence, shall be the complex conjugate of the +positive norm ones. +• The quantum field can be expanded in terms of the modes, where the annihilation +and creation operators are associated with the positive and negative norm modes +respectively. +• The set of positive norm modes is not unique, and hence, there are different sets of +annihilation and creation operators, which can be related to each other by Bogoliubov +transformations. Moreover, an annihilation operator of one set can be a combination +of annihilation and creation operators of the other set. +• The vacuum in the theory is defined as a state which is annihilated by all annihilation +operators. In general vacua of two sets of modes are distinct, unless the annihila- +tion operators of one set can be written in terms of just annihilation operators (not +annihilation and creation) of another one. +Unruh [3] introduced a thought-provoking field mode, later named Unruh-Minkowski +mode, in order to investigate the relation between the usual Minkowski plane wave and +Rindler modes. The mode’s vacuum is Minkowski, but its form is similar to the Rindler +mode. Furthermore, Unruh studied the behavior of a particle detector undergoing a uniform +acceleration in the Minkowski vacuum. The particle detector first was considered as a box +with discrete energy levels by Unruh and subsequently was simplified more by DeWitt +[16] by considering the first two energy levels of the box, hence is named Unruh-DeWitt +detector. The detector, starting from the ground state, gets excited and emits a photon +in an Unruh-Minkowski mode. The bizarre feature of the mode is it resides mostly in the +opposite wedge of where the detector is as emphasized by Unruh and Wald [17]. More +recently, two Unruh-DeWitt detectors were envisaged in [18, 19] and it was shown despite +the apparent violation of causality, the latter is upheld. +In contrary to the more conventional QFT in Minkowski spacetime, where the plane +wave is the mode one usually conceives, in fact an infinite number of different field modes +exist, and the inner product should be exploited in order to associate the annihilation and +– 2 – + +creation operators to these modes appropriately. Recently, we [20] introduced a general +mode, named κ-mode, which is a combination of opposite sign norm Rindler modes in the +right and left Rindler wedges. The fascinating feature of κ-modes is their distinct vacua, +which are parameterized by a real positive parameter κ. Two well-known vacua, Minkowski +and Rindler, are special cases of κ-vacuum for κ = 1 and κ → ∞ respectively. +The thermofield double state [3, 21] is ubiquitous in theoretical physics, e.g., it is a key +ingredient of the AdS/CFT dictionary [22–24], as depicted in Maldacena’s proposal [25] +about the duality of an eternal AdS black hole in the bulk and two copies of boundary CFT +in the thermofield double state. Moreover, thermofield double state appears in quantum +optics under the name of squeezed state [26]. We generalize the usual thermofield double +state to a relation between κ-vacuum and κ′-vacuum. As a matter of fact, we find a non- +thermofield double state by relating a general κ-vacuum to the Rindler vacuum. The famous +thermofield double state is then a special case (κ = 1) of this generalized non-thermofield +double state. +The rest of the paper is organized as follows. In section 2 we study the Klein-Gordon +inner product, and we show why the positive norm mode is associated with the annihilation +operator by relating the inner product between modes to the commutation relations between +operators. In section 3 we study the κ-mode and elaborate more on what was reported in +[20]. In section 4 we show that it is not possible to get a mode by combining the same sign +norm Rindler modes. In section 5 we find the generalization of thermofield double state to +generalized non-thermofield double state. Finally we conclude in the last section 6. +2 +The Klein-Gordon inner product +The Klein-Gordon inner product is the main tool to distinguish positive and negative norm +modes, associated with annihilation and creation operators respectively. The inner product +of two modes is constant in time; this is a great feature of the inner product. In this section, +we show the interconnection between the inner product and the commutation relations. +Furthermore, we explicitly find the Minkowski plane wave and Rindler modes utilizing the +inner product. +Finally we demonstrate how to generate a new positive norm mode by +combining a given positive norm mode and its complex conjugate. +2.1 +The definition and properties of the inner product +According to standard quantum field theory, a quantum field may be written as an infinite +superposition of solutions of the equation of motions, i.e., modes, with operator coefficients +which turn out to be annihilation and creation operators. The question then arises how to +distinguish between modes associate to these operators. In other words, one may write a +field as +Φ(x) = +� +dΩ +� +Φ(x, Ω) aΩ + Φ∗(x, Ω) a† +Ω +� +. +(2.1) +– 3 – + +To interpret aΩ and a† +Ω as annihilation and creation operators respectively, they have to +satisfy the standard commutation relation +[aΩ, a† +Ω′] = δ(Ω − Ω′) . +(2.2) +Henceforth, it is crucial to choose a correct mode Φ(x, Ω) associated with the annihila- +tion operator, while its complex conjugate Φ∗(x, Ω) is associated with the creation one. The +key concept for distinguishing these two modes is the inner product. For the Klein-Gordon +scalar field the inner product reads +⟨φ1, φ2⟩ = −i +� +Σ +√−g dΣµ (φ∗ +1 ∂µφ2 − ∂µφ∗ +1 φ2) , +(2.3) +where Σ is the appropriate Cauchy hypersurface. Note, we have used the convention of +(−, +, · · · , +) for the Minkowski metric. +If one were to choose mostly negative signa- +ture for the metric, then there would be an overall minus sign difference, namely ⟨f, g⟩ = +i +� +Σ +√−g dΣµ (f∗ ∂µg − ∂µf∗ g). +It is worthwhile here to note about different conventions on the Klein-Gordon inner +product among the early investigators. We consider four of them in the following: +• Hawking [2]: ⟨φ1, φ2⟩H = i +2 +� +Σ [φ1∂µφ∗ +2 − (∂µφ1) φ∗ +2] dSµ . +• DeWitt [27]: ⟨φ1, φ2⟩DeW = −i +� +Σ [φ∗ +1∂µφ2 − (∂µφ∗ +1) φ2] dSµ . +• Wald [28]: ⟨φ1, φ2⟩Wald = i +� +Σ [φ∗ +1∂µφ2 − (∂µφ∗ +1) φ2] dSµ . +• Unruh and Wald [17]: ⟨φ1, φ2⟩UW = i +2 +� +Σ [φ∗ +1∂µφ2 − (∂µφ∗ +1) φ2] dSµ . +The inner product is anti-linear in the second argument only in Hawking’s notation +and is linear in the rest. While all of the above authors used the mostly positive signature +for the metric, the sign of the inner product is not consistent among them. Of course, there +is a factor of one half discrepancy too. +Some useful relations in Klein-Gordon inner product are as follows: +⟨f, αg + βh⟩ = α ⟨f, g⟩ + β ⟨f, h⟩ , +⟨f, g⟩∗ = ⟨g, f⟩ , +⟨f∗, g∗⟩ = − ⟨f, g⟩∗ . +(2.4) +Note, the above Klein-Gordon “inner product” is not actually an inner product, strictly +speaking, since +⟨f∗, f∗⟩ = − ⟨f, f⟩ , +(2.5) +and therefore the positivity of inner product has been violated. However, this property is +very crucial in distinguishing between positive and negative norm modes. One may associate +the annihilation operator to a positive norm mode, while the mode’s complex conjugate, +with a negative norm, is associated to the creation one. Actually, the inner product is +defined so as to satisfy the following properties: +� +Φ(x, Ω), Φ(x, Ω′) +� += [aΩ, a† +Ω′] = δ(Ω − Ω′) , +� +Φ(u, Ω), Φ∗(u, Ω′) +� += − [aΩ, aΩ′] = 0 . +(2.6) +– 4 – + +We prove the above relations in the next subsection. Also, using (2.4), then (2.6) implies +� +Φ∗(x, Ω), Φ∗(x, Ω′) +� += −δ(Ω − Ω′) . +(2.7) +In a D + 1 dimensional Minkowski spacetime with the usual convention for the com- +ponent, i.e., 0 and i represent time and space components, and for a constant time Cauchy +surface, one has n0 = 1, ni = 0, n0 = −1, and ni = 0. Here nµ represents the unit normal +vector to the manifold. Then (2.3) indicates +⟨f, g⟩ = i +� +dDx (f∗ ∂tg − ∂tf∗ g) . +(2.8) +Here we have dΣµ = δµ0 n0dDx = −dDx for µ = 0 and zero for the rest of indices. Note +mostly minus sign convention for the metric yields n0 = 1, and hence to keep (2.8), one +should start off from ⟨f, g⟩ = i +� +Σ +√−g dΣµ (f∗ ∂µg − ∂µf∗ g) as we have emphasized. +In 1 + 1 Minkowski spacetime, the metric is ds2 = −dt2 + dx2 = −du dv. Here, we set +c = 1, and consider the (−, +) convention for the metric. The light-cone coordinates are +u = t − x , v = t + x. Thus for a manifold of constant u, one has nv = −2 , nu = 0; while +for a manifold of constant v, one has nu = −2 , nv = 0. Since √−g = 1 +2, then the inner +product (2.3) becomes +⟨f, g⟩ = i +� ∞ +−∞ +dv +� +f∗ ∂ +∂vg − ∂ +∂vf∗g +� +, +⟨f, g⟩ = i +� ∞ +−∞ +du +� +f∗ ∂ +∂ug − ∂ +∂uf∗g +� +, +(2.9) +where a constant u, and a constant v manifolds were chosen in the above relation respec- +tively. +2.2 +The inner product and commutation relations +Here in this subsection we show the connection between the inner products and the com- +mutation relations. Namely, +[aΩ, a† +Ω′] = +� +Φ(x, Ω), Φ(x, Ω′) +� +, +[aΩ, aΩ′] = − +� +Φ(u, Ω), Φ∗(u, Ω′) +� +. +(2.10) +In order to prove the above relations, one may start from the following: +aΩ = ⟨Φ(x, Ω), Φ(x)⟩ , +a† +Ω = − ⟨Φ∗(x, Ω), Φ(x)⟩ , +(2.11) +where they can be found from the field mode expansion (2.1), and using the properties of +the inner product (2.4). Next, we present a very useful lemma as follows. +Lemma: For any modes f(u, Ω) and g(u, Ω) one has the following relation: +� +⟨f(u, Ω), Φ(u)⟩, +� +g +� +u′, Ω′� +, Φ +� +u′��� += − +� +f(u, Ω), g∗(u, Ω′) +� +. +(2.12) +The proof is given in the appendix A. +Having used the lemma (2.12), one can now prove the above mentioned (2.10) inter- +– 5 – + +connection between the commutation relations and the inner products. Namely, +� +aΩ, a† +Ω′ +� += +� +⟨Φ(u, Ω), Φ(u)⟩, − +� +Φ∗ � +u′, Ω′� +, Φ +� +u′��� += +� +Φ(u, Ω), Φ(u, Ω′) +� +, +[aΩ, aΩ′] = +� +⟨Φ(u, Ω), Φ(u)⟩, +� +Φ +� +u′, Ω′� +, Φ +� +u′��� += − +� +Φ(u, Ω), Φ∗(u, Ω′) +� +, (2.13) +where we have used (2.11) and the lemma (2.12). +2.3 +Some examples: Minkowski plane wave and Rindler +In this subsection, we derive Minkowski plane wave and Rindler positive norm modes by +utilizing the inner product. It is more convenient to work with the light-cone coordinate +(u, v). We consider a massless Klein-Gordon field in 1 + 1 dimensions. The field equation +is □Φ = 0, or in terms of light-cone coordinates +∂ +∂u +∂ +∂vΦ = 0, and can be solved simply by +Φ(u, v) = Φ(u) + Ψ(v), where Φ(u) and Ψ(v) are general functions indicating right- and +left-moving waves respectively. Consider the change of coordinates +u = − 1 +ae−a(τ−ξ) , +v = 1 +aea(τ+ξ) . +(2.14) +One may call (τ, ξ) Rindler coordinates [29] and the metric shall be written as ds2 = +e2a ξ (−dτ 2 + dξ2). For positive (negative) a, we have u < 0 (u > 0) and v > 0 (v < 0) and +the associated region in spacetime diagram is called Rindler right (left) wedge. +Also since Φ(u, v) = Φ(u) + Ψ(v), without loss of generality, we consider the right +moving wave, i.e., Φ(u). +Minkowski plane wave +Right moving Minkowski plane wave reads Φ(u, Ω) = f(Ω) e−i uΩ. +Imposing the Klein- +Gordon inner product (2.9) yields +� +Φ(u, Ω), Φ(u, Ω′) +� += i f∗(Ω)f(Ω′) +� ∞ +−∞ +du +� +ei uΩ ∂ +∂u e−i uΩ′ − ∂ +∂uei uΩ e−i uΩ′� +(2.15) += f∗(Ω)f(Ω′)(Ω + Ω′) +� ∞ +−∞ +du ei u(Ω−Ω′) = 4πΩ|f(Ω)|2δ(Ω − Ω′) . +Hence ⟨Φ(u, Ω), Φ(u, Ω′)⟩ = δ(Ω − Ω′) yields the important fact that Ω should be a positive +real number, and f(Ω) = +1 +√ +4πΩ, and thus the positive norm mode in right moving Minkowski +plane wave reads +Φ(u, Ω) = +1 +√ +4πΩ +e−i uΩ , +Ω > 0 . +(2.16) +It is easy to check the inner product of a mode and its conjugate vanishes. +Rindler +Rindler mode can be found in Rindler right (u < 0, v > 0), or left (u > 0, v < 0) wedges. +Again, considering the right moving wave, we may write the mode as +Φ(u, Ω) = θ(u) f(Ω) ui Ω , +Φ(u, Ω) = θ(−u) g(Ω) (−u)i Ω , +(2.17) +– 6 – + +for the left and right wedges respectively. Here Ω is a real number, not necessarily a positive +one. The inner product for the left Rindler wedge reads +� +Φ(u, Ω), Φ(u, Ω′) +� += i f∗(Ω)f(Ω′) +� ∞ +−∞ +θ(u)du +� +u−i Ω ∂ +∂u ui Ω′ − ∂ +∂uu−i Ω ui Ω′� += −f∗(Ω)f(Ω′)(Ω + Ω′) +� ∞ +−∞ +θ(u)du u−i (Ω−Ω′)−1 += −4πΩ|f(Ω)|2δ(Ω − Ω′) , +(2.18) +therefore, the orthonormality condition implies Ω < 0 and f(Ω) = +1 +√−4πΩ. +Similarly for the right wedge, one has +� +Φ(u, Ω), Φ(u, Ω′) +� += i g∗(Ω)g(Ω′) +� ∞ +−∞ +θ(−u)du +� +(−u)−i Ω ∂ +∂u (−u)i Ω′ − ∂ +∂u(−u)−i Ω (−u)i Ω′� += g∗(Ω)g(Ω′)(Ω + Ω′) +� ∞ +−∞ +θ(−u)du (−u)−i (Ω−Ω′)−1 += 4πΩ|f(Ω)|2δ(Ω − Ω′) , +(2.19) +therefore, the orthonormality condition implies Ω > 0 and f(Ω) = +1 +√ +4πΩ. Consequently the +positive norm Rindler mode for right moving wave is +Left wedge: +Φ(u, Ω) = θ(u) +1 +√ +4πΩ +u−i Ω , +Ω > 0 , +Right wedge: +Φ(u, Ω) = θ(−u) +1 +√ +4πΩ +(−u)i Ω , +Ω > 0 . +(2.20) +Note ⟨f∗, f∗⟩ = − ⟨f, f⟩ implies the following relations for all real Ω: +� +θ(u)ui Ω, θ(u)ui Ω′� += −4πΩ δ(Ω′ − Ω) , +� +θ(−u)(−u)i Ω, θ(−u)(−u)i Ω′� += 4πΩ δ(Ω′ − Ω) , +� +θ(u)ui Ω, θ(−u)(−u)i Ω′� += 0 , +(2.21) +where the last relation is obvious since θ(u)θ(−u) = 0. +2.4 +Combining positive and negative norm modes +One way of obtaining new modes is a linear combination of positive and negative norm +modes. This can be performed such that the new mode should satisfy the inner product +relations. +The new mode Ψ(u, Ω) may be defined as +Ψ(u, Ω) = α(Ω)Φ(u, Ω) + β(Ω)Φ∗(u, Ω) , +(2.22) +– 7 – + +where Φ(u, Ω) is the initial mode, and satisfies the following inner product relations +� +Φ(u, Ω), Φ(u, Ω′) +� += δ(Ω − Ω′) , +� +Φ(u, Ω), Φ∗(u, Ω′) +� += 0 . +(2.23) +Here α(Ω) and β(Ω) are general complex coefficients. The inner product for two modes +Ψ(u, Ω) reads +� +Ψ(u, Ω), Ψ(u, Ω′) +� += +� +α(Ω)Φ(u, Ω) + β(Ω)Φ∗(u, Ω), α(Ω′)Φ(u, Ω′) + β(Ω′)Φ∗(u, Ω′) +� +=α∗(Ω)α +� +Ω′� � +Φ(u, Ω), Φ +� +u, Ω′�� ++ α∗(Ω)β +� +Ω′� � +Φ(u, Ω), Φ∗ � +u, Ω′�� ++ β∗(Ω)α +� +Ω′� � +Φ∗(u, Ω), Φ(u, Ω′) +� ++ β∗ � +Ω)β +� +Ω′� � +Φ∗(u, Ω), Φ∗ � +u, Ω′�� += +� +|α(Ω)|2 − |β(Ω)|2� +δ +� +Ω − Ω′� +. +(2.24) +Requiring the answer to be δ (Ω − Ω′) yields the following constraint: +|α(Ω)|2 − |β(Ω)|2 = 1 . +(2.25) +Furthermore, the inner product of a mode Ψ(u, Ω) and its conjugate should be zero. +Namely, +� +Ψ(u, Ω), Ψ∗(u, Ω′) +� += +� +α(Ω)Φ(u, Ω) + β(Ω)Φ∗(u, Ω), α∗(Ω′)Φ∗(u, Ω′) + β∗(Ω′)Φ(u, Ω′) +� += +� +α∗(Ω)β∗ (Ω) − β∗(Ω)α∗ (Ω) +� +δ +� +Ω − Ω′� +. +(2.26) +However, this is identically zero and it does not impose any constraint on α(Ω) and β(Ω). +Henceforth, the only constraint for the coefficients is |α(Ω)|2 − |β(Ω)|2 = 1, which can +be solved as +α(Ω) = cosh (κΩ) e +i γ +2 , +β(Ω) = sinh (κΩ) e− i γ +2 , +(2.27) +where κ and γ are real parameters. +3 +Kappa mode: combining opposite sign norm Rindler modes +In addition to the strategy introduced in section (2.4), one may consider another type of +combination of Rindler modes, namely, combining Rindler modes of opposite wedges. There +are two ways to proceed: combining the same and the opposite sign norm modes of opposite +wedges. In this section we address the latter and in the next section we study the former. +3.1 +Constructing Kappa modes +The ansatz for the opposite sign norm is +Φ(u, Ω) = α(Ω)θ(−u)(−u)i Ω + β(Ω)θ(u)ui Ω . +(3.1) +Note for positive (negative) Ω, the first (second) term has positive norm, while the second +(first) term has negative norm. +– 8 – + +The inner product of two modes reads +⟨Φ(u, Ω),Φ(u, Ω′)⟩ += +� +α(Ω)θ(−u)(−u)i Ω + β(Ω)θ(u)ui Ω, α(Ω′)θ(−u)(−u)i Ω′ + β(Ω′)θ(u)ui Ω′� += α∗(Ω)α(Ω′) +� +θ(−u)(−u)i Ω, θ(−u)(−u)i Ω′� ++ β∗(Ω)β(Ω′) +� +θ(u)ui Ω, θ(u)ui Ω′� += 4πΩ +� +|α(Ω)|2 − |β(Ω)|2� +δ(Ω − Ω′) . +(3.2) +Therefore, the positivity of the norm indicates +4πΩ +� +|α(Ω)|2 − |β(Ω)|2� += 1 . +(3.3) +Note Ω now can be either a positive or a negative real number. +Furthermore, the inner product of the mode and its conjugate becomes +⟨Φ(u, Ω), Φ∗(u, Ω′)⟩ +(3.4) += +� +α(Ω)θ(−u)(−u)i Ω + β(Ω)θ(u)ui Ω, α∗(Ω′)θ(−u)(−u)−i Ω′ + β∗(Ω′)θ(u)u−i Ω′� += α∗(Ω)α∗(Ω′) +� +θ(−u)(−u)i Ω, θ(−u)(−u)−i Ω′� ++ β∗(Ω)β∗(Ω′) +� +θ(u)ui Ω, θ(u)u−i Ω′� +. +By using (2.21) one has +� +θ(u)ui Ω, θ(u)u−i Ω′� += − +� +θ(−u)(−u)i Ω, θ(−u)(−u)−i Ω′� += −4πΩ δ(Ω + Ω′) . +(3.5) +Thus the inner product (3.4) then becomes +⟨Φ(u, Ω),Φ∗(u, Ω′)⟩ = 4πΩ δ(Ω + Ω′) +� +α∗(Ω)α∗(Ω′) − β∗(Ω)β∗(Ω′) +� +. +(3.6) +This should be zero; however, in contrast to the previous case of subsection (2.4), since Ω +and Ω′ can be any real numbers, the inner product is not automatically vanishing, but (3.6) +imposes a constraint on α(Ω) and β(Ω). Namely, +� +α∗(Ω)α∗(Ω′) − β∗(Ω)β∗(Ω′) +� +δ(Ω + Ω′) = +� +α∗(Ω)α∗(−Ω) − β∗(Ω)β∗(−Ω) +� +δ(Ω + Ω′) = 0 , +(3.7) +or simply, +α(Ω)α(−Ω) − β(Ω)β(−Ω) = 0 , +(3.8) +for all Ω. +Consequently, the inner product imposes two constraints. Let us first start with (3.3), +and introduce an ansatz +α(Ω) = +e +κπΩ +2 ++i θ +� +8πΩ sinh (κπΩ) +, +β(Ω) = +e− κπΩ +2 ++i φ +� +8πΩ sinh (κπΩ) +, +(3.9) +where κ is an arbitrary positive real number, and θ and φ are any real numbers. Note κ +– 9 – + +should be chosen as a positive real number so the term 8πΩ sinh (κπΩ) is always positive +for any real Ω. +Next, imposing the second constraint (3.8) implies +θ = φ . +(3.10) +Thus, ei θ is simply an overall phase which can be ignored. Therefore, we have +α(Ω) = +e +κπΩ +2 +� +8πΩ sinh (κπΩ) +, +β(Ω) = +e− κπΩ +2 +� +8πΩ sinh (κπΩ) +. +(3.11) +Consequently, one may write the following final result for the new mode: +Φ(u, Ω, κ) = +1 +� +8πΩ sinh (κπΩ) +� +θ(−u)(−u)i Ω e +κπΩ +2 ++ θ(u)ui Ω e− κπΩ +2 +� +. +(3.12) +Since it is classified by a positive number κ, we call it κ-mode. The field can be written as +Φ(u) = +� ∞ +−∞ +dΩ +� +Φ(u, Ω, κ)AΩ,κ + Φ∗(u, Ω, κ)A† +Ω,κ +� +, +(3.13) +where we have used AΩ,κ to denote the annihilation operator for a κ-mode with the fre- +quency Ω (Note Ω is any real number). Explicitly, using (3.12) one has +Φ(u) = θ(−u) +� ∞ +−∞ +dΩ +1 +� +8πΩ sinh (κπΩ) +� +(−u)i Ω e +κπΩ +2 +AΩ,κ + (−u)−i Ω e +κπΩ +2 +A† +Ω,κ +� ++θ(u) +� ∞ +−∞ +dΩ +1 +� +8πΩ sinh (κπΩ) +� +ui Ω e− κπΩ +2 +AΩ,κ + u−i Ω e− κπΩ +2 +A† +Ω,κ +� +. (3.14) +3.2 +Rindler and Unruh-Minkowski as special cases +In this section we find Rindler and Unruh-Minkowski modes as special cases of the κ-mode. +Rindler +Let κ → ∞ in (3.12). For positive Ω, the first term of (3.12) survives, indicating the Rindler +mode in the right wedge. Thus the κ-mode in this special case reads +Φ(u, κ → ∞) = +1 +√ +4πΩ +θ(−u) (−u)i Ω . +(3.15) +Similarly, considering κ → ∞ with negative Ω in (3.12), the second term of (3.12) +survives, indicating the Rindler mode in the left wedge. Namely, +Φ(u, κ → ∞) = +1 +√ +4πΩ +θ(u) u−i Ω , +(3.16) +where we changed Ω → −Ω, and hence Ω > 0 above. +– 10 – + +Unruh-Minkowski +It is simple to observe Unruh-Minkowski as a special case of the κ-mode. Set κ = 1 in +(3.12). One has +Φ(u, Ω, κ = 1) = +1 +� +8πΩ sinh (πΩ) +� +θ(−u)(−u)i Ω e +πΩ +2 + θ(u)ui Ω e− πΩ +2 +� +. +(3.17) +3.3 +Bogoliubov transformation between κ-modes +The goal is to find the Bogoliubov transformation between distinct κ-modes. Using (3.14), +one has +Φ(u) = θ(−u) +� ∞ +−∞ +dΩ +1 +� +8πΩ sinh (κπΩ) +� +(−u)i Ω e +κπΩ +2 +AΩ,κ + (−u)−i Ω e +κπΩ +2 +A† +Ω,κ +� ++θ(u) +� ∞ +−∞ +dΩ +1 +� +8πΩ sinh (κπΩ) +� +ui Ω e− κπΩ +2 +AΩ,κ + u−i Ω e− κπΩ +2 +A† +Ω,κ +� += θ(−u) +� ∞ +−∞ +dΩ +1 +� +8πΩ sinh (κ′πΩ) +� +(−u)i Ω e +κ′πΩ +2 +AΩ,κ′ + (−u)−i Ω e +κ′πΩ +2 +A† +Ω,κ′ +� ++θ(u) +� ∞ +−∞ +dΩ +1 +� +8πΩ sinh (κ′πΩ) +� +ui Ω e− κ′πΩ +2 +AΩ,κ′ + u−i Ω e− κ′πΩ +2 +A† +Ω,κ′ +� +.(3.18) +Next, we compare the factors of θ(−u)(−u)i Λ, and θ(u)ui Λ in (3.18). Comparing θ(−u)(−u)i Λ +factor indicates +1 +� +8πΛ sinh (κπΛ) +� +e +κπΛ +2 AΛ,κ + e +−κπΛ +2 +A† +−Λ,κ +� += +1 +� +8πΛ sinh (κ′πΛ) +� +e +κ′πΛ +2 +AΛ,κ′ + e +−κ′πΛ +2 +A† +−Λ,κ′ +� +. +(3.19) +One has to notice that since −∞ < Ω < ∞, then θ(−u)(−u)i Λ appears both in the first +and second term of the first line of (3.18). Also, comparing θ(u)ui Λ factors in (3.18) yields +1 +� +8πΛ sinh (κπΛ) +� +e− κπΛ +2 AΛ,κ + e +κπΛ +2 A† +−Λ,κ +� += +1 +� +8πΛ sinh (κ′πΛ) +� +e− κ′πΛ +2 +AΛ,κ′ + e +κ′πΛ +2 +A† +−Λ,κ′ +� +. +(3.20) +It is useful to find a transformation of the pair +� +AΛ,κ +A† +−Λ,κ +� +. From (3.19) and (3.20), one +has +� +AΛ,κ′ +A† +−Λ,κ′ +� += +sgn (Λ) +� +sinh (κπΛ) sinh (κ′πΛ) +� +sinh +� (κ+κ′)πΛ +2 +� +sinh +� (κ′−κ)πΛ +2 +� +sinh +� (κ′−κ)πΛ +2 +� +sinh +� (κ+κ′)πΛ +2 +� +� � +AΛ,κ +A† +−Λ,κ +� +. +(3.21) +The above relation is very crucial. It clearly shows, since the off-diagonal elements of +the above matrix are non-vanishing for κ ̸= κ′, that the annihilation operator in a mode κ′ +– 11 – + +depends upon both annihilation and creation operators of a mode κ, meaning that these +modes have different vacua. +One may check explicitly the transformation between Rindler and Unruh-Minkowski +operators. Namely, with κ = ∞, κ′ = 1, one has +AΛ = +1 +√ +1 − e−2πΛ +� +bRΛ − e−πΛ b† +LΛ +� +, +A−Λ = +1 +√ +1 − e−2πΛ +� +bLΛ − e−πΛ b† +RΛ +� +, +(3.22) +where Λ > 0. This is in agreement with eqs (2.18) and (2.20) of the Unruh-Wald paper [17]. +Note here we adopt the following convention for Unruh-Minkowski and Rindler operators +AΩ,1 = AΩ , +Ω ∈ R , +AΩ,∞ = bRΩ , +A−Ω,∞ = bLΩ , +Ω > 0 . +(3.23) +where AΩ, bRΩ, and bLΩ are the annihilation operators for Unruh-Minkowski, Rindler right +wedge, and Rindler left wedge respectively. +One can get Rindler in terms of Unruh-Minkowski by either finding the inverse of (3.22), +or by setting κ = 1, κ′ = ∞ in (3.21). It reads +bRΛ = +1 +√ +1 − e−2πΛ +� +AΛ + e−πΛ A† +−Λ +� +, +bLΛ = +1 +√ +1 − e−2πΛ +� +A−Λ + e−πΛ A† +Λ +� +, +(3.24) +again, recovering eq (2.24) of Unruh-Wald. +3.4 +Bogoliubov transformation between a κ-mode and the Minkowski plane +wave +The goal is to find the Bogoliubov transformation between a κ-mode and the Minkowski +plane wave . Using (3.14), one has +Φ(u) = θ(−u) +� ∞ +−∞ +dΩ +1 +� +8πΩ sinh (κπΩ) +� +(−u)i Ω e +κπΩ +2 +AΩ + (−u)−i Ω e +κπΩ +2 +A† +Ω +� ++θ(u) +� ∞ +−∞ +dΩ +1 +� +8πΩ sinh (κπΩ) +� +ui Ω e− κπΩ +2 +AΩ + u−i Ω e− κπΩ +2 +A† +Ω +� += +� ∞ +0 +dν +√ +4πν +� +aν e−i ν u + a† +ν ei ν u� +. +(3.25) +– 12 – + +Next, one may find aν by calculating +� ∞ +−∞ du ei ν uΦ(u). It reads +aν = +� +ν +π +� ∞ +−∞ +dΩ +i +� +8πΩ sinh (κπΩ) +(3.26) +� +− e +κπΩ +2 +e +πΩ +2 ν−(1+i Ω) Γ(1 + i Ω) AΩ + e +−κπΩ +2 +e− πΩ +2 ν−(1+i Ω) Γ(1 + i Ω) AΩ +−e +κπΩ +2 +e− πΩ +2 ν−(1−i Ω) Γ(1 − i Ω) A† +Ω + e +−κπΩ +2 +e +πΩ +2 ν−(1−i Ω) Γ(1 − i Ω) A† +Ω +� +, +where we have used the following useful integrals: +� +∞ +−∞ +du ei νu (−u)i Ωθ(−u) = +� ∞ +0 +du e−i νu ui Ω = −i ν−(1+i Ω)e +πΩ +2 Γ(1 + i Ω) , +� +∞ +−∞ +du ei νu ui Ωθ(u) = +� ∞ +0 +du ei νuui Ω = i ν−(1+i Ω)e− πΩ +2 Γ(1 + i Ω) . +(3.27) +Therefore, the final answer for the Bogoliubov transformation between κ-mode and plane +wave Minkowski reads +aν = +� +ν +π +� ∞ +−∞ +dΩ +i +� +2πΩ sinh (κπΩ) +� +− ν−(1+i Ω) Γ(1 + i Ω) sinh +�(κ + 1)πΩ +2 +� +AΩ +− ν−(1−i Ω) Γ(1 − i Ω) sinh +�(κ − 1)πΩ +2 +� +A† +Ω +� +. +(3.28) +It is clear from the above relation that for κ = 1 , i.e., the Unruh-Minkowski mode, the +pre-factor of the creation operator vanishes. It indicates the vacuum is the Minkowski one, +as we have expected from the Unruh-Minkowski mode. +3.5 +Commutation relations for different κ +Let’s find the commutation relation between different κ. To do so, one has to find the inner +product of the modes with different κ. Using (3.12) we find the inner product between two +positive norm modes as follows: +� +Φ(u, Ω, κ), Φ(u, Ω′, κ′) +� += +1 +� +8πΩ sinh (κπΩ) +1 +� +8πΩ′ sinh (κ′πΩ′) +(3.29) +� +θ(−u)(−u)i Ω e +κπΩ +2 ++ θ(u)ui Ω e− κπΩ +2 , θ(−u)(−u)i Ω′ e +κ′πΩ′ +2 ++ θ(u)ui Ω′ e− κ′πΩ′ +2 +� += +sgn (Λ) +� +sinh (κπΛ) sinh (κ′πΛ) +sinh +� (κ+κ′)πΛ +2 +� +δ(Ω − Ω′) , +– 13 – + +where we have used (2.21). Therefore the commutation relation between annihilation and +creation operators with different κ and κ′ is +� +AΩ,κ, A† +Ω′,κ′ +� += +� +Φ(u, Ω, κ), Φ(u, Ω′, κ′) +� += sgn (Λ) sinh +� (κ+κ′)πΛ +2 +� +� +sinh (κπΛ) sinh (κ′πΛ) +δ(Ω − Ω′) . +(3.30) +Note the case of κ = κ′ yields the standard relation +� +AΩ,κ, A† +Ω′,κ +� += δ(Ω − Ω′). +Next, the inner product of the positive and negative norm modes with different κ and +κ′ reads +� +Φ(u, Ω, κ), Φ∗(u, Ω′, κ′) +� += +1 +� +8πΩ sinh (κπΩ) +1 +� +8πΩ′ sinh (κ′πΩ′) +(3.31) +� +θ(−u)(−u)i Ω e +κπΩ +2 ++ θ(u)ui Ω e− κπΩ +2 , θ(−u)(−u)−i Ω′ e +κ′πΩ′ +2 ++ θ(u)u−i Ω′ e− κ′πΩ′ +2 +� += +sgn (Λ) +� +sinh (κπΛ) sinh (κ′πΛ) +sinh +� (κ−κ′)πΛ +2 +� +δ(Ω + Ω′) , +where again (2.21) has been used. The commutation relation between annihilation operators +with different κ and κ′ is thus +� +AΩ,κ, AΩ′,κ′� += − +� +Φ(u, Ω, κ), Φ∗(u, Ω′, κ′) +� += − sgn (Λ) sinh +� (κ−κ′)πΛ +2 +� +� +sinh (κπΛ) sinh (κ′πΛ) +δ(Ω + Ω′) . +(3.32) +Again the case of κ = κ′ yields +� +AΩ,κ, AΩ′,κ +� += 0, as we expected. +It is interesting to note the Bogoliubov relation (3.21) can be found using the above +commutation relation results. Providing that +� +AΛ,κ′ +A† +−Λ,κ′ +� +can be written in terms of a matrix +multiplying +� +AΛ,κ +A† +−Λ,κ +� +, the matrix elements can be found using the relations (3.30) and +(3.32). +One then consequently obtains (3.21) by employing the standard commutation +relations +� +AΩ,κ, A† +Ω′,κ +� += δ(Ω − Ω′), and +� +AΩ,κ, AΩ′,κ +� += 0. +4 +Combining same sign norm Rindler modes +The same sign norm ansatz can be written as follows: +Φ(u, Ω) = α(Ω)θ(−u)(−u)i Ω + β(Ω)θ(u)u−i Ω . +(4.1) +Note for a positive Ω both terms above have positive norm, while for a negative Ω, both of +them have negative one. +The new mode should satisfy the following inner products: +� +Φ(u, Ω), Φ(u, Ω′) +� += δ(Ω − Ω′) , +� +Φ(u, Ω), Φ∗(u, Ω′) +� += 0 . +(4.2) +– 14 – + +Hence, one may check +⟨Φ(u, Ω),Φ(u, Ω′)⟩ += +� +α(Ω)θ(−u)(−u)i Ω + β(Ω)θ(u)u−i Ω, α(Ω′)θ(−u)(−u)i Ω′ + β(Ω′)θ(u)u−i Ω′� += α∗(Ω)α(Ω′) +� +θ(−u)(−u)i Ω, θ(−u)(−u)i Ω′� ++ β∗(Ω)β(Ω′) +� +θ(u)u−i Ω, θ(u)u−i Ω′� += 4πΩ +� +|α(Ω)|2 + |β(Ω)|2� +δ(Ω − Ω′) , +(4.3) +where we have used (2.21). The value of the above inner product, providing Φ(u, Ω) is a +positive norm mode, should be δ(Ω − Ω′), and hence +4πΩ +� +|α(Ω)|2 + |β(Ω)|2� += 1 . +(4.4) +This implies Ω should be positive. +Also, the inner product of the mode and its conjugate reads +⟨Φ(u, Ω), Φ∗(u, Ω′)⟩ +(4.5) += +� +α(Ω)θ(−u)(−u)i Ω + β(Ω)θ(u)u−i Ω, α∗(Ω′)θ(−u)(−u)−i Ω′ + β∗(Ω′)θ(u)ui Ω′� += α∗(Ω)α∗(Ω′) +� +θ(−u)(−u)i Ω, θ(−u)(−u)−i Ω′� ++ β∗(Ω)β∗(Ω′) +� +θ(u)u−i Ω, θ(u)ui Ω′� +, +where by appropriate change of sign of Ω and Ω′ in (2.21) one has +� +θ(u)u−i Ω, θ(u)ui Ω′� += +� +θ(−u)(−u)i Ω, θ(−u)(−u)−i Ω′� += 4πΩ δ(Ω + Ω′) . +(4.6) +Thus, the inner product (4.5) becomes +⟨Φ(u, Ω), Φ∗(u, Ω′)⟩ = +� +α∗(Ω)α∗(Ω′) + β∗(Ω)β∗(Ω′) +� +4πΩ δ(Ω + Ω′) . +(4.7) +The above term is zero automatically, since Ω and Ω′ are both positive. Therefore, the inner +product of positive and negative norm modes asserts no restriction on coefficients α(Ω) and +β(Ω). Thus, (4.4) is the only constraint on α(Ω) and β(Ω). One may solve this constraint +as follows: +α(Ω) = +e +κπΩ +2 ++ i γ +2 +� +8πΩ cosh (κπΩ) +, +β(Ω) = +e− κπΩ +2 +− i γ +2 +� +8πΩ cosh (κπΩ) +. +(4.8) +Therefore, one may write the following final result for the mode: +Φ(u, Ω, κ, γ) = +1 +� +8πΩ cosh (κπΩ) +� +θ(−u)(−u)i Ω e +κπΩ +2 ++ i γ +2 + θ(u)u−i Ω e +−κπΩ +2 +− i γ +2 +� +. (4.9) +– 15 – + +The field can be written as +Φ(u) = +� ∞ +0 +dΩ +� +Φ(u, Ω, κ, γ)AΩ,κ,γ + Φ∗(u, Ω, κ, γ)A† +Ω,κ,γ +� +, +(4.10) +where we have used AΩ,κ,γ to denote the annihilation operator for the κ-mode. It is more +convenient to drop (κ, γ). Explicitly, using (3.12), one has +Φ(u) = θ(−u) +� ∞ +0 +dΩ +1 +� +8πΩ cosh (κπΩ) +� +(−u)i Ω e +κπΩ +2 +e +i γ +2 AΩ,κ,γ + (−u)−i Ω e +κπΩ +2 +e− i γ +2 A† +Ω,κ,γ +� ++ θ(u) +� ∞ +0 +dΩ +1 +� +8πΩ cosh (κπΩ) +� +u−i Ω e− κπΩ +2 +e− i γ +2 AΩ,κ,γ + ui Ω e− κπΩ +2 +e +i γ +2 A† +Ω,κ,γ +� +. +(4.11) +One may wonder how to modify the mode, in order to include negative Ω as well. +One possibility is mixing the positive and negative norms together in the ansatz instead of +adding the same sign norm modes as we have done in (4.1). +4.1 +Bogoliubov transformation between different modes +The goal is to find the Bogoliubov transformation between κ-modes with different κ and γ. +using (4.11), one has +Φ(u) = θ(−u) +� ∞ +0 +dΩ +1 +� +8πΩ cosh (κπΩ) +� +(−u)i Ω e +κπΩ +2 +e +i γ +2 AΩ + (−u)−i Ω e +κπΩ +2 +e− i γ +2 A† +Ω +� ++θ(u) +� ∞ +0 +dΩ +1 +� +8πΩ cosh (κπΩ) +� +u−i Ω e− κπΩ +2 +e− i γ +2 AΩ + ui Ω e− κπΩ +2 +e +i γ +2 A† +Ω +� +(4.12) += θ(−u) +� ∞ +0 +dΩ +1 +� +8πΩ cosh (κ′πΩ) +� +(−u)i Ω e +κ′πΩ +2 +e +i γ′ +2 A′Ω + (−u)−i Ω e +κ′πΩ +2 +e− i γ′ +2 A′† +Ω +� ++θ(u) +� ∞ +0 +dΩ +1 +� +8πΩ cosh (κ′πΩ) +� +u−i Ω e− κ′πΩ +2 +e− i γ′ +2 A′Ω + ui Ω e− κ′πΩ +2 +e +i γ′ +2 A′† +Ω +� +, +where AΩ ≡ AΩ,κ,γ, and A′Ω ≡ AΩ,κ′,γ′. Comparing θ(−u)(−u)i Λ and θ(u)ui Λ in (4.12) +indicates +1 +� +8πΛ cosh (κπΛ) +e +κπΛ +2 e +i γ +2 AΛ = +1 +� +8πΛ cosh (κ′πΛ) +e +κ′πΛ +2 +e +i γ′ +2 A′Λ , +1 +� +8πΛ cosh (κπΛ) +e− κπΛ +2 e− i γ +2 AΛ = +1 +� +8πΛ cosh (κ′πΛ) +e− κ′πΛ +2 +e− i γ′ +2 A′Λ . +(4.13) +The above expressions simply yield κ = κ′ and γ = γ′. Consequently if there exists a mode +in the same sign norm, then κ and γ would be unique. +– 16 – + +4.2 +Bogoliubov transformation between the same sign norm and the opposite +sign norm modes +So far we have found if there were any mode in the same sign norm scenario, it would be +just a unique (κ, γ) mode. Here in this subsection, we find the Bogoliubov transformation +between the latter mode and the previous case of the opposite sign norm mode. Using +(3.14) and (4.11) one has +Φ(u) = θ(−u) +� ∞ +0 +dΩ +1 +� +8πΩ cosh (κπΩ) +� +(−u)i Ω e +κπΩ +2 +e +i γ +2 AΩ,κ,γ + (−u)−i Ω e +κπΩ +2 +e− i γ +2 A† +Ω,κ,γ +� ++θ(u) +� ∞ +0 +dΩ +1 +� +8πΩ cosh (κπΩ) +� +u−i Ω e− κπΩ +2 +e− i γ +2 AΩ,κ,γ + ui Ω e− κπΩ +2 +e +i γ +2 A† +Ω,κ,γ +� += θ(−u) +� ∞ +−∞ +dΩ +1 +� +8πΩ sinh (κ′πΩ) +� +(−u)i Ω e +κ′πΩ +2 +AΩ,κ′ + (−u)−i Ω e +κ′πΩ +2 +A† +Ω,κ′ +� ++θ(u) +� ∞ +−∞ +dΩ +1 +� +8πΩ sinh (κ′πΩ) +� +ui Ω e− κ′πΩ +2 +AΩ,κ′ + u−i Ω e− κ′πΩ +2 +A† +Ω,κ′ +� +. +(4.14) +Now, comparing θ(u)ui Λ for Λ > 0 in the above relation yields +e +−κπΛ +2 ++ i γ +2 +� +8πΛ cosh (κπΛ) +A† +Λ,κ,γ = +1 +� +8πΛ sinh (κ′πΛ) +� +e− κ′πΛ +2 +AΛ,κ′ + e +κ′πΛ +2 +A† +−Λ,κ′ +� +. +(4.15) +Also comparing θ(−u)(−u)i Λ for Λ > 0 in (4.14) shows +e +κπΛ +2 + i γ +2 +� +8πΛ cosh (κπΛ) +AΛ,κ,γ = +1 +� +8πΛ sinh (κ′πΛ) +� +e +κ′πΛ +2 +AΛ,κ′ + e +−κ′πΛ +2 +A† +−Λ,κ′ +� +. +(4.16) +Now it is clear while the left-hand sides of (4.16) and Hermitian conjugate of (4.15) are +proportional, the right-hand sides are not. Therefore, one concludes the same sign norm +mode cannot exist even for a unique value of κ and γ. +5 +Generalized non-thermofield double states +Unruh [3] and Israel [21], both in 1976, found the thermofield double state. The easiest +way to obtain this mode is using the Bogoliubov transformation between Rindler and plane +wave Minkowski. Namely, +� +bLω − e− πω +a b† +Rω +� +|0M⟩ = 0 , +� +bRω − e− πω +a b† +Lω +� +|0M⟩ = 0 . +(5.1) +where bRω and bLω denoting the Rindler annihilation operators for the right and left wedges +with a frequency ω respectively. Also a represents the constant acceleration of a particle. +The Minkowski vacuum is denoted by |0M⟩. +Then the Minkowski vacuum can be written in terms of entangled Rindler right-left +– 17 – + +wedges state as follows +|0M⟩ = +1 +√ +Z +exp +�� ∞ +0 +dω e− βω +2 b† +Rωb† +Lω +� +|0R⟩ ⊗ |0L⟩ , +(5.2) +where Z is the partition function, β = 1 +T = 2πckB +ℏa +is an inverse Unruh temperature, and +|0R⟩ and |0L⟩ are the Rindler vacuum in the right and left wedges respectively. +5.1 +Relating different κ-vacua +To find a generalized non-thermofield double state, first note AΩ,κ′ |0κ′⟩ = 0. Then by +exploiting the Bogoliubov transformation (3.21), one may observe +� +AΩ,κ − ηA† +−Ω,κ +� +|0κ′⟩ = 0 , +(5.3) +where +ηκ,κ′,Ω = sinh +� 1 +2(κ − κ′)πΩ +� +sinh +� 1 +2(κ + κ′)πΩ +� . +(5.4) +Following the same argument as in the case of Rindler-Minkowski, we have the relation +between κ and κ′ vacua as follows: +|0κ′⟩ = +1 +√Zκκ′ exp +�� ∞ +0 +dΩ ηκ,κ′,Ω A† +Ω,κ A† +−Ω,κ +� +|0κ⟩ , +(5.5) +where Zκκ′ is the normalization factor which depends upon κ and κ′. Note in the above +relation although Ω can be both positive and negative, but we have to integrate just the pos- +itive frequency. Otherwise, We may include all positive and negative frequencies, however, +a factor of half should be included in the integral, since ηκ,κ′,Ω = ηκ,κ′,−Ω. +5.2 +κ-vacuum in terms of Rindler +It is useful to find a special case of (5.5), where κ′-vacuum is written in terms of κ-vacuum, +with κ → ∞, i.e., Rindler vacuum. +One can easily obtain from (5.5), (5.4), and the +convention in (3.23), the following relation +|0κ⟩ = +1 +√Zκ +exp +�� ∞ +0 +dω e− κβω +2 b† +Rωb† +Lω +� +|0R⟩ ⊗ |0L⟩ . +(5.6) +The above expression is a generalization of the thermofield double state to a non- +thermofield double state. Obviously κ ̸= 1 ruins the thermality of the state, that’s why we +call it non-thermofield double state. +For retrieving the familiar case of the Minkowski vacuum in terms of the Rindler one, +i.e., thermofield double state, we should consider a κ-vacuum as the Minkowski vacuum, +namely κ = 1. The thermofield double state appears trivially by setting κ = 1 in (5.6). +5.3 +κ-vacuum in terms of Minkowski +The last special case is writing a general κ-vacuum in terms of the Minkowski vacuum. +This can be observed readily by setting κ = 1 in (5.5) and using the convention in (3.23). +– 18 – + +Namely, +|0κ⟩ = +1 +√Zκ +exp +�� ∞ +0 +dΩ ηκ,Ω A† +Ω A† +−Ω +� +|0M⟩ , +(5.7) +where +ηκ,Ω = −sinh +� 1 +2(κ − 1)πΩ +� +sinh +� 1 +2(κ + 1)πΩ +� , +(5.8) +is a special case of (5.4) for κ = 1 and relabeling κ′ as κ. +6 +Conclusion +In this paper, we discuss the importance of the Klein-Gordon inner product with respect to +appropriately defining a mode associated with annihilation and creation operators. Essen- +tially (2.6) relates the inner product of modes to the commutator relations, hence, to have +the standard commutation relations, the inner product between modes should be as stated +in (2.6). This requirement imposes a strong constraint on the form of modes. Two famous +modes, plane wave Minkowski and Rindler, were worked out as an example. +Furthermore, the inner product can be used to find new modes from a given mode. +Specifically, we worked this procedure out for Rindler-like modes. In other words, a new +set of modes can be obtained by a combination of two Rindler modes in the right and left +wedges. One can perform this in two distinct ways where the modes have the same and the +opposite sign norm. Interestingly, while the latter scenario yields an infinite set of modes +parameterized by a real positive parameter κ, the former case yields no valid mode. This +new κ-mode, inspired by the work of Unruh [3], yields the Unruh-Minkowski and Rindler +modes for special cases of κ = 1 and κ → ∞ respectively. +Moreover, the well-known thermofield double state, relating the Minkowski to the +Rindler vacuum, is generalized to include the κ-vacua. +Namely, a κ-vacuum is written +in terms of another, say κ′-vacuum. The relation is similar to that of the thermofield dou- +ble, with a modified coefficient in the exponential, as well as using the κ-mode annihilation +and creation operators instead of the Rindler ones. Of course, the generalized expression +for the thermofield double state reduces to the usual one if one considers (κ, κ′) = (∞, 1) +in (5.5). Two special cases of this generalization, i.e., κ-vacuum in terms of Rindler and +κ-vacuum in terms of Minkowski are outlined. The former case (5.6) resembles the famous +thermofield double state, however, an important difference is that it is no longer a thermal +state for a general κ ̸= 1. +The thermofield double state has been an indispensable part of AdS/CFT program +since Maldacena’s proposal of the equivalence of the eternal AdS black hole in the bulk and +the thermofield double state in the boundary. Now the generalized non-thermofield double +state may shed more light on the bulk physics, since, there is another degree of freedom κ. +Acknowledgments +AA is grateful to Girish Agarwal, Jonathan Ben-Benjamin, David Lee, Yusef Malek, Ana- +toly Svidzinsky, and especially Marlan Scully and Bill Unruh for illuminating discussions. +– 19 – + +AA thanks Reed Nessler for his careful reading of the manuscript and providing helpful +comments. +This work was supported by the Robert A. Welch Foundation (Grant No. A-1261), +and the National Science Foundation (Grant No. PHY-2013771). +A +Proof of the lemma (2.12) +To prove the lemma (2.12), one may start from the definition of the inner product as follows +� +⟨f(u, Ω), Φ(u)⟩, +� +g +� +u′, Ω′� +, Φ +� +u′��� +(A.1) += − +� +∞ +−∞ +du +� +∞ +−∞ +du′ +� +f∗(u, Ω)g∗(u′, Ω′) [∂uΦ(u), ∂u′Φ(u′)] +− f∗(u, Ω)∂u′g∗(u′, Ω′) [∂uΦ(u), Φ(u′)] +− ∂uf∗(u, Ω)g∗(u′, Ω′)[Φ(u), ∂u′Φ(u′)] ++ ∂uf∗(u, Ω)∂u′g∗(u′, Ω′)[Φ(u), Φ(u′)] +� +. +There are four different commutation relations involved above where they are expressed +as follows +[Φ(u), Φ(u′)] = i +4 sgn(u′ − u) , +[∂uΦ(u), Φ(u′)] = − i +2δ(u′ − u) , +[Φ(u), ∂u′Φ(u′)] = i +2δ(u′ − u) , +[∂uΦ(u), ∂u′Φ(u′)] = i +2δ′(u′ − u) . +(A.2) +Next, one has to use the following properties of Dirac delta function +� +∞ +−∞ +dx f(x)δ(x) = f(0) , +(A.3) +� +∞ +−∞ +dx f(x)δ′(x) = +� +∞ +−∞ +dx +� d +dx +� +f(x)δ(x) +� +− df +dxδ(x) +� += − +� +∞ +−∞ +dxf′(x)δ(x) . +Plugging (A.2) and (A.3) in (A.1), we have +� +⟨f(u, Ω), Φ(u)⟩, +� +g +� +u′, Ω′� +, Φ +� +u′��� += +− i +2 +� +∞ +−∞ +du f∗(u, Ω)∂ug∗ � +u, Ω′� +− i +2 +� +∞ +−∞ +du +� +f∗(u, Ω)∂ug∗ � +u, Ω′� +− ∂uf∗(u, Ω)g∗ � +u, Ω′� � +− i +4 +� +∞ +−∞ +dudu′ ∂uf∗(u, Ω)∂u′g∗(u′, Ω′) sgn(u′ − u) . +(A.4) +– 20 – + +To evaluate the last term above, one may proceed as follows +� +∞ +−∞ +dudu′ ∂uf∗(u, Ω)∂u′g∗(u′, Ω′) sgn(u′ − u) += +� +∞ +−∞ +du +� +− +� u +−∞ +du′∂uf∗ (u, Ω) ∂u′g∗ � +u′, Ω′� ++ +� ∞ +u +du′∂uf∗ (u, Ω) ∂u′g∗ � +u′, Ω′� +� += +� +∞ +−∞ +du +� +− ∂uf∗ (u, Ω) +� +g∗ � +u, Ω′� +− g∗ � +−∞, Ω′� � ++ ∂uf∗ (u, Ω) +� +g∗ � +∞, Ω′� +− g∗ � +u, Ω′� �� += −2 +� +∞ +−∞ +du ∂uf∗ (u, Ω) g∗ � +u, Ω′� +, +(A.5) +where we have assumed the field is zero at infinities. Using the above relation, the commu- +tation relation (A.4) can be written finally as +� +⟨f(u, Ω), Φ(u)⟩, +� +g +� +u′, Ω′� +, Φ +� +u′��� += −i +� +∞ +−∞ +du +� +f∗(u, Ω)∂ug∗ � +u, Ω′� +− ∂uf∗(u, Ω)g∗ � +u, Ω′� � += − +� +f(u, Ω), g∗(u, Ω′) +� +. +(A.6) +This completes the proof. +References +[1] E. Witten, Why does quantum field theory in curved spacetime make sense? and what +happens to the algebra of observables in the thermodynamic limit?, in Dialogues Between +Physics and Mathematics, pp. 241–284. Springer, 2022. +[2] S. W. Hawking, Particle Creation by Black Holes, Commun. Math. Phys. 43 (1975) 199–220. +[Erratum: Commun.Math.Phys. 46, 206 (1976)]. +[3] W. G. Unruh, Notes on black hole evaporation, Phys. Rev. D 14 (1976) 870. +[4] A. Iorio and G. Lambiase, The Hawking-Unruh phenomenon on graphene, Phys. Lett. B 716 +(2012) 334–337, [arXiv:1108.2340]. +[5] A. Bhardwaj and D. E. Sheehy, Unruh Effect and Takagi’s Statistics Inversion in Strained +Graphene, arXiv:2209.08053. +[6] M. O. Scully, V. V. Kocharovsky, A. Belyanin, E. Fry, and F. Capasso, Enhancing +Acceleration Radiation from Ground-State Atoms via Cavity Quantum Electrodynamics, +Phys. Rev. Lett. 91 (2003) 243004, [quant-ph/0305178]. +[7] A. Belyanin, V. V. Kocharovsky, F. Capasso, E. Fry, M. S. Zubairy, and M. O. Scully, +Quantum electrodynamics of accelerated atoms in free space and in cavities, Phys. Rev. A 74 +(Aug, 2006) 023807. +[8] D. E. Bruschi, J. Louko, E. Martín-Martínez, A. Dragan, and I. Fuentes, Unruh effect in +– 21 – + +quantum information beyond the single-mode approximation, Physical Review A 82 (2010), +no. 4 042332. +[9] K. Bradler, P. Hayden, and P. Panangaden, Private information via the Unruh effect, JHEP +08 (2009) 074, [arXiv:0807.4536]. +[10] S. W. Hawking, Breakdown of Predictability in Gravitational Collapse, Phys. Rev. D 14 +(1976) 2460–2473. +[11] G. Penington, Entanglement Wedge Reconstruction and the Information Paradox, JHEP 09 +(2020) 002, [arXiv:1905.08255]. +[12] A. Almheiri, N. Engelhardt, D. Marolf, and H. Maxfield, The entropy of bulk quantum fields +and the entanglement wedge of an evaporating black hole, JHEP 12 (2019) 063, +[arXiv:1905.08762]. +[13] G. Penington, S. H. Shenker, D. Stanford, and Z. Yang, Replica wormholes and the black hole +interior, JHEP 03 (2022) 205, [arXiv:1911.11977]. +[14] A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian, and A. Tajdini, Replica Wormholes +and the Entropy of Hawking Radiation, JHEP 05 (2020) 013, [arXiv:1911.12333]. +[15] S. A. Fulling, Nonuniqueness of canonical field quantization in Riemannian space-time, Phys. +Rev. D 7 (1973) 2850–2862. +[16] S. W. Hawking and W. Israel, General Relativity: An Einstein Centenary Survey. Univ. Pr., +Cambridge, UK, 1979. +[17] W. G. Unruh and R. M. Wald, What happens when an accelerating observer detects a rindler +particle, Phys. Rev. D 29 (Mar, 1984) 1047–1056. +[18] A. Svidzinsky, A. Azizi, J. S. Ben-Benjamin, M. O. Scully, and W. Unruh, Unruh and +cherenkov radiation from a negative frequency perspective, Phys. Rev. Lett. 126 (Feb, 2021) +063603. +[19] A. Svidzinsky, A. Azizi, J. S. Ben-Benjamin, M. O. Scully, and W. Unruh, Causality in +quantum optics and entanglement of minkowski vacuum, Phys. Rev. Res. 3 (Mar, 2021) +013202. +[20] A. Azizi, Kappa vacua: Infinite number of new vacua in two-dimensional quantum field +theory, arXiv:2212.03781. +[21] W. Israel, Thermo field dynamics of black holes, Phys. Lett. A 57 (1976) 107–110. +[22] J. M. Maldacena, The Large N limit of superconformal field theories and supergravity, Adv. +Theor. Math. Phys. 2 (1998) 231–252, [hep-th/9711200]. +[23] E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253–291, +[hep-th/9802150]. +[24] S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, Gauge theory correlators from noncritical +string theory, Phys. Lett. B 428 (1998) 105–114, [hep-th/9802109]. +[25] J. M. Maldacena, Eternal black holes in anti-de Sitter, JHEP 04 (2003) 021, +[hep-th/0106112]. +[26] M. O. Scully and M. S. Zubairy, Quantum Optics. Cambridge University Press, 1997. +[27] B. S. DeWitt, Quantum Field Theory in Curved Space-Time, Phys. Rept. 19 (1975) 295–357. +[28] R. M. Wald, On Particle Creation by Black Holes, Commun. Math. Phys. 45 (1975) 9–34. +– 22 – + +[29] W. Rindler, Kruskal Space and the Uniformly Accelerated Frame, Am. J. Phys. 34 (1966) +1174. +– 23 – + diff --git a/O9FRT4oBgHgl3EQf5TiT/content/tmp_files/load_file.txt b/O9FRT4oBgHgl3EQf5TiT/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b1ee64bdd269933daa55a8e2dbbf79778513c9f4 --- /dev/null +++ b/O9FRT4oBgHgl3EQf5TiT/content/tmp_files/load_file.txt @@ -0,0 +1,676 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf,len=675 +page_content='Kappa vacua: A generalization of the thermofield double state Arash Azizia aThe Institute for Quantum Science and Engineering, Texas A&M University, College Station, TX 77843, USA E-mail: sazizi@tamu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='edu Abstract: We elaborate more on κ-mode, a mode that was found by a combination of the opposite sign norm Rindler modes in the right and left Rindler wedges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Especially, we show how the thermofield double state can be extended to a generalized non-thermofield double state by considering a relation between κ-vacua, similar to the Minkowski-Rindler vacua relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' A general κ ̸= 1 vacuum, in contrast to the well-known case of the Minkowski vacuum, is no longer thermal when reduced to a specific Rindler wedge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='13672v1 [hep-th] 31 Jan 2023 Contents 1 Introduction 1 2 The Klein-Gordon inner product 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='1 The definition and properties of the inner product 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='2 The inner product and commutation relations 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='3 Some examples: Minkowski plane wave and Rindler 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='4 Combining positive and negative norm modes 7 3 Kappa mode: combining opposite sign norm Rindler modes 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='1 Constructing Kappa modes 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='2 Rindler and Unruh-Minkowski as special cases 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='3 Bogoliubov transformation between κ-modes 11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='4 Bogoliubov transformation between a κ-mode and the Minkowski plane wave 12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='5 Commutation relations for different κ 13 4 Combining same sign norm Rindler modes 14 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='1 Bogoliubov transformation between different modes 16 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='2 Bogoliubov transformation between the same sign norm and the opposite sign norm modes 17 5 Generalized non-thermofield double states 17 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='1 Relating different κ-vacua 18 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='2 κ-vacuum in terms of Rindler 18 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='3 κ-vacuum in terms of Minkowski 18 6 Conclusion 19 A Proof of the lemma (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='12) 20 1 Introduction Quantum field theory (QFT) in curved spacetime1 gives us intriguing results such as Hawk- ing radiation [2] and the Unruh effect [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' These phenomena have not only far-reaching consequences, where any consistent theory of quantum gravity should reproduce them at appropriate limits, but also appear in diverse research fields such as theoretical condensed matter [4, 5], quantum optics [6, 7], and quantum information [8, 9], in addition to their 1For a recent review on mathematical background see [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' – 1 – original research program, high energy theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' The latter is especially reflected in the re- cent developments on resolving the Hawking information paradox [10] which have been conducted by two groups in [11–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' The quintessence of Hawking radiation and the Unruh effect is the fact that generally there is not a unique vacuum in QFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Fulling [15] was one of the first scholars who realized this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' The canonical way of describing QFT is as follows: Consider a field theory of desired spin in a given manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' The curved spacetime is treated classically (no back reaction from quantum fields on the background).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' The so- lutions of the equation of motion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=', field modes, can be derived from the Lagrangian of the theory in the curved background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' The positive norm modes, with respect to the appropriate inner product, can be computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' The negative norm modes, hence, shall be the complex conjugate of the positive norm ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' The quantum field can be expanded in terms of the modes, where the annihilation and creation operators are associated with the positive and negative norm modes respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' The set of positive norm modes is not unique, and hence, there are different sets of annihilation and creation operators, which can be related to each other by Bogoliubov transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Moreover, an annihilation operator of one set can be a combination of annihilation and creation operators of the other set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' The vacuum in the theory is defined as a state which is annihilated by all annihilation operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' In general vacua of two sets of modes are distinct, unless the annihila- tion operators of one set can be written in terms of just annihilation operators (not annihilation and creation) of another one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Unruh [3] introduced a thought-provoking field mode, later named Unruh-Minkowski mode, in order to investigate the relation between the usual Minkowski plane wave and Rindler modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' The mode’s vacuum is Minkowski, but its form is similar to the Rindler mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Furthermore, Unruh studied the behavior of a particle detector undergoing a uniform acceleration in the Minkowski vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' The particle detector first was considered as a box with discrete energy levels by Unruh and subsequently was simplified more by DeWitt [16] by considering the first two energy levels of the box, hence is named Unruh-DeWitt detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' The detector, starting from the ground state, gets excited and emits a photon in an Unruh-Minkowski mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' The bizarre feature of the mode is it resides mostly in the opposite wedge of where the detector is as emphasized by Unruh and Wald [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' More recently, two Unruh-DeWitt detectors were envisaged in [18, 19] and it was shown despite the apparent violation of causality, the latter is upheld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' In contrary to the more conventional QFT in Minkowski spacetime, where the plane wave is the mode one usually conceives, in fact an infinite number of different field modes exist, and the inner product should be exploited in order to associate the annihilation and – 2 – creation operators to these modes appropriately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Recently, we [20] introduced a general mode, named κ-mode, which is a combination of opposite sign norm Rindler modes in the right and left Rindler wedges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' The fascinating feature of κ-modes is their distinct vacua, which are parameterized by a real positive parameter κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Two well-known vacua, Minkowski and Rindler, are special cases of κ-vacuum for κ = 1 and κ → ∞ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' The thermofield double state [3, 21] is ubiquitous in theoretical physics, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=', it is a key ingredient of the AdS/CFT dictionary [22–24], as depicted in Maldacena’s proposal [25] about the duality of an eternal AdS black hole in the bulk and two copies of boundary CFT in the thermofield double state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Moreover, thermofield double state appears in quantum optics under the name of squeezed state [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' We generalize the usual thermofield double state to a relation between κ-vacuum and κ′-vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' As a matter of fact, we find a non- thermofield double state by relating a general κ-vacuum to the Rindler vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' The famous thermofield double state is then a special case (κ = 1) of this generalized non-thermofield double state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' In section 2 we study the Klein-Gordon inner product, and we show why the positive norm mode is associated with the annihilation operator by relating the inner product between modes to the commutation relations between operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' In section 3 we study the κ-mode and elaborate more on what was reported in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' In section 4 we show that it is not possible to get a mode by combining the same sign norm Rindler modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' In section 5 we find the generalization of thermofield double state to generalized non-thermofield double state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Finally we conclude in the last section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' 2 The Klein-Gordon inner product The Klein-Gordon inner product is the main tool to distinguish positive and negative norm modes, associated with annihilation and creation operators respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' The inner product of two modes is constant in time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' this is a great feature of the inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' In this section, we show the interconnection between the inner product and the commutation relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Furthermore, we explicitly find the Minkowski plane wave and Rindler modes utilizing the inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Finally we demonstrate how to generate a new positive norm mode by combining a given positive norm mode and its complex conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='1 The definition and properties of the inner product According to standard quantum field theory, a quantum field may be written as an infinite superposition of solutions of the equation of motions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=', modes, with operator coefficients which turn out to be annihilation and creation operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' The question then arises how to distinguish between modes associate to these operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' In other words, one may write a field as Φ(x) = � dΩ � Φ(x, Ω) aΩ + Φ∗(x, Ω) a† Ω � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='1) – 3 – To interpret aΩ and a† Ω as annihilation and creation operators respectively, they have to satisfy the standard commutation relation [aΩ, a† Ω′] = δ(Ω − Ω′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='2) Henceforth, it is crucial to choose a correct mode Φ(x, Ω) associated with the annihila- tion operator, while its complex conjugate Φ∗(x, Ω) is associated with the creation one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' The key concept for distinguishing these two modes is the inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' For the Klein-Gordon scalar field the inner product reads ⟨φ1, φ2⟩ = −i � Σ √−g dΣµ (φ∗ 1 ∂µφ2 − ∂µφ∗ 1 φ2) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='3) where Σ is the appropriate Cauchy hypersurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Note, we have used the convention of (−, +, · · · , +) for the Minkowski metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' If one were to choose mostly negative signa- ture for the metric, then there would be an overall minus sign difference, namely ⟨f, g⟩ = i � Σ √−g dΣµ (f∗ ∂µg − ∂µf∗ g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' It is worthwhile here to note about different conventions on the Klein-Gordon inner product among the early investigators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' We consider four of them in the following: Hawking [2]: ⟨φ1, φ2⟩H = i 2 � Σ [φ1∂µφ∗ 2 − (∂µφ1) φ∗ 2] dSµ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' DeWitt [27]: ⟨φ1, φ2⟩DeW = −i � Σ [φ∗ 1∂µφ2 − (∂µφ∗ 1) φ2] dSµ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Wald [28]: ⟨φ1, φ2⟩Wald = i � Σ [φ∗ 1∂µφ2 − (∂µφ∗ 1) φ2] dSµ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Unruh and Wald [17]: ⟨φ1, φ2⟩UW = i 2 � Σ [φ∗ 1∂µφ2 − (∂µφ∗ 1) φ2] dSµ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' The inner product is anti-linear in the second argument only in Hawking’s notation and is linear in the rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' While all of the above authors used the mostly positive signature for the metric, the sign of the inner product is not consistent among them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Of course, there is a factor of one half discrepancy too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Some useful relations in Klein-Gordon inner product are as follows: ⟨f, αg + βh⟩ = α ⟨f, g⟩ + β ⟨f, h⟩ , ⟨f, g⟩∗ = ⟨g, f⟩ , ⟨f∗, g∗⟩ = − ⟨f, g⟩∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='4) Note, the above Klein-Gordon “inner product” is not actually an inner product, strictly speaking, since ⟨f∗, f∗⟩ = − ⟨f, f⟩ , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='5) and therefore the positivity of inner product has been violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' However, this property is very crucial in distinguishing between positive and negative norm modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' One may associate the annihilation operator to a positive norm mode, while the mode’s complex conjugate, with a negative norm, is associated to the creation one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Actually, the inner product is defined so as to satisfy the following properties: � Φ(x, Ω), Φ(x, Ω′) � = [aΩ, a† Ω′] = δ(Ω − Ω′) , � Φ(u, Ω), Φ∗(u, Ω′) � = − [aΩ, aΩ′] = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='6) – 4 – We prove the above relations in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Also, using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='4), then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='6) implies � Φ∗(x, Ω), Φ∗(x, Ω′) � = −δ(Ω − Ω′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='7) In a D + 1 dimensional Minkowski spacetime with the usual convention for the com- ponent, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=', 0 and i represent time and space components, and for a constant time Cauchy surface, one has n0 = 1, ni = 0, n0 = −1, and ni = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Here nµ represents the unit normal vector to the manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='3) indicates ⟨f, g⟩ = i � dDx (f∗ ∂tg − ∂tf∗ g) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='8) Here we have dΣµ = δµ0 n0dDx = −dDx for µ = 0 and zero for the rest of indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Note mostly minus sign convention for the metric yields n0 = 1, and hence to keep (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='8), one should start off from ⟨f, g⟩ = i � Σ √−g dΣµ (f∗ ∂µg − ∂µf∗ g) as we have emphasized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' In 1 + 1 Minkowski spacetime, the metric is ds2 = −dt2 + dx2 = −du dv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Here, we set c = 1, and consider the (−, +) convention for the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' The light-cone coordinates are u = t − x , v = t + x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Thus for a manifold of constant u, one has nv = −2 , nu = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' while for a manifold of constant v, one has nu = −2 , nv = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Since √−g = 1 2, then the inner product (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='3) becomes ⟨f, g⟩ = i � ∞ −∞ dv � f∗ ∂ ∂vg − ∂ ∂vf∗g � , ⟨f, g⟩ = i � ∞ −∞ du � f∗ ∂ ∂ug − ∂ ∂uf∗g � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='9) where a constant u, and a constant v manifolds were chosen in the above relation respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='2 The inner product and commutation relations Here in this subsection we show the connection between the inner products and the com- mutation relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Namely, [aΩ, a† Ω′] = � Φ(x, Ω), Φ(x, Ω′) � , [aΩ, aΩ′] = − � Φ(u, Ω), Φ∗(u, Ω′) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='10) In order to prove the above relations, one may start from the following: aΩ = ⟨Φ(x, Ω), Φ(x)⟩ , a† Ω = − ⟨Φ∗(x, Ω), Φ(x)⟩ , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='11) where they can be found from the field mode expansion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='1), and using the properties of the inner product (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Next, we present a very useful lemma as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Lemma: For any modes f(u, Ω) and g(u, Ω) one has the following relation: � ⟨f(u, Ω), Φ(u)⟩, � g � u′, Ω′� , Φ � u′��� = − � f(u, Ω), g∗(u, Ω′) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='12) The proof is given in the appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Having used the lemma (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='12), one can now prove the above mentioned (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='10) inter- – 5 – connection between the commutation relations and the inner products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Namely, � aΩ, a† Ω′ � = � ⟨Φ(u, Ω), Φ(u)⟩, − � Φ∗ � u′, Ω′� , Φ � u′��� = � Φ(u, Ω), Φ(u, Ω′) � , [aΩ, aΩ′] = � ⟨Φ(u, Ω), Φ(u)⟩, � Φ � u′, Ω′� , Φ � u′��� = − � Φ(u, Ω), Φ∗(u, Ω′) � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='13) where we have used (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='11) and the lemma (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='3 Some examples: Minkowski plane wave and Rindler In this subsection, we derive Minkowski plane wave and Rindler positive norm modes by utilizing the inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' It is more convenient to work with the light-cone coordinate (u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' We consider a massless Klein-Gordon field in 1 + 1 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' The field equation is □Φ = 0, or in terms of light-cone coordinates ∂ ∂u ∂ ∂vΦ = 0, and can be solved simply by Φ(u, v) = Φ(u) + Ψ(v), where Φ(u) and Ψ(v) are general functions indicating right- and left-moving waves respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Consider the change of coordinates u = − 1 ae−a(τ−ξ) , v = 1 aea(τ+ξ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='14) One may call (τ, ξ) Rindler coordinates [29] and the metric shall be written as ds2 = e2a ξ (−dτ 2 + dξ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' For positive (negative) a, we have u < 0 (u > 0) and v > 0 (v < 0) and the associated region in spacetime diagram is called Rindler right (left) wedge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Also since Φ(u, v) = Φ(u) + Ψ(v), without loss of generality, we consider the right moving wave, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=', Φ(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Minkowski plane wave Right moving Minkowski plane wave reads Φ(u, Ω) = f(Ω) e−i uΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Imposing the Klein- Gordon inner product (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='9) yields � Φ(u, Ω), Φ(u, Ω′) � = i f∗(Ω)f(Ω′) � ∞ −∞ du � ei uΩ ∂ ∂u e−i uΩ′ − ∂ ∂uei uΩ e−i uΩ′� (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='15) = f∗(Ω)f(Ω′)(Ω + Ω′) � ∞ −∞ du ei u(Ω−Ω′) = 4πΩ|f(Ω)|2δ(Ω − Ω′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Hence ⟨Φ(u, Ω), Φ(u, Ω′)⟩ = δ(Ω − Ω′) yields the important fact that Ω should be a positive real number, and f(Ω) = 1 √ 4πΩ, and thus the positive norm mode in right moving Minkowski plane wave reads Φ(u, Ω) = 1 √ 4πΩ e−i uΩ , Ω > 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='16) It is easy to check the inner product of a mode and its conjugate vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Rindler Rindler mode can be found in Rindler right (u < 0, v > 0), or left (u > 0, v < 0) wedges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Again, considering the right moving wave, we may write the mode as Φ(u, Ω) = θ(u) f(Ω) ui Ω , Φ(u, Ω) = θ(−u) g(Ω) (−u)i Ω , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='17) – 6 – for the left and right wedges respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Here Ω is a real number, not necessarily a positive one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' The inner product for the left Rindler wedge reads � Φ(u, Ω), Φ(u, Ω′) � = i f∗(Ω)f(Ω′) � ∞ −∞ θ(u)du � u−i Ω ∂ ∂u ui Ω′ − ∂ ∂uu−i Ω ui Ω′� = −f∗(Ω)f(Ω′)(Ω + Ω′) � ∞ −∞ θ(u)du u−i (Ω−Ω′)−1 = −4πΩ|f(Ω)|2δ(Ω − Ω′) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='18) therefore, the orthonormality condition implies Ω < 0 and f(Ω) = 1 √−4πΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Similarly for the right wedge, one has � Φ(u, Ω), Φ(u, Ω′) � = i g∗(Ω)g(Ω′) � ∞ −∞ θ(−u)du � (−u)−i Ω ∂ ∂u (−u)i Ω′ − ∂ ∂u(−u)−i Ω (−u)i Ω′� = g∗(Ω)g(Ω′)(Ω + Ω′) � ∞ −∞ θ(−u)du (−u)−i (Ω−Ω′)−1 = 4πΩ|f(Ω)|2δ(Ω − Ω′) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='19) therefore, the orthonormality condition implies Ω > 0 and f(Ω) = 1 √ 4πΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Consequently the positive norm Rindler mode for right moving wave is Left wedge: Φ(u, Ω) = θ(u) 1 √ 4πΩ u−i Ω , Ω > 0 , Right wedge: Φ(u, Ω) = θ(−u) 1 √ 4πΩ (−u)i Ω , Ω > 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='20) Note ⟨f∗, f∗⟩ = − ⟨f, f⟩ implies the following relations for all real Ω: � θ(u)ui Ω, θ(u)ui Ω′� = −4πΩ δ(Ω′ − Ω) , � θ(−u)(−u)i Ω, θ(−u)(−u)i Ω′� = 4πΩ δ(Ω′ − Ω) , � θ(u)ui Ω, θ(−u)(−u)i Ω′� = 0 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='21) where the last relation is obvious since θ(u)θ(−u) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='4 Combining positive and negative norm modes One way of obtaining new modes is a linear combination of positive and negative norm modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' This can be performed such that the new mode should satisfy the inner product relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' The new mode Ψ(u, Ω) may be defined as Ψ(u, Ω) = α(Ω)Φ(u, Ω) + β(Ω)Φ∗(u, Ω) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='22) – 7 – where Φ(u, Ω) is the initial mode, and satisfies the following inner product relations � Φ(u, Ω), Φ(u, Ω′) � = δ(Ω − Ω′) , � Φ(u, Ω), Φ∗(u, Ω′) � = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='23) Here α(Ω) and β(Ω) are general complex coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' The inner product for two modes Ψ(u, Ω) reads � Ψ(u, Ω), Ψ(u, Ω′) � = � α(Ω)Φ(u, Ω) + β(Ω)Φ∗(u, Ω), α(Ω′)Φ(u, Ω′) + β(Ω′)Φ∗(u, Ω′) � =α∗(Ω)α � Ω′� � Φ(u, Ω), Φ � u, Ω′�� + α∗(Ω)β � Ω′� � Φ(u, Ω), Φ∗ � u, Ω′�� + β∗(Ω)α � Ω′� � Φ∗(u, Ω), Φ(u, Ω′) � + β∗ � Ω)β � Ω′� � Φ∗(u, Ω), Φ∗ � u, Ω′�� = � |α(Ω)|2 − |β(Ω)|2� δ � Ω − Ω′� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='24) Requiring the answer to be δ (Ω − Ω′) yields the following constraint: |α(Ω)|2 − |β(Ω)|2 = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='25) Furthermore, the inner product of a mode Ψ(u, Ω) and its conjugate should be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Namely, � Ψ(u, Ω), Ψ∗(u, Ω′) � = � α(Ω)Φ(u, Ω) + β(Ω)Φ∗(u, Ω), α∗(Ω′)Φ∗(u, Ω′) + β∗(Ω′)Φ(u, Ω′) � = � α∗(Ω)β∗ (Ω) − β∗(Ω)α∗ (Ω) � δ � Ω − Ω′� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='26) However, this is identically zero and it does not impose any constraint on α(Ω) and β(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Henceforth, the only constraint for the coefficients is |α(Ω)|2 − |β(Ω)|2 = 1, which can be solved as α(Ω) = cosh (κΩ) e i γ 2 , β(Ω) = sinh (κΩ) e− i γ 2 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='27) where κ and γ are real parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' 3 Kappa mode: combining opposite sign norm Rindler modes In addition to the strategy introduced in section (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='4), one may consider another type of combination of Rindler modes, namely, combining Rindler modes of opposite wedges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' There are two ways to proceed: combining the same and the opposite sign norm modes of opposite wedges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' In this section we address the latter and in the next section we study the former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='1 Constructing Kappa modes The ansatz for the opposite sign norm is Φ(u, Ω) = α(Ω)θ(−u)(−u)i Ω + β(Ω)θ(u)ui Ω .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='1) Note for positive (negative) Ω, the first (second) term has positive norm, while the second (first) term has negative norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' – 8 – The inner product of two modes reads ⟨Φ(u, Ω),Φ(u, Ω′)⟩ = � α(Ω)θ(−u)(−u)i Ω + β(Ω)θ(u)ui Ω, α(Ω′)θ(−u)(−u)i Ω′ + β(Ω′)θ(u)ui Ω′� = α∗(Ω)α(Ω′) � θ(−u)(−u)i Ω, θ(−u)(−u)i Ω′� + β∗(Ω)β(Ω′) � θ(u)ui Ω, θ(u)ui Ω′� = 4πΩ � |α(Ω)|2 − |β(Ω)|2� δ(Ω − Ω′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='2) Therefore, the positivity of the norm indicates 4πΩ � |α(Ω)|2 − |β(Ω)|2� = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='3) Note Ω now can be either a positive or a negative real number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Furthermore, the inner product of the mode and its conjugate becomes ⟨Φ(u, Ω), Φ∗(u, Ω′)⟩ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='4) = � α(Ω)θ(−u)(−u)i Ω + β(Ω)θ(u)ui Ω, α∗(Ω′)θ(−u)(−u)−i Ω′ + β∗(Ω′)θ(u)u−i Ω′� = α∗(Ω)α∗(Ω′) � θ(−u)(−u)i Ω, θ(−u)(−u)−i Ω′� + β∗(Ω)β∗(Ω′) � θ(u)ui Ω, θ(u)u−i Ω′� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' By using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='21) one has � θ(u)ui Ω, θ(u)u−i Ω′� = − � θ(−u)(−u)i Ω, θ(−u)(−u)−i Ω′� = −4πΩ δ(Ω + Ω′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='5) Thus the inner product (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='4) then becomes ⟨Φ(u, Ω),Φ∗(u, Ω′)⟩ = 4πΩ δ(Ω + Ω′) � α∗(Ω)α∗(Ω′) − β∗(Ω)β∗(Ω′) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='6) This should be zero;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' however, in contrast to the previous case of subsection (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='4), since Ω and Ω′ can be any real numbers, the inner product is not automatically vanishing, but (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='6) imposes a constraint on α(Ω) and β(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Namely, � α∗(Ω)α∗(Ω′) − β∗(Ω)β∗(Ω′) � δ(Ω + Ω′) = � α∗(Ω)α∗(−Ω) − β∗(Ω)β∗(−Ω) � δ(Ω + Ω′) = 0 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='7) or simply, α(Ω)α(−Ω) − β(Ω)β(−Ω) = 0 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='8) for all Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Consequently, the inner product imposes two constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Let us first start with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='3), and introduce an ansatz α(Ω) = e κπΩ 2 +i θ � 8πΩ sinh (κπΩ) , β(Ω) = e− κπΩ 2 +i φ � 8πΩ sinh (κπΩ) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='9) where κ is an arbitrary positive real number, and θ and φ are any real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Note κ – 9 – should be chosen as a positive real number so the term 8πΩ sinh (κπΩ) is always positive for any real Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Next, imposing the second constraint (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='8) implies θ = φ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='10) Thus, ei θ is simply an overall phase which can be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Therefore, we have α(Ω) = e κπΩ 2 � 8πΩ sinh (κπΩ) , β(Ω) = e− κπΩ 2 � 8πΩ sinh (κπΩ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='11) Consequently, one may write the following final result for the new mode: Φ(u, Ω, κ) = 1 � 8πΩ sinh (κπΩ) � θ(−u)(−u)i Ω e κπΩ 2 + θ(u)ui Ω e− κπΩ 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='12) Since it is classified by a positive number κ, we call it κ-mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' The field can be written as Φ(u) = � ∞ −∞ dΩ � Φ(u, Ω, κ)AΩ,κ + Φ∗(u, Ω, κ)A† Ω,κ � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='13) where we have used AΩ,κ to denote the annihilation operator for a κ-mode with the fre- quency Ω (Note Ω is any real number).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Explicitly, using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='12) one has Φ(u) = θ(−u) � ∞ −∞ dΩ 1 � 8πΩ sinh (κπΩ) � (−u)i Ω e κπΩ 2 AΩ,κ + (−u)−i Ω e κπΩ 2 A† Ω,κ � +θ(u) � ∞ −∞ dΩ 1 � 8πΩ sinh (κπΩ) � ui Ω e− κπΩ 2 AΩ,κ + u−i Ω e− κπΩ 2 A† Ω,κ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='14) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='2 Rindler and Unruh-Minkowski as special cases In this section we find Rindler and Unruh-Minkowski modes as special cases of the κ-mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Rindler Let κ → ∞ in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' For positive Ω, the first term of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='12) survives, indicating the Rindler mode in the right wedge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Thus the κ-mode in this special case reads Φ(u, κ → ∞) = 1 √ 4πΩ θ(−u) (−u)i Ω .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='15) Similarly, considering κ → ∞ with negative Ω in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='12), the second term of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='12) survives, indicating the Rindler mode in the left wedge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Namely, Φ(u, κ → ∞) = 1 √ 4πΩ θ(u) u−i Ω , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='16) where we changed Ω → −Ω, and hence Ω > 0 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' – 10 – Unruh-Minkowski It is simple to observe Unruh-Minkowski as a special case of the κ-mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Set κ = 1 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' One has Φ(u, Ω, κ = 1) = 1 � 8πΩ sinh (πΩ) � θ(−u)(−u)i Ω e πΩ 2 + θ(u)ui Ω e− πΩ 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='17) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='3 Bogoliubov transformation between κ-modes The goal is to find the Bogoliubov transformation between distinct κ-modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='14), one has Φ(u) = θ(−u) � ∞ −∞ dΩ 1 � 8πΩ sinh (κπΩ) � (−u)i Ω e κπΩ 2 AΩ,κ + (−u)−i Ω e κπΩ 2 A† Ω,κ � +θ(u) � ∞ −∞ dΩ 1 � 8πΩ sinh (κπΩ) � ui Ω e− κπΩ 2 AΩ,κ + u−i Ω e− κπΩ 2 A† Ω,κ � = θ(−u) � ∞ −∞ dΩ 1 � 8πΩ sinh (κ′πΩ) � (−u)i Ω e κ′πΩ 2 AΩ,κ′ + (−u)−i Ω e κ′πΩ 2 A† Ω,κ′ � +θ(u) � ∞ −∞ dΩ 1 � 8πΩ sinh (κ′πΩ) � ui Ω e− κ′πΩ 2 AΩ,κ′ + u−i Ω e− κ′πΩ 2 A† Ω,κ′ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='18) Next, we compare the factors of θ(−u)(−u)i Λ, and θ(u)ui Λ in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Comparing θ(−u)(−u)i Λ factor indicates 1 � 8πΛ sinh (κπΛ) � e κπΛ 2 AΛ,κ + e −κπΛ 2 A† −Λ,κ � = 1 � 8πΛ sinh (κ′πΛ) � e κ′πΛ 2 AΛ,κ′ + e −κ′πΛ 2 A† −Λ,κ′ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='19) One has to notice that since −∞ < Ω < ∞, then θ(−u)(−u)i Λ appears both in the first and second term of the first line of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Also, comparing θ(u)ui Λ factors in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='18) yields 1 � 8πΛ sinh (κπΛ) � e− κπΛ 2 AΛ,κ + e κπΛ 2 A† −Λ,κ � = 1 � 8πΛ sinh (κ′πΛ) � e− κ′πΛ 2 AΛ,κ′ + e κ′πΛ 2 A† −Λ,κ′ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='20) It is useful to find a transformation of the pair � AΛ,κ A† −Λ,κ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='19) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='20), one has � AΛ,κ′ A† −Λ,κ′ � = sgn (Λ) � sinh (κπΛ) sinh (κ′πΛ) � sinh � (κ+κ′)πΛ 2 � sinh � (κ′−κ)πΛ 2 � sinh � (κ′−κ)πΛ 2 � sinh � (κ+κ′)πΛ 2 � � � AΛ,κ A† −Λ,κ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='21) The above relation is very crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' It clearly shows, since the off-diagonal elements of the above matrix are non-vanishing for κ ̸= κ′, that the annihilation operator in a mode κ′ – 11 – depends upon both annihilation and creation operators of a mode κ, meaning that these modes have different vacua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' One may check explicitly the transformation between Rindler and Unruh-Minkowski operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Namely, with κ = ∞, κ′ = 1, one has AΛ = 1 √ 1 − e−2πΛ � bRΛ − e−πΛ b† LΛ � , A−Λ = 1 √ 1 − e−2πΛ � bLΛ − e−πΛ b† RΛ � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='22) where Λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' This is in agreement with eqs (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='18) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='20) of the Unruh-Wald paper [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Note here we adopt the following convention for Unruh-Minkowski and Rindler operators AΩ,1 = AΩ , Ω ∈ R , AΩ,∞ = bRΩ , A−Ω,∞ = bLΩ , Ω > 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='23) where AΩ, bRΩ, and bLΩ are the annihilation operators for Unruh-Minkowski, Rindler right wedge, and Rindler left wedge respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' One can get Rindler in terms of Unruh-Minkowski by either finding the inverse of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='22), or by setting κ = 1, κ′ = ∞ in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' It reads bRΛ = 1 √ 1 − e−2πΛ � AΛ + e−πΛ A† −Λ � , bLΛ = 1 √ 1 − e−2πΛ � A−Λ + e−πΛ A† Λ � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='24) again, recovering eq (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='24) of Unruh-Wald.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='4 Bogoliubov transformation between a κ-mode and the Minkowski plane wave The goal is to find the Bogoliubov transformation between a κ-mode and the Minkowski plane wave .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='14), one has Φ(u) = θ(−u) � ∞ −∞ dΩ 1 � 8πΩ sinh (κπΩ) � (−u)i Ω e κπΩ 2 AΩ + (−u)−i Ω e κπΩ 2 A† Ω � +θ(u) � ∞ −∞ dΩ 1 � 8πΩ sinh (κπΩ) � ui Ω e− κπΩ 2 AΩ + u−i Ω e− κπΩ 2 A† Ω � = � ∞ 0 dν √ 4πν � aν e−i ν u + a† ν ei ν u� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='25) – 12 – Next, one may find aν by calculating � ∞ −∞ du ei ν uΦ(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' It reads aν = � ν π � ∞ −∞ dΩ i � 8πΩ sinh (κπΩ) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='26) � − e κπΩ 2 e πΩ 2 ν−(1+i Ω) Γ(1 + i Ω) AΩ + e −κπΩ 2 e− πΩ 2 ν−(1+i Ω) Γ(1 + i Ω) AΩ −e κπΩ 2 e− πΩ 2 ν−(1−i Ω) Γ(1 − i Ω) A† Ω + e −κπΩ 2 e πΩ 2 ν−(1−i Ω) Γ(1 − i Ω) A† Ω � , where we have used the following useful integrals: � +∞ −∞ du ei νu (−u)i Ωθ(−u) = � ∞ 0 du e−i νu ui Ω = −i ν−(1+i Ω)e πΩ 2 Γ(1 + i Ω) , � +∞ −∞ du ei νu ui Ωθ(u) = � ∞ 0 du ei νuui Ω = i ν−(1+i Ω)e− πΩ 2 Γ(1 + i Ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='27) Therefore, the final answer for the Bogoliubov transformation between κ-mode and plane wave Minkowski reads aν = � ν π � ∞ −∞ dΩ i � 2πΩ sinh (κπΩ) � − ν−(1+i Ω) Γ(1 + i Ω) sinh �(κ + 1)πΩ 2 � AΩ − ν−(1−i Ω) Γ(1 − i Ω) sinh �(κ − 1)πΩ 2 � A† Ω � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='28) It is clear from the above relation that for κ = 1 , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=', the Unruh-Minkowski mode, the pre-factor of the creation operator vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' It indicates the vacuum is the Minkowski one, as we have expected from the Unruh-Minkowski mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='5 Commutation relations for different κ Let’s find the commutation relation between different κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' To do so, one has to find the inner product of the modes with different κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='12) we find the inner product between two positive norm modes as follows: � Φ(u, Ω, κ), Φ(u, Ω′, κ′) � = 1 � 8πΩ sinh (κπΩ) 1 � 8πΩ′ sinh (κ′πΩ′) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='29) � θ(−u)(−u)i Ω e κπΩ 2 + θ(u)ui Ω e− κπΩ 2 , θ(−u)(−u)i Ω′ e κ′πΩ′ 2 + θ(u)ui Ω′ e− κ′πΩ′ 2 � = sgn (Λ) � sinh (κπΛ) sinh (κ′πΛ) sinh � (κ+κ′)πΛ 2 � δ(Ω − Ω′) , – 13 – where we have used (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Therefore the commutation relation between annihilation and creation operators with different κ and κ′ is � AΩ,κ, A† Ω′,κ′ � = � Φ(u, Ω, κ), Φ(u, Ω′, κ′) � = sgn (Λ) sinh � (κ+κ′)πΛ 2 � � sinh (κπΛ) sinh (κ′πΛ) δ(Ω − Ω′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='30) Note the case of κ = κ′ yields the standard relation � AΩ,κ, A† Ω′,κ � = δ(Ω − Ω′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Next, the inner product of the positive and negative norm modes with different κ and κ′ reads � Φ(u, Ω, κ), Φ∗(u, Ω′, κ′) � = 1 � 8πΩ sinh (κπΩ) 1 � 8πΩ′ sinh (κ′πΩ′) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='31) � θ(−u)(−u)i Ω e κπΩ 2 + θ(u)ui Ω e− κπΩ 2 , θ(−u)(−u)−i Ω′ e κ′πΩ′ 2 + θ(u)u−i Ω′ e− κ′πΩ′ 2 � = sgn (Λ) � sinh (κπΛ) sinh (κ′πΛ) sinh � (κ−κ′)πΛ 2 � δ(Ω + Ω′) , where again (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='21) has been used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' The commutation relation between annihilation operators with different κ and κ′ is thus � AΩ,κ, AΩ′,κ′� = − � Φ(u, Ω, κ), Φ∗(u, Ω′, κ′) � = − sgn (Λ) sinh � (κ−κ′)πΛ 2 � � sinh (κπΛ) sinh (κ′πΛ) δ(Ω + Ω′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='32) Again the case of κ = κ′ yields � AΩ,κ, AΩ′,κ � = 0, as we expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' It is interesting to note the Bogoliubov relation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='21) can be found using the above commutation relation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Providing that � AΛ,κ′ A† −Λ,κ′ � can be written in terms of a matrix multiplying � AΛ,κ A† −Λ,κ � , the matrix elements can be found using the relations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='30) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' One then consequently obtains (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='21) by employing the standard commutation relations � AΩ,κ, A† Ω′,κ � = δ(Ω − Ω′), and � AΩ,κ, AΩ′,κ � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' 4 Combining same sign norm Rindler modes The same sign norm ansatz can be written as follows: Φ(u, Ω) = α(Ω)θ(−u)(−u)i Ω + β(Ω)θ(u)u−i Ω .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='1) Note for a positive Ω both terms above have positive norm, while for a negative Ω, both of them have negative one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' The new mode should satisfy the following inner products: � Φ(u, Ω), Φ(u, Ω′) � = δ(Ω − Ω′) , � Φ(u, Ω), Φ∗(u, Ω′) � = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='2) – 14 – Hence, one may check ⟨Φ(u, Ω),Φ(u, Ω′)⟩ = � α(Ω)θ(−u)(−u)i Ω + β(Ω)θ(u)u−i Ω, α(Ω′)θ(−u)(−u)i Ω′ + β(Ω′)θ(u)u−i Ω′� = α∗(Ω)α(Ω′) � θ(−u)(−u)i Ω, θ(−u)(−u)i Ω′� + β∗(Ω)β(Ω′) � θ(u)u−i Ω, θ(u)u−i Ω′� = 4πΩ � |α(Ω)|2 + |β(Ω)|2� δ(Ω − Ω′) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='3) where we have used (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' The value of the above inner product, providing Φ(u, Ω) is a positive norm mode, should be δ(Ω − Ω′), and hence 4πΩ � |α(Ω)|2 + |β(Ω)|2� = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='4) This implies Ω should be positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Also, the inner product of the mode and its conjugate reads ⟨Φ(u, Ω), Φ∗(u, Ω′)⟩ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='5) = � α(Ω)θ(−u)(−u)i Ω + β(Ω)θ(u)u−i Ω, α∗(Ω′)θ(−u)(−u)−i Ω′ + β∗(Ω′)θ(u)ui Ω′� = α∗(Ω)α∗(Ω′) � θ(−u)(−u)i Ω, θ(−u)(−u)−i Ω′� + β∗(Ω)β∗(Ω′) � θ(u)u−i Ω, θ(u)ui Ω′� , where by appropriate change of sign of Ω and Ω′ in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='21) one has � θ(u)u−i Ω, θ(u)ui Ω′� = � θ(−u)(−u)i Ω, θ(−u)(−u)−i Ω′� = 4πΩ δ(Ω + Ω′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='6) Thus, the inner product (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='5) becomes ⟨Φ(u, Ω), Φ∗(u, Ω′)⟩ = � α∗(Ω)α∗(Ω′) + β∗(Ω)β∗(Ω′) � 4πΩ δ(Ω + Ω′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='7) The above term is zero automatically, since Ω and Ω′ are both positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Therefore, the inner product of positive and negative norm modes asserts no restriction on coefficients α(Ω) and β(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Thus, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='4) is the only constraint on α(Ω) and β(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' One may solve this constraint as follows: α(Ω) = e κπΩ 2 + i γ 2 � 8πΩ cosh (κπΩ) , β(Ω) = e− κπΩ 2 − i γ 2 � 8πΩ cosh (κπΩ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='8) Therefore, one may write the following final result for the mode: Φ(u, Ω, κ, γ) = 1 � 8πΩ cosh (κπΩ) � θ(−u)(−u)i Ω e κπΩ 2 + i γ 2 + θ(u)u−i Ω e −κπΩ 2 − i γ 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='9) – 15 – The field can be written as Φ(u) = � ∞ 0 dΩ � Φ(u, Ω, κ, γ)AΩ,κ,γ + Φ∗(u, Ω, κ, γ)A† Ω,κ,γ � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='10) where we have used AΩ,κ,γ to denote the annihilation operator for the κ-mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' It is more convenient to drop (κ, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Explicitly, using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='12), one has Φ(u) = θ(−u) � ∞ 0 dΩ 1 � 8πΩ cosh (κπΩ) � (−u)i Ω e κπΩ 2 e i γ 2 AΩ,κ,γ + (−u)−i Ω e κπΩ 2 e− i γ 2 A† Ω,κ,γ � + θ(u) � ∞ 0 dΩ 1 � 8πΩ cosh (κπΩ) � u−i Ω e− κπΩ 2 e− i γ 2 AΩ,κ,γ + ui Ω e− κπΩ 2 e i γ 2 A† Ω,κ,γ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='11) One may wonder how to modify the mode, in order to include negative Ω as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' One possibility is mixing the positive and negative norms together in the ansatz instead of adding the same sign norm modes as we have done in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='1 Bogoliubov transformation between different modes The goal is to find the Bogoliubov transformation between κ-modes with different κ and γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='11), one has Φ(u) = θ(−u) � ∞ 0 dΩ 1 � 8πΩ cosh (κπΩ) � (−u)i Ω e κπΩ 2 e i γ 2 AΩ + (−u)−i Ω e κπΩ 2 e− i γ 2 A† Ω � +θ(u) � ∞ 0 dΩ 1 � 8πΩ cosh (κπΩ) � u−i Ω e− κπΩ 2 e− i γ 2 AΩ + ui Ω e− κπΩ 2 e i γ 2 A† Ω � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='12) = θ(−u) � ∞ 0 dΩ 1 � 8πΩ cosh (κ′πΩ) � (−u)i Ω e κ′πΩ 2 e i γ′ 2 A′Ω + (−u)−i Ω e κ′πΩ 2 e− i γ′ 2 A′† Ω � +θ(u) � ∞ 0 dΩ 1 � 8πΩ cosh (κ′πΩ) � u−i Ω e− κ′πΩ 2 e− i γ′ 2 A′Ω + ui Ω e− κ′πΩ 2 e i γ′ 2 A′† Ω � , where AΩ ≡ AΩ,κ,γ, and A′Ω ≡ AΩ,κ′,γ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Comparing θ(−u)(−u)i Λ and θ(u)ui Λ in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='12) indicates 1 � 8πΛ cosh (κπΛ) e κπΛ 2 e i γ 2 AΛ = 1 � 8πΛ cosh (κ′πΛ) e κ′πΛ 2 e i γ′ 2 A′Λ , 1 � 8πΛ cosh (κπΛ) e− κπΛ 2 e− i γ 2 AΛ = 1 � 8πΛ cosh (κ′πΛ) e− κ′πΛ 2 e− i γ′ 2 A′Λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='13) The above expressions simply yield κ = κ′ and γ = γ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Consequently if there exists a mode in the same sign norm, then κ and γ would be unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' – 16 – 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='2 Bogoliubov transformation between the same sign norm and the opposite sign norm modes So far we have found if there were any mode in the same sign norm scenario, it would be just a unique (κ, γ) mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Here in this subsection, we find the Bogoliubov transformation between the latter mode and the previous case of the opposite sign norm mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='14) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='11) one has Φ(u) = θ(−u) � ∞ 0 dΩ 1 � 8πΩ cosh (κπΩ) � (−u)i Ω e κπΩ 2 e i γ 2 AΩ,κ,γ + (−u)−i Ω e κπΩ 2 e− i γ 2 A† Ω,κ,γ � +θ(u) � ∞ 0 dΩ 1 � 8πΩ cosh (κπΩ) � u−i Ω e− κπΩ 2 e− i γ 2 AΩ,κ,γ + ui Ω e− κπΩ 2 e i γ 2 A† Ω,κ,γ � = θ(−u) � ∞ −∞ dΩ 1 � 8πΩ sinh (κ′πΩ) � (−u)i Ω e κ′πΩ 2 AΩ,κ′ + (−u)−i Ω e κ′πΩ 2 A† Ω,κ′ � +θ(u) � ∞ −∞ dΩ 1 � 8πΩ sinh (κ′πΩ) � ui Ω e− κ′πΩ 2 AΩ,κ′ + u−i Ω e− κ′πΩ 2 A† Ω,κ′ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='14) Now, comparing θ(u)ui Λ for Λ > 0 in the above relation yields e −κπΛ 2 + i γ 2 � 8πΛ cosh (κπΛ) A† Λ,κ,γ = 1 � 8πΛ sinh (κ′πΛ) � e− κ′πΛ 2 AΛ,κ′ + e κ′πΛ 2 A† −Λ,κ′ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='15) Also comparing θ(−u)(−u)i Λ for Λ > 0 in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='14) shows e κπΛ 2 + i γ 2 � 8πΛ cosh (κπΛ) AΛ,κ,γ = 1 � 8πΛ sinh (κ′πΛ) � e κ′πΛ 2 AΛ,κ′ + e −κ′πΛ 2 A† −Λ,κ′ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='16) Now it is clear while the left-hand sides of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='16) and Hermitian conjugate of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='15) are proportional, the right-hand sides are not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Therefore, one concludes the same sign norm mode cannot exist even for a unique value of κ and γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' 5 Generalized non-thermofield double states Unruh [3] and Israel [21], both in 1976, found the thermofield double state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' The easiest way to obtain this mode is using the Bogoliubov transformation between Rindler and plane wave Minkowski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Namely, � bLω − e− πω a b† Rω � |0M⟩ = 0 , � bRω − e− πω a b† Lω � |0M⟩ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='1) where bRω and bLω denoting the Rindler annihilation operators for the right and left wedges with a frequency ω respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Also a represents the constant acceleration of a particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' The Minkowski vacuum is denoted by |0M⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Then the Minkowski vacuum can be written in terms of entangled Rindler right-left – 17 – wedges state as follows |0M⟩ = 1 √ Z exp �� ∞ 0 dω e− βω 2 b† Rωb† Lω � |0R⟩ ⊗ |0L⟩ , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='2) where Z is the partition function, β = 1 T = 2πckB ℏa is an inverse Unruh temperature, and |0R⟩ and |0L⟩ are the Rindler vacuum in the right and left wedges respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='1 Relating different κ-vacua To find a generalized non-thermofield double state, first note AΩ,κ′ |0κ′⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Then by exploiting the Bogoliubov transformation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='21), one may observe � AΩ,κ − ηA† −Ω,κ � |0κ′⟩ = 0 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='3) where ηκ,κ′,Ω = sinh � 1 2(κ − κ′)πΩ � sinh � 1 2(κ + κ′)πΩ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='4) Following the same argument as in the case of Rindler-Minkowski, we have the relation between κ and κ′ vacua as follows: |0κ′⟩ = 1 √Zκκ′ exp �� ∞ 0 dΩ ηκ,κ′,Ω A† Ω,κ A† −Ω,κ � |0κ⟩ , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='5) where Zκκ′ is the normalization factor which depends upon κ and κ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Note in the above relation although Ω can be both positive and negative, but we have to integrate just the pos- itive frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Otherwise, We may include all positive and negative frequencies, however, a factor of half should be included in the integral, since ηκ,κ′,Ω = ηκ,κ′,−Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='2 κ-vacuum in terms of Rindler It is useful to find a special case of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='5), where κ′-vacuum is written in terms of κ-vacuum, with κ → ∞, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=', Rindler vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' One can easily obtain from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='5), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='4), and the convention in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='23), the following relation |0κ⟩ = 1 √Zκ exp �� ∞ 0 dω e− κβω 2 b† Rωb† Lω � |0R⟩ ⊗ |0L⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='6) The above expression is a generalization of the thermofield double state to a non- thermofield double state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Obviously κ ̸= 1 ruins the thermality of the state, that’s why we call it non-thermofield double state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' For retrieving the familiar case of the Minkowski vacuum in terms of the Rindler one, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=', thermofield double state, we should consider a κ-vacuum as the Minkowski vacuum, namely κ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' The thermofield double state appears trivially by setting κ = 1 in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='3 κ-vacuum in terms of Minkowski The last special case is writing a general κ-vacuum in terms of the Minkowski vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' This can be observed readily by setting κ = 1 in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='5) and using the convention in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' – 18 – Namely, |0κ⟩ = 1 √Zκ exp �� ∞ 0 dΩ ηκ,Ω A† Ω A† −Ω � |0M⟩ , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='7) where ηκ,Ω = −sinh � 1 2(κ − 1)πΩ � sinh � 1 2(κ + 1)πΩ � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='8) is a special case of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='4) for κ = 1 and relabeling κ′ as κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' 6 Conclusion In this paper, we discuss the importance of the Klein-Gordon inner product with respect to appropriately defining a mode associated with annihilation and creation operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Essen- tially (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='6) relates the inner product of modes to the commutator relations, hence, to have the standard commutation relations, the inner product between modes should be as stated in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' This requirement imposes a strong constraint on the form of modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Two famous modes, plane wave Minkowski and Rindler, were worked out as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Furthermore, the inner product can be used to find new modes from a given mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Specifically, we worked this procedure out for Rindler-like modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' In other words, a new set of modes can be obtained by a combination of two Rindler modes in the right and left wedges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' One can perform this in two distinct ways where the modes have the same and the opposite sign norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Interestingly, while the latter scenario yields an infinite set of modes parameterized by a real positive parameter κ, the former case yields no valid mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' This new κ-mode, inspired by the work of Unruh [3], yields the Unruh-Minkowski and Rindler modes for special cases of κ = 1 and κ → ∞ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Moreover, the well-known thermofield double state, relating the Minkowski to the Rindler vacuum, is generalized to include the κ-vacua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Namely, a κ-vacuum is written in terms of another, say κ′-vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' The relation is similar to that of the thermofield dou- ble, with a modified coefficient in the exponential, as well as using the κ-mode annihilation and creation operators instead of the Rindler ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Of course, the generalized expression for the thermofield double state reduces to the usual one if one considers (κ, κ′) = (∞, 1) in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Two special cases of this generalization, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=', κ-vacuum in terms of Rindler and κ-vacuum in terms of Minkowski are outlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' The former case (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='6) resembles the famous thermofield double state, however, an important difference is that it is no longer a thermal state for a general κ ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' The thermofield double state has been an indispensable part of AdS/CFT program since Maldacena’s proposal of the equivalence of the eternal AdS black hole in the bulk and the thermofield double state in the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Now the generalized non-thermofield double state may shed more light on the bulk physics, since, there is another degree of freedom κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Acknowledgments AA is grateful to Girish Agarwal, Jonathan Ben-Benjamin, David Lee, Yusef Malek, Ana- toly Svidzinsky, and especially Marlan Scully and Bill Unruh for illuminating discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' – 19 – AA thanks Reed Nessler for his careful reading of the manuscript and providing helpful comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' This work was supported by the Robert A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Welch Foundation (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' A-1261), and the National Science Foundation (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' PHY-2013771).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' A Proof of the lemma (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='12) To prove the lemma (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='12), one may start from the definition of the inner product as follows � ⟨f(u, Ω), Φ(u)⟩, � g � u′, Ω′� , Φ � u′��� (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='1) = − � +∞ −∞ du � +∞ −∞ du′ � f∗(u, Ω)g∗(u′, Ω′) [∂uΦ(u), ∂u′Φ(u′)] − f∗(u, Ω)∂u′g∗(u′, Ω′) [∂uΦ(u), Φ(u′)] − ∂uf∗(u, Ω)g∗(u′, Ω′)[Φ(u), ∂u′Φ(u′)] + ∂uf∗(u, Ω)∂u′g∗(u′, Ω′)[Φ(u), Φ(u′)] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' There are four different commutation relations involved above where they are expressed as follows [Φ(u), Φ(u′)] = i 4 sgn(u′ − u) , [∂uΦ(u), Φ(u′)] = − i 2δ(u′ − u) , [Φ(u), ∂u′Φ(u′)] = i 2δ(u′ − u) , [∂uΦ(u), ∂u′Φ(u′)] = i 2δ′(u′ − u) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='2) Next, one has to use the following properties of Dirac delta function � +∞ −∞ dx f(x)δ(x) = f(0) , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='3) � +∞ −∞ dx f(x)δ′(x) = � +∞ −∞ dx � d dx � f(x)δ(x) � − df dxδ(x) � = − � +∞ −∞ dxf′(x)δ(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Plugging (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='2) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='3) in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='1), we have � ⟨f(u, Ω), Φ(u)⟩, � g � u′, Ω′� , Φ � u′��� = − i 2 � +∞ −∞ du f∗(u, Ω)∂ug∗ � u, Ω′� − i 2 � +∞ −∞ du � f∗(u, Ω)∂ug∗ � u, Ω′� − ∂uf∗(u, Ω)g∗ � u, Ω′� � − i 4 � +∞ −∞ dudu′ ∂uf∗(u, Ω)∂u′g∗(u′, Ω′) sgn(u′ − u) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='4) – 20 – To evaluate the last term above, one may proceed as follows � +∞ −∞ dudu′ ∂uf∗(u, Ω)∂u′g∗(u′, Ω′) sgn(u′ − u) = � +∞ −∞ du � − � u −∞ du′∂uf∗ (u, Ω) ∂u′g∗ � u′, Ω′� + � ∞ u du′∂uf∗ (u, Ω) ∂u′g∗ � u′, Ω′� � = � +∞ −∞ du � − ∂uf∗ (u, Ω) � g∗ � u, Ω′� − g∗ � −∞, Ω′� � + ∂uf∗ (u, Ω) � g∗ � ∞, Ω′� − g∗ � u, Ω′� �� = −2 � +∞ −∞ du ∂uf∗ (u, Ω) g∗ � u, Ω′� , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='5) where we have assumed the field is zero at infinities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Using the above relation, the commu- tation relation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='4) can be written finally as � ⟨f(u, Ω), Φ(u)⟩, � g � u′, Ω′� , Φ � u′��� = −i � +∞ −∞ du � f∗(u, Ω)∂ug∗ � u, Ω′� − ∂uf∗(u, Ω)g∗ � u, Ω′� � = − � f(u, Ω), g∗(u, Ω′) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='6) This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' References [1] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Witten, Why does quantum field theory in curved spacetime make sense?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' and what happens to the algebra of observables in the thermodynamic limit?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=', in Dialogues Between Physics and Mathematics, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' 241–284.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Springer, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' [2] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Hawking, Particle Creation by Black Holes, Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' 43 (1975) 199–220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' [Erratum: Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' 46, 206 (1976)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' [3] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Unruh, Notes on black hole evaporation, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' D 14 (1976) 870.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' [4] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Iorio and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Lambiase, The Hawking-Unruh phenomenon on graphene, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' B 716 (2012) 334–337, [arXiv:1108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='2340].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' [5] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Bhardwaj and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Sheehy, Unruh Effect and Takagi’s Statistics Inversion in Strained Graphene, arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='08053.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' [6] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Scully, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Kocharovsky, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Belyanin, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Fry, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Capasso, Enhancing Acceleration Radiation from Ground-State Atoms via Cavity Quantum Electrodynamics, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' 91 (2003) 243004, [quant-ph/0305178].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' [7] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Belyanin, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Kocharovsky, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Capasso, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Fry, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Zubairy, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Scully, Quantum electrodynamics of accelerated atoms in free space and in cavities, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' A 74 (Aug, 2006) 023807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' [8] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Bruschi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Louko, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Martín-Martínez, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Dragan, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Fuentes, Unruh effect in – 21 – quantum information beyond the single-mode approximation, Physical Review A 82 (2010), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' 4 042332.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' [9] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Bradler, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Hayden, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Panangaden, Private information via the Unruh effect, JHEP 08 (2009) 074, [arXiv:0807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='4536].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' [10] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Hawking, Breakdown of Predictability in Gravitational Collapse, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' D 14 (1976) 2460–2473.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' [11] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Penington, Entanglement Wedge Reconstruction and the Information Paradox, JHEP 09 (2020) 002, [arXiv:1905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='08255].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' [12] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Almheiri, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Engelhardt, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Marolf, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Maxfield, The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole, JHEP 12 (2019) 063, [arXiv:1905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='08762].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' [13] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Penington, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Shenker, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Stanford, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Yang, Replica wormholes and the black hole interior, JHEP 03 (2022) 205, [arXiv:1911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='11977].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' [14] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Almheiri, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Hartman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Maldacena, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Shaghoulian, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Tajdini, Replica Wormholes and the Entropy of Hawking Radiation, JHEP 05 (2020) 013, [arXiv:1911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='12333].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' [15] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Fulling, Nonuniqueness of canonical field quantization in Riemannian space-time, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' D 7 (1973) 2850–2862.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' [16] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Hawking and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Israel, General Relativity: An Einstein Centenary Survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Pr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=', Cambridge, UK, 1979.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' [17] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Unruh and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Wald, What happens when an accelerating observer detects a rindler particle, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' D 29 (Mar, 1984) 1047–1056.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' [18] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Svidzinsky, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Azizi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Ben-Benjamin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Scully, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Unruh, Unruh and cherenkov radiation from a negative frequency perspective, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' 126 (Feb, 2021) 063603.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' [19] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Svidzinsky, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Azizi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Ben-Benjamin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Scully, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Unruh, Causality in quantum optics and entanglement of minkowski vacuum, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' 3 (Mar, 2021) 013202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' [20] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Azizi, Kappa vacua: Infinite number of new vacua in two-dimensional quantum field theory, arXiv:2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content='03781.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' [21] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Israel, Thermo field dynamics of black holes, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' A 57 (1976) 107–110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' [22] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Maldacena, The Large N limit of superconformal field theories and supergravity, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' 2 (1998) 231–252, [hep-th/9711200].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' [23] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Witten, Anti-de Sitter space and holography, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' 2 (1998) 253–291, [hep-th/9802150].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' [24] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Gubser, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Klebanov, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Polyakov, Gauge theory correlators from noncritical string theory, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' B 428 (1998) 105–114, [hep-th/9802109].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' [25] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Maldacena, Eternal black holes in anti-de Sitter, JHEP 04 (2003) 021, [hep-th/0106112].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' [26] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Scully and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Zubairy, Quantum Optics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Cambridge University Press, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' [27] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' DeWitt, Quantum Field Theory in Curved Space-Time, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' 19 (1975) 295–357.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' [28] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Wald, On Particle Creation by Black Holes, Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' 45 (1975) 9–34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' – 22 – [29] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Rindler, Kruskal Space and the Uniformly Accelerated Frame, Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' 34 (1966) 1174.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} +page_content=' – 23 –' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQf5TiT/content/2301.13672v1.pdf'} diff --git a/ONAyT4oBgHgl3EQf7Pol/content/tmp_files/2301.00834v1.pdf.txt b/ONAyT4oBgHgl3EQf7Pol/content/tmp_files/2301.00834v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2d9de15e6f3c2c087ef4e3a1cfbee3f697972227 --- /dev/null +++ b/ONAyT4oBgHgl3EQf7Pol/content/tmp_files/2301.00834v1.pdf.txt @@ -0,0 +1,1994 @@ +Draft version January 4, 2023 +Typeset using LATEX twocolumn style in AASTeX631 +Things that might go bump in the night: Assessing structure in the binary black hole mass spectrum +Amanda M. Farah +,1 Bruce Edelman +,2 Michael Zevin +,3, 4 Maya Fishbach +,5 Jose Mar´ıa Ezquiaga +,6 +Ben Farr +,2 and Daniel E. Holz +1, 3, 4 +1Department of Physics, University of Chicago, Chicago, IL 60637, USA +2Institute for Fundamental Science, Department of Physics, University of Oregon, Eugene, OR 97403, USA +3Kavli Institute for Cosmological Physics, The University of Chicago, 5640 South Ellis Avenue, Chicago, Illinois 60637, USA +4Enrico Fermi Institute, The University of Chicago, 933 East 56th Street, Chicago, Illinois 60637, USA +5Canadian Institute for Theoretical Astrophysics, David A. Dunlap Department of Astronomy and Astrophysics, and Department of +Physics, 60 St George St, University of Toronto, Toronto, ON M5S 3H8, Canada +6Niels Bohr International Academy, Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen, Denmark +ABSTRACT +Several features in the mass spectrum of merging binary black holes (BBHs) have been identified +using data from the Third Gravitational Wave Transient Catalog (GWTC-3). +These features are +of particular interest as they may encode the uncertain mechanism of BBH formation. +We assess +if the features are statistically significant or the result of Poisson noise due to the finite number of +observed events. We simulate realistic catalogs of BBHs whose underlying distribution does not have +the features of interest, apply the analysis previously performed on GWTC-3, and determine how often +such features are spuriously found. We find that two of the features found in GWTC-3, the peaks +at ∼ 10 M⊙ and ∼ 35 M⊙, cannot be explained by Poisson noise alone: peaks as significant occur in +< 0.33% of catalogs generated from a featureless population. These peaks are therefore likely to be of +astrophysical origin. However, additional structure beyond a power law, such as the purported dip at +∼ 14 M⊙, can be explained by Poisson noise. We also provide a publicly-available package, GWMockCat, +that creates simulated catalogs of BBH events with realistic measurement uncertainty and selection +effects according to user-specified underlying distributions and detector sensitivities. +1. INTRODUCTION +Gravitational waves (GWs) from more than 70 merg- +ers of compact objects have now been detected in the +data of the LIGO (Aasi et al. 2015) and Virgo (Ac- +ernese et al. 2014) detectors. A cumulative catalog of +these events and their properties has been produced by +the LIGO-Virgo-KAGRA (LVK) collaborations. +This +collection of all detections to date is called the “Third +Gravitational-Wave Transient Catalog” (Abbott et al. +2021, GWTC-3), and has enabled several insights into +the nature of gravity (Abbott et al. 2021) , the local +expansion of the universe (Abbott et al. 2021), and the +population of GW sources (Abbott et al. 2021a). +The underlying population of GW sources holds infor- +mation about the astrophysical processes that give rise +to merging binaries of compact objects. The mass spec- +afarah@uchicago.edu +bedelman@uoregon.edu +trum of binary black holes (BBHs), for example, encodes +information about numerous physical processes underly- +ing massive-star evolution, supernova physics, compact +object formation, and binary interactions. +For exam- +ple, the presence or dearth of black holes with masses +between ∼ 2–5 M⊙ (¨Ozel et al. 2010; Farr et al. 2011; +Fishbach et al. 2020; Farah et al. 2022) may unveil the +maximum neutron star mass, the stability of mass trans- +fer, and the timescales relevant for the engines that drive +supernova explosions (e.g., Fryer et al. 2012; Zevin et al. +2020; Mandel & M¨uller 2020; Li et al. 2021; van Son et al. +2022a; Patton et al. 2022; Siegel et al. 2022). On the +high mass end, a sharp decrease in the mass spectrum +for black holes with masses ≳ 50 M⊙ (Fishbach & Holz +2017; Edelman et al. 2021) would be a strong indication +that the pair instability process is at play and limiting +the core mass of massive stars (Fowler & Hoyle 1964; +Barkat et al. 1967; Heger & Woosley 2002; Heger et al. +2003; Woosley & Heger 2015; Belczynski et al. 2016; +Woosley 2017, 2019; Marchant et al. 2019; Renzo et al. +arXiv:2301.00834v1 [astro-ph.HE] 2 Jan 2023 + +ID2 +Farah, Edelman, et al. +2020), with the location of the decrease in the differential +merger rate acting to constrain relevant nuclear reaction +rates (Farmer et al. 2020). Other overdensities and un- +derdensities in the observed mass distribution (Edelman +et al. 2022; Tiwari & Fairhurst 2021; Tiwari 2022; Edel- +man et al. 2022), as well as the evolution of the mass +distribution with redshift (Fishbach et al. 2021; van Son +et al. 2022b; Karathanasis et al. 2022a; van Son et al. +2022a), will further inform the dominant BBH formation +channels, binary evolution physics, and the metallicity +evolution of the universe. +All of the parameters that are measurable from the +signal of a binary merger can provide insight into for- +mation mechanisms of merging binaries, especially when +used in a population analysis (Stevenson et al. 2015; +Zevin et al. 2017). However, the masses of the objects +in the merging system are the best measured and span +the largest dynamic range. Additionally, the mass dis- +tribution of compact objects can be used to measure cos- +mological parameters using the “spectral siren” method, +provided there is structure in the distribution beyond a +boundless power law (Chernoff & Finn 1993; Messen- +ger & Read 2012; Taylor et al. 2012; Farr et al. 2019; +Ezquiaga & Holz 2021; Ezquiaga & Holz 2022; Abbott +et al. 2021), such as edges, gaps, peaks, or changes in +the power law slope. Multiple features must be present +to disentangle redshift evolution of the mass spectrum +from cosmology, and more features further aid in break- +ing this degeneracy (Ezquiaga & Holz 2022). +There- +fore, considerable effort in the field of GW astronomy +has gone towards understanding the mass distribution of +GW sources. There are currently many more detected +BBH mergers than binary neutron star (BNS) or neu- +tron star-black hole (NSBH) mergers, so much of the +activity has been on population properties of the BBH +distribution, though the mass distribution of BNSs and +NSBHs has also been been considered (Fishbach et al. +2020; Landry & Read 2021; Farah et al. 2022; Ye & +Fishbach 2022; Biscoveanu et al. 2022b). +The BBH mass distribution is typically parameter- +ized by the primary mass m1, the larger of the two +component masses in the binary, and the mass ratio +q = m2/m1, the ratio of the less massive object’s mass to +the primary mass, though other parameterizations are +possible and valid (e.g., Farah et al. 2022; Fishbach & +Holz 2020a; Tiwari & Fairhurst 2021). The community +has thus far gained a robust understanding of the large- +scale features of the BBH mass distribution, and is just +beginning to resolve its finer details. After the release +of the First Gravitational-Wave Transient Catalog (Ab- +bott et al. 2019, GWTC-1), minimum and maximum +masses at ∼ 5 M⊙ and ∼ 40 M⊙ were identified in the +10 +30 +50 +70 +90 +m1 [MØ] +10°2 +100 +dR +dm1[Gpc°3 yr°1 M°1 +Ø ] +Smoothed power law +Power Law + Peak +Power Law + Spline +minimum mass +maximum mass +global +maximum +additional structure? +Figure 1. +Distribution of primary BBH masses inferred +using GWTC-3 and three different population models. The +smoothed power law model (grey) consists of a single power +law slope between a minimum and maximum mass, with the +merger rate set to exactly zero outside of those bounds. It +also includes a smoothing parameter at the low-mass end +that allows for an offset between the minimum BH mass and +the global maximum of the distribution. The Power Law ++ Peak model is similar to the smoothed power law, but +also includes a Gaussian component. The Power Law + +Spline model adds a cubic spline modulation to a smoothed +power law to allow for additional substructure. We seek to +determine if the perturbations beyond a power law found by +Power Law + Spline and other semi-parametric models +can be explained by random associations in the data due to +a finite number of observations, or if they are features of the +true underlying distribution. +BBH primary mass distribution, but it was not yet pos- +sible to distinguish between a uniform distribution and a +power law between those two bounds (Fishbach & Holz +2017; Talbot & Thrane 2018; Abbott et al. 2019). The +Second Gravitational-Wave Transient Catalog (Abbott +et al. 2021b, GWTC-2) brought dozens of additional +events, and the BBH mass distribution was found to +have a global maximum at ∼ 8 M⊙ and an excess of +BHs between ∼ 30 M⊙–40 M⊙ followed by a steep, al- +though not infinitely sharp, drop off in the rate at higher +masses extending to ∼ 80 M⊙ (instead of sharp cutoff at +∼ 40 M⊙). At the time, there were not enough observa- +tions to determine whether the mass distribution had a +local maximum at ∼ 35 M⊙, represented by a Gaussian +peak on top of a power law, or whether the steepening +towards higher masses was better described as a break +in the power law (Abbott et al. 2021). +At the end of the third LIGO–Virgo observing run, the +same two features at ∼ 8 M⊙ and ∼ 35 M⊙ remained, +and the feature at 35 M⊙ was classified as a peak rather +than a break in the power law (Abbott et al. 2021a). +Additionally, non-parametric (Mandel et al. 2017; Ri- +naldi & Del Pozzo 2022; Sadiq et al. 2022; Payne & +Thrane 2022; Edelman et al. 2022) and semi-parametric +(Edelman et al. 2022) analyses found robust evidence + +3 +for an additional peak at ∼ 10 M⊙, the same peak at +∼ 35 M⊙, as well as modest evidence for a paucity of +events near ∼ 14 M⊙ (Abbott et al. 2021a). These fea- +tures in the primary mass distribution correspond to +similar ones in the chirp mass distribution, occurring at +∼ 9 M⊙, ∼ 11 M⊙, and ∼ 26 M⊙, respectively (Tiwari & +Fairhurst 2021; Tiwari 2022). The current picture of the +BBH mass distribution is therefore a decreasing power +law from low to high masses, with a global maximum at +m1 ∼ 10 M⊙, a potential underdensity at m1 ∼ 14 M⊙, +and an overdensity at m1 ∼ 35 M⊙. This can be see +in Figure 1, where we plot the results of fitting two pa- +rameteric models and one semi-parametric model to the +BBHs in GWTC-3. +While the existence of this substructure in the current +data set appears robust, its interpretation is less clear. +Plausible explanations for this substructure include (1) +Poisson noise, (2) modeling systematics, or (3) astro- +physical signatures from one or several formation chan- +nels. We aim to disentangle the first two possibilities +from the third using the Power Law + Spline model +(Edelman et al. 2022), one of the semi-parametric mod- +els used to identify the substructure reported in Abbott +et al. (2021a). +Poisson noise would be caused by the fact that the +fiducial BBH analysis in Abbott et al. (2021a) includes +only 69 events over a mass range that spans more than +an order of magnitude, so observations may appear to +be clumped at some masses even if the underlying distri- +bution is smooth. We first determine if this explanation +accounts for the data by simulating realistic catalogs +of BBHs whose underlying distribution does not have +the features of interest, applying the analysis previously +performed on GWTC-3, and determining how often such +features are spuriously found. We develop several met- +rics comparing observations to simulated data in order +to assess the statistical significance of the “bumps” in +the primary mass distribution found by Abbott et al. +(2021a); Edelman et al. (2022). All of the metrics de- +rived in this work answer the same general question: +how often do we infer the existence of a feature when an- +alyzing observations of a true population without that +feature? In this sense, these metrics are analogous to +frequentist p-values, as lower values correspond to more +significant features in the data. Readers familiar with +gravitational wave data analysis might find it useful to +think of these metrics as false alarm rates because they +quantify how often noise resembles the observed signal. +A similar frequentist analysis on a large number of +mock catalogs was performed by Sadiq et al. (2022) on +the peak at ∼ 35 M⊙ using an adaptive kernel density es- +timator (aKDE) to find features in samples drawn from +featureless mass models, as well as from a model with a +single peak. They account for selection effects, but not +measurement uncertainty. They find that an aKDE is +able to identify peaks in the data, and that the peak +at ∼ 35 M⊙ found in GWTC-2 is statistically significant +within the aKDE model. +The second effect mentioned above, model system- +atics, could also plausibly cause spurious inference of +features beyond a power law. +It is potentially con- +cerning that the models considered in Abbott et al. +(2021a) that find peaks and troughs in the mass dis- +tribution are inherently “bumpy”: both Power Law ++ Peak (Talbot & Thrane 2018) and Multi source +employ a smoothed power law with a Gaussian compo- +nent (Wysocki & O’Shaughnessy 2021), Flexible Mix- +tures is a linear combination of Gaussian components, +and Power Law + Spline employs a smoothed power +law under a cubic spline modulation. The question is +then whether these “bumpy” models can recover sharp +features or if they instead create peaks and troughs that +are morphologically dissimilar to the true distribution. +This is most easily addressed by cross-checking with in- +dependent models such as Broken Power Law (Ab- +bott et al. 2021; Abbott et al. 2021a) and the auto- +regressive model presented in Callister & Farr (in prep.). +Inaccuracies in the selection function are also known +to cause systematic biases when inferring the underly- +ing population (e.g. Malmquist 1922, 1925). These bi- +ases could, in principle, also cause an incorrect inference +of structure in the astrophysical distribution of BBH +masses. However, selection effects in GW detectors are +remarkably well-characterized, so we expect this effect to +be subdominant to Poisson uncertainty. As the number +of events grows, so will our accuracy in the estimation of +the selection function (Farr 2019; Essick & Farr 2022). +We provide posterior samples from our simulated cata- +logs in an accompanying data release, and also provide a +publicly-available python package, GWMockCat, to create +similar samples according to user-defined populations.1 +Section 2 provides a demonstrative example: it fore- +goes a full fit to the astrophysical population of sources, +and compares the observed distribution of masses to pos- +sible observed distributions given an underlying power +law in primary mass, (incorrectly) assuming no mea- +surement uncertainty. This analysis suggests that the +observed peak at ∼ 35 M⊙ is statistically significant, +but that all other features beyond a simple power law +might be explainable by Poisson noise. This motivates a +1The data release can be found at https://zenodo.org/record/ +7411991, and GWMockCat can be installed at https://git.ligo.org/ +amanda.farah/mock-PE . + +4 +Farah, Edelman, et al. +thorough study using a full hierarchical Bayesian anal- +ysis on simulated event posteriors, which we carry out +in Section 3. Section 4 summarizes our conclusions and +discusses their implications for the astrophysical origin +of the gravitational waves observed thus far by the LVK. +Readers primarily interested in the significance of fea- +tures in the mass distribution may wish to skip to Sec- +tion 3.3, whereas those interested in using the package +GWMockCat can find details in Appendices A and B. +2. MOTIVATION +To construct a simple test of feature significance and +motivate further study, we first avoid a fit to the mass +distribution and instead consider the observed distribu- +tion of primary masses and its resemblance to one that +would result from a simple power law. +The observed +population differs significantly from the astrophysical +one, as current gravitational wave detectors are subject +to selection biases that favor the detection of closer and +more massive systems, as well as measurement error that +affects each system differently. We construct plausible +observed mass distributions that could occur from de- +tecting 69 BBHs whose astrophysical distribution is a +featureless power law in primary mass. To do this, we +use the samples provided by LIGO Scientific Collabora- +tion et al. (2021a), which were created for sensitivity es- +timation for the LVK’s GWTC-3 analysis. Each of these +samples comes with a probability of being drawn from +an assumed underlying distribution and a false alarm +rate (FAR) assigned by each search used by the LVK. +We can then re-weight these samples to our desired pop- +ulation model (in this case, a power law in m1, q, and +z) using the draw probability, and apply the same FAR +threshold used in Abbott et al. (2021a) to select “found +injections.” Of the ∼ 6 × 104 found injections, we re- +sample to N = 104 independent sets of 69 draws each +to directly compare to observations. +We then histogram each set of these found injections, +thereby obtaining a distribution of bin heights for our +mock populations. Using several thousand realizations +of found injection sets enables us to construct a null dis- +tribution of bin heights and characterize the effect of +Poisson noise on the shape of the observed distribution. +We compare these null distributions with the observed +distribution of BBH masses in GWTC-32 by assuming +2For all comparisons to real observations, we use the pub- +licly available posterior samples for the GWTC-2.1 and GWTC- +3 data releases (LIGO Scientific Collaboration & Virgo Collabo- +ration 2022; LIGO Scientific Collaboration et al. 2021b, respec- +tively). We use samples generated with the IMRPhenomXPHM +waveform and a prior proportional to the square of the luminosity +distances (i.e. the samples were not “cosmologically reweighted”). +0.00 +0.02 +0.04 +0.06 +p(m1) [M−1 +⊙ ] +Medians of GWTC-3 BBHs +Found Injections, α = 3.25 +Found Injections, α = 2.7 +10 +20 +30 +40 +50 +60 +70 +80 +90 +m1 [M⊙] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +CDF +Figure 2. +Observed source-frame primary mass distribu- +tions. Black solid lines contain the median a posteriori val- +ues for the binary black holes in GWTC-3. Pink and blue +bands indicate the 90% credible interval on the observed dis- +tributions predicted from astrophysical distributions that are +power laws in primary mass with spectral index α = 3.25 +and α = 2.7, respectively. The top panel shows a histogram +of observed primary masses. +For GWTC-3’s distribution +to be consistent with the null distributions, we expect its +bin heights in the top panel to be within the 90% credi- +ble intervals in 18 out of the 20 bins. The uncertainties in +these predicted distributions are due only to Poisson noise +resulting from a finite number of observations, rather than +modeling uncertainty or uncertainty in parameter estima- +tion. Therefore, the cumulative distribution functions in the +bottom panel are similar to a conventional posterior predic- +tive check, but with only one source of uncertainty. The large +deviations of the black curve from the shaded bands in some +regions indicate the difficulty that a single power law with +Poisson shot noise has in fully explaining the observations. +However, many of the apparent excursions from a power law +are well-contained within the predicted bands. +To make the most direct comparison with Abbott et al. (2021a), + +5 +the primary masses are measured perfectly and using +the median a posteriori values of their primary masses +as point estimates. The result is shown in the top panel +of Figure 2, which plots the 90% credible interval on the +observed null distributions, along with the distribution +of median primary masses of GWTC-3’s BBHs. For the +null distributions, we consider two power law spectral in- +dices as representative examples: α = 2.7 and α = 3.25. +These are chosen to represent a range of plausible values +for the BBHs in GWTC-3: a power law fit to GWTC-3 +yields α = 2.98+0.16 +−0.28, where the bounds represent 1-σ +deviations. +To obtain a more quantitative measure, we compare +bin heights from the found injections, hinj, to the bin +heights of observed events in GWTC-3, hGWTC-3, ob- +taining for each bin i the fraction of simulated bin +heights that are lower than those of GWTC-3 BBHs. +Explicitly, +ri +h = 1 +N +N +� +j +� +� +� +1 +if hj +i,inj < hi,GWTC-3 +0 +if hj +i,inj ≥ hi,GWTC-3 +, +(1) +where the sum is over the N = 104 sets of found injec- +tions, and rh is defined for each bin. For example, if +rh = 0.95 for a given bin, the observed distribution in +that bin is larger than would be expected from a feature- +less power law 95% of the time. A value of rh approach- +ing unity corresponds to a “bump” in the observed mass +distribution, and a value of rh approaching zero is in- +dicative of a “dip.” +Note that the comparison between the null distribu- +tions and GWTC-3 are occurring at each bin, rather +than across all bins. We do this because the magnitude +of Poisson noise depends on the value of m1: since the +underlying distribution is not uniform, fewer events are +expected at very high m1 and therefore the relative stan- +dard deviation is larger. This is also a consequence of +Eddington bias (Eddington 1913). Making comparisons +at specific points in m1 does not, however, properly cor- +rect for the look-elsewhere effect. We will address this +effect in Section 3. +The three most significant values of ri +h in the case +of α = 3.25 are r15.6 M⊙ +h += 0.033, r27.9 M⊙ +h += 0.036, +r36.1 M⊙ +h +> 0.999, where the superscripts indicate the +centers of the bins at which r was calculated. +This +means that less than 0.1% of mock populations had +more events near 36.1 M⊙ than GWTC-3 does, 3.3% of +mock populations had fewer events near 15.6 M⊙ than +we keep events with secondary mass larger than 3 M⊙ and FAR +less than 1 yr−1, resulting in 69 events. +GWTC-3, and at 27.9 M⊙, 3.6% of mock populations +had fewer events. +Repeating the exercise for α = 2.7, we find the three +most significant values of ri +hto be r40.2 M⊙ +h += 0.935, +r27.9 M⊙ +h += 0.020, r36.1 M⊙ +h +> 0.999. +The locations of +the significant features differ when the assumed under- +lying distribution changes. In either case, the bump at +∼ 35 M⊙ is unlikely to be due to Poisson noise, but other +features may be. +To avoid the need to arbitrarily choose bins, we ad- +ditionally construct a cumulative distribution function +(CDF) of the primary masses and compare it to the +CDFs of the null distributions, shown in the bottom +panel of Figure 2. This comparison is akin to a poste- +rior predictive check in that it can highlight where the +model fails to predict the data. Importantly, though, it +differs from the conventional posterior predictive check +because we have purposefully left out the effects of mod- +eling uncertainty and measurement uncertainty in order +to isolate the effects of Poisson noise. The prior distri- +butions are therefore also not included, since each event +is assumed to be measured with perfect accuracy. +If α = 3.25, the null distributions are consistent with +the data below ∼ 18 M⊙ and above ∼ 35 M⊙, but not +between them, meaning that the ∼ 10 M⊙ and ∼ 35 M⊙ +peaks can be explained by Poisson noise, but the under- +density between them could not be. On the other hand, +if α = 2.7, the null distributions are consistent with the +data everywhere except for above ∼ 40 M⊙, suggesting +that under this scenario, Poisson noise can explain all +features except for the ∼ 35 M⊙ peak. +For both spectral indices considered, two of the three +features found by Abbott et al. (2021a) can be explained +by Poisson noise from a finite number of observations. +However, this does not mean that exactly two of the fea- +tures are the result of Poisson noise, just that no more +than two can be caused by the phenomenon. Addition- +ally, it is not clear which of the features are more likely +to have physical origin, as this method offers no quan- +titative way to determine which power law slope is pre- +ferred. +Importantly, this methodology does not account for +the effects of measurement error, which can cause signif- +icant biases near the edges of sharp distributions when +not properly accounted for (Fishbach et al. 2020). We +therefore turn to a full hierarchical Bayesian analysis of +simulated catalogs, which will allow us to fit for popula- +tion model parameters, take measurement uncertainty +into account, and directly compare to metrics used in +Abbott et al. (2021a). +3. FULL ANALYSIS + +6 +Farah, Edelman, et al. +We determine how often the features inferred in the +mass distribution of BBHs would be spuriously found +in data whose underlying distribution does not have +those features. To do this, we construct a null distri- +bution by simulating BBH observations that would oc- +cur if the underlying astrophysical distribution was a +single power law with no substructure in a finite range. +The procedure for creating synthetic BBH observations +is described in Appendix A. Mock observations are com- +bined with corresponding sensitivity estimates in a hier- +archical Bayesian analysis, described in Loredo (2009); +Mandel et al. (2019); Thrane & Talbot (2019). We an- +alyze these simulations in the same way as the BBHs in +GWTC-3 to determine how often the features observed +in GWTC-3 would be found from an underlying distri- +bution without those features. +3.1. Power Law + Spline Mass Model +We use the Power Law + Spline semi-parametric +primary mass model as a flexible model that is easily +capable of finding peaks and valleys in the mass distri- +bution (Edelman et al. 2022; Abbott et al. 2021a). This +model parameterizes perturbations or deviations from a +simpler underlying distribution with flexible cubic spline +functions. Specifically, given an underlying hyper-prior +for primary mass, p(m1|Λ), the Power Law + Spline +model describes the primary mass distribution as: +pspline(m1|Λ, {mi}, {fi}) +∝ p(m1|Λ) exp(f(m1|{mi}, {fi})) +(2) +where f(m1|{mi}, {fi}) is the function describing the +perturbations, which we model with a cubic spline +function interpolated by introduced hyper-parameters, +{mi}, the locations of spline knots in mass space, and +{fi}, the height of the perturbation function at each +knot. This describes a semi-parametric model as it in- +cludes a simple “parametric” component (the underly- +ing distribution) in addition to a non-parametric com- +ponent that models the perturbation around the simple +description. For this study we use the simplest primary +mass model for the underlying description, which is the +Truncated model, describing a power law with sharp +cutoffs at the lower and upper mass bounds (Fishbach & +Holz 2017; Edelman et al. 2022). While this model has +been shown to insufficiently describe the primary mass +distribution, it captures the majority of the broadest +features (Abbott et al. 2021; Abbott et al. 2021a). +To assess the significance of peaks or valleys found +with the Power Law + Spline model one can look +at the posterior distribution of the perturbation heights +as a function of mass. This tells us how far “off” the +simple power law description is from accounting for the +data. Specifically we can find what percentile f(m1) = 0 +falls in the posterior distribution as a function of mass. +For data exactly distributed as a power law (the un- +derlying population), the inferred perturbation function +should be symmetric about 0 with widths determined by +the prior distributions on the knot heights. At masses +where the percentile of zero perturbation approaches +100% (0%) we can say there is an over (under) density +of events at these masses, compared to the underlying +power law distribution. This is identical to the analysis +done by Abbott et al. (2021a), who use the percentile +at which the perturbation function excludes zero at a +given location as a metric for how significant a feature +is at that location. +3.2. Metrics of Feature Significance +As described in Section 3.1, the Power Law + +Spline model makes use of a perturbation function con- +structed from cubic splines. The height of the pertur- +bation function, f(m1), at a point in primary mass, m1, +is then a direct measure of the deviation from a power +law at that point. +We can determine how often one +would find spurious evidence for substructure by simu- +lating catalogs from a power law, fitting them with the +Power Law + Spline model, and examining the re- +sulting perturbation function. +If the mock catalogs produce perturbation functions +with similar amplitudes to those seen for GWTC-3, the +structure in the GWTC-3 fit might be described by Pois- +son noise. On the other hand, if the perturbation func- +tions produced by fits to the mock catalogs are always +lower in amplitude to that of the GWTC-3 fit, the struc- +ture in the GWTC-3 data is likely to be present in the +underlying distribution. +For a given mock catalog, we find the m1 value where +the perturbation function is maximal. We obtain the +posterior distribution of perturbation function ampli- +tudes at that location, g(fmax). We repeat this for all +mock catalogs, obtaining a set of maximal perturbation +function distributions, {gj(fmax)}. These are plotted in +light grey on the left panels of Figure 4. The locations +of the three maximal perturbation function amplitudes +in the GWTC-3 fit are, from least to most significant, +13.5 M⊙, 10.2 M⊙, and 34.6 M⊙. +The posterior distri- +butions of perturbation function heights at these loca- +tions are gGWTC-3(f(13.5 M⊙)), gGWTC-3(f(10.2 M⊙)), +and gGWTC-3(f(34.6 M⊙)), and are plotted in orange in +the left panels of Figure 4. The amplitude of the per- +turbation function at 13.5 M⊙ is negative (i.e. it is a dip +rather than a bump), so we flip its distribution about +zero for more direct comparison. The same is done for + +7 +all g(fmax) whose medians are negative, as the pertur- +bation function’s prior is symmetric about zero. +3.3. Simulation Study +To determine whether the features in the mass spec- +trum of GWTC-3 BBHs are the result of Poisson noise +of a finite number of observations drawn from a feature- +less power law, we compare Power Law + Spline fits +using the GWTC-3 catalog and 300 mock catalogs gen- +erated from a “featureless” power law. The mock cat- +alogs considered in this section are all generated from +the same underlying distribution: a truncated power +law in primary mass, mass ratio, and redshift, with a +smoothing at low component masses to ensure the peak +of the mass distribution is not in the same location as +the minimum mass. The explicit form of the mock cat- +alogs’ population model, including values of all of its +hyperparameters, can be found in Appendix B. Despite +knowing the parameters of the underlying population +for the mock catalogs, we allow all hyperparameters to +vary when fitting Power Law + Spline to the mock +catalogs. +The resulting perturbation functions are shown in Fig- +ure 3 for 10 randomly chosen mock catalogs and GWTC- +3. The perturbation functions deviate from their prior +distribution in the mass range where detections exist +(above ∼ 5 M⊙ and below ∼ 85 M⊙), even in the case +of mock catalogs. +This means that the perturbation +functions are informed by the mock data despite the +mock data not inherently requiring a deviation from +a power law. +The question still remains whether the +perturbation function heights inferred from mock cata- +logs with no substructure are larger than those inferred +from GWTC-3. While nonzero values of the perturba- +tion function are common in the 10 mock catalog fits +shown in Figure 3, only a few amplitudes appear com- +parable in height to the three largest amplitudes of the +GWTC-3 perturbation function. +To verify this, we isolate the largest amplitude per- +turbations for all 300 mock catalog fits and compare +them to the three largest amplitude perturbations for +the GWTC-3 fit. These are plotted in the leftmost pan- +els of Figure 4. +The light grey curves are the poste- +rior distributions of largest perturbation function am- +plitudes {gj(fmax)} for each simulated catalog j. These +appear to have the same general shape as one another, +though with noticeable scatter. The orange curves in +each panel are the posterior distributions of GWTC-3’s +perturbation function gGWTC-3(f(m1)) at its three max- +imal locations: m1 = 13.5 M⊙, 10.2 M⊙, and 34.6 M⊙. +The distribution for the ∼ 14 M⊙ dip appears qual- +itatively similar to that of the simulated catalogs, the +∼ 10 M⊙ peak appears to be slightly shifted with respect +to most of the simulated catalogs but still within their +range, and the ∼ 35 M⊙ peak is noticeably shifted to- +wards higher values relative to the bulk of the simulated +catalog distributions. This suggests that the ∼ 35 M⊙ +peak is unlikely to be the result of Poisson noise or mod- +eling systematics, while other features could plausibly be +explained by those effects. +3.3.1. Maximum Perturbation Amplitude +To obtain a more quantitative measure, we derive sev- +eral metrics from the distributions of maximal pertur- +bation function amplitudes. +The first uses the Kol- +mogorov–Smirnov (KS) test: we compute the KS di- +vergence D between each of the {gj(fmax)} values to +obtain a null distribution of KS divergences, shown in +the solid black curve in the middle panels of Figure 4. +We then perform a KS test between the {gj(fmax)} val- +ues and gGWTC-3(f(m1)) and obtain the orange curves +in the middle panels of Figure 4. +From this, we find +that the KS divergences for GWTC-3 are larger than +those of the mock catalogs 15%, 10%, and 4% of the +time for the 14 M⊙, 10 M⊙, and 35 M⊙ features, respec- +tively. This means, for example, that mock catalogs can +produce perturbation function posteriors as tall as the +one inferred from GWTC-3 at ∼ 35 M⊙ only 4% of the +time. In terms of g(f), gGWTC-3(f(14 M⊙)) ̸= gj(fmax) +to 16%, gGWTC-3(f(10 M⊙)) ̸= gj(fmax) to 9%, and +gGWTC-3(f(35 M⊙)) ̸= gj(fmax) to 3%. +Though none +of these percentages are convincingly small, this indi- +cates that the orange histograms are more statistically +distinct from the black histograms in the case of the +∼ 35 M⊙ peak than they are in the cases of the features +at 10 M⊙ and 14 M⊙. +The second metric is obtained by quantifying the shift +of gGWTC-3(f(m1)) relative to the set of {gj(fmax)}. For +each point in gGWTC-3(f(m1)), we calculate the per- +centile in which it lies in each of the {gj(fmax)}, ob- +taining the orange bands in the rightmost panels of Fig- +ure 4. For comparison, we do the same for each of the +{gj(fmax)} relative to each other, constructing the grey +bands in the rightmost panels of Figure 4. We then take +the mean of the set of light orange bands and light black +bands to obtain the solid orange and solid black curves, +respectively. +The black bands serve as null distribu- +tions, so large deviations from those indicate significant +shifts. We observe a large deviation for the ∼ 35 M⊙ +peak, a moderate deviation for the ∼ 10 M⊙ peak, and +only a slight deviation for the ∼ 14 M⊙ dip. +Quan- +titatively, gGWTC-3(f(35 M⊙)) ≥ gj(fmax) to 94+6 +−80% +(90% credible interval), meaning that the ∼ 35 M⊙ +peak lies in the 94+6 +−80th percentile of the mock cata- + +8 +Farah, Edelman, et al. +−3 +−2 +−1 +0 +1 +2 +Perturbation Function +f(m1) +Mock Catalogs +GWTC-3 +5 +10 +20 +30 +40 +50 +60 +70 80 90 +m1 [M⊙] +−3 +−2 +−1 +0 +1 +2 +Perturbation Function +f(m1) +Figure 3. Median (top panel) and 90% credible interval (bottom panel) of the perturbation function resulting from the Power +Law + Spline fit to the primary masses in GWTC-3 (orange) and in 10 mock catalogs (grey). The perturbation function +multiplies a smoothed power law in primary mass to add modulations to an otherwise monotonic distribution, making it a +direct measure of deviations from a power law. It is a cubic spline with knots fixed at the locations indicated by the black +vertical tick marks. +The prior on the perturbation heights is the unit normal distribution, as can be seen below ∼ 5 M⊙ +where there are no detections to constrain the likelihood and the posterior reverts to the prior. The perturbation function +corresponding to GWTC-3 events appears large in amplitude in three locations: ∼ 10 M⊙, ∼ 14 M⊙, and ∼ 35 M⊙. While the +medians of the perturbation function at these distributions are comparable in amplitude, the posterior distribution at ∼ 35 M⊙ +(∼ 14 M⊙) is the most (least) tightly constrained. +logs’ largest perturbation heights. +For the other fea- +tures, gGWTC-3(f(10 M⊙)) ≥ gj(fmax) to 78+22 +−69% and +gGWTC-3(f(14 M⊙)) ≥ gj(fmax) to 68+31 +−61%. In compari- +son, the corresponding statistic for the null distributions +is gj(fmax) ≥ gi(fmax) to 50+47 +−46%. +It is not possible to draw firm conclusions from these +large uncertainties, especially since all features are con- +sistent with being in both the 100th and 50th percentiles +of mock catalog perturbation functions. However, the +central values indicate that the ∼ 35 M⊙ peak is notice- +ably shifted relative to the mock catalogs’ perturbation +functions, while the other features are not shifted as sig- +nificantly. +3.3.2. Inconsistency With a Power Law +The final metric we consider is inspired by the statis- +tic presented in Abbott et al. (2021a), which states that +“the inferred perturbation f(m1) strongly disfavors zero +at both the 10 M⊙ and 35 M⊙ peak.” We therefore turn +from considering the full distribution of perturbation +function heights at a given location to the percentile +at which it excludes zero. A perturbation function am- +plitude of zero is a useful reference point for several rea- +sons. The most intuitive is that it causes the population + +9 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Density +13.5 M⊙ +Mock Catalogs +GWTC-3 +0 +1 +2 +3 +4 +Density +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Cumulative Density +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Density +10.2 M⊙ +0 +1 +2 +3 +4 +Density +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Cumulative Density +0 +5 +Largest Perturbation +f(m1)|max +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Density +34.6 M⊙ +0.0 +0.2 +0.4 +0.6 +KS divergence D +0 +1 +2 +3 +4 +Density +0 +50 +100 +Percentile +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Cumulative Density +Figure 4. Three largest deviations from a power law observed in GWTC-3 compared to mock catalogs. Left column: The +posterior distribution of perturbation function heights at the location where the posterior distribution is maximal for mock +catalogs (light grey) and GWTC-3 (solid orange). Middle column: Null distribution (black) and GWTC-3 distribution (orange) +of Kolmogorov–Smirnov (KS) divergences between the individual distributions in the left column. Smaller values of the KS +divergence indicate more similar distributions. Right column: Null distribution (black) and GWTC-3 distribution (orange) of +percentiles. Large deviations from the diagonal indicate a more significant rightward shift of the GWTC-3 distribution relative +to the mock catalogs. Each row corresponds to a different local extremum for GWTC-3: m1 = 13.5 M⊙ (top), m1 = 10.2 M⊙ +(middle), and m1 = 34.6 M⊙ (bottom), while the global extrema for each mock catalog are shown in all rows, along with the +aggregated distribution across all mock catalogs (solid black). The ∼ 35 M⊙ peak is an outlier with respect to both the KS and +percentile statistics, but the other two features are more ambiguous. + +10 +Farah, Edelman, et al. +0.01 0.03 +0.1 0.2 0.5 1 +2 +5 10 20 +Zero-exclusion Percentile [%] +0 +20 +40 +60 +Density +13.5 M⊙ +10.2 M⊙ +34.6 M⊙ +Mock Catalogs +Figure 5. Percentile at which the posterior distribution of +the perturbation function excludes zero for GWTC-3 (or- +ange vertical lines) and catalogs drawn from a featureless +distribution (black histogram). +For GWTC-3, we evaluate +the perturbation function’s posterior distribution at primary +mass (m1) values of 10.2 M⊙ (dashed), 13.5 M⊙ (dotted) and +34.6 M⊙ (solid). +For mock catalogs, we find the primary +mass value at which the perturbation function is maximal +and evaluate its posterior distribution there. The values re- +ported here are the percentage of the posterior distribution +that is greater than zero at those values in m1. The 13.5 M⊙ +feature excludes zero to a level comparable to some of the +mock catalogs, but the other two features exclude zero to a +level not reproducible by any mock catalogs. +model to behave like a featureless power law, so a pos- +terior that excludes zero to high credibility indicates an +inconsistency with a power law. Zero is also the mean +of the prior predictive distribution for the perturbation +function: the prior allows for equal upwards and down- +wards fluctuations, symmetric about zero perturbation. +Similarly, a vanishing perturbation function amplitude +is the state to which we expect the posterior predictive +distribution to asymptote in the limit of infinite detec- +tions from an underlying power law distribution. +We +therefore plot the percentile at which each mock catalog +excludes zero perturbation in Figure 5 +We calculate how often a simulated catalog’s pertur- +bation function excludes zero to the same credibility as +that of GWTC-3. None of the 300 {gj(fmax)} exclude +zero to the same percentile as gGWTC-3(f(35 M⊙)) or +gGWTC-3(f(10 M⊙)), and 1.3% of the {gj(fmax)} exclude +zero to the same percentile as gGWTC-3(f(14 M⊙))3. +3The fact that gGWTC-3(f(14 M⊙)) < 0 to 0.51% but 1.3% of +mock catalogs have a similar or smaller statistic is due to the differ- +ence between Bayesian credible intervals and frequentist p-values, +and because our metric corrects for the look-elsewhere effect by +comparing GWTC-3’s perturbation function at specific locations +to all possible locations in the mock catalogs. +This, combined with the metrics presented in Sec- +tion 3.3.1, lead us to conclude that the peaks at ∼ 10 M⊙ +and ∼ 35 M⊙ are difficult to reproduce with featureless +catalogs, but it is possible that the dip at ∼ 14 M⊙ is a +large fluctuation. +In summary, even though featureless catalogs can pro- +duce perturbations as tall as the ∼ 10 M⊙ peak, they +cannot create perturbations constrained away from zero +with the same confidence. This means that the ampli- +tude of the ∼ 10 M⊙ peak can be reproduced by Pois- +son fluctuations, but its inconsistency with a power law +cannot. +The dip at ∼ 14 M⊙ could be a Poisson fluc- +tuation because fits to featureless catalogs can easily +produce perturbations as large, and can sometimes pro- +duce fluctuations as confidently constrained away from +zero perturbation. The peak at ∼ 35 M⊙ cannot be re- +produced by mock catalogs in any way: its perturbation +amplitude is too large and too confidently constrained +away from zero. +The fact that we find one of the features explainable by +Poisson noise is consistent with Section 2, which suggests +that up to two of the excursions from a power law can be +explained by Poisson fluctuations. Our conclusions are +also in broad agreement with those presented in Abbott +et al. (2021a), as they report confident detections for +the two largest peaks in the mass distribution but only +modest evidence for the dip at ∼ 14 M⊙. +4. DISCUSSION +Previous analyses of the BBH mass spectrum by the +LVK and others have found evidence for structure be- +yond a simple power law (Abbott et al. 2021; Abbott +et al. 2021a). +There has been considerable work ex- +ploring possible astrophysical causes of these identified +features. Our aim is instead to determine, from a statis- +tical viewpoint, whether astrophysical arguments need +be invoked at all. +We first demonstrate that it is only possible for up +to two of the three deviations from a power law to be +explained by Poisson noise about a single power law dis- +tribution. Therefore, at least one feature must be added +on top of a power law to describe the data. +We then perform a more thorough analysis, simulat- +ing thousands of BBHs with realistic measurement un- +certainty, selection effects, and a known underlying dis- +tribution. We fit the Power Law + Spline model to +the resulting catalogs and find that the data is incon- +sistent with a single power law, agreeing with the LVK +result. However, we find that one of the previously iden- +tified features, an underdensity at ∼ 14 M⊙, may not be +present in the true astrophysical distribution. Instead, it +may have been the result of a Poisson fluctuation around + +11 +a simple power law, or an artifact of the models used to +fit the mass spectrum. The metrics constructed in this +work differ from those previously used to assess the sig- +nificance of features in the mass distribution because, +by virtue of comparing to several simulated catalogs, +they correct for the look-elsewhere effect. This is only +in mild tension with the conclusions reached by Abbott +et al. (2021a), as they report “modest evidence” in favor +of a dip at 14 M⊙. +We find the other two previously identified peaks, at ∼ +10 M⊙ and ∼ 35 M⊙, unlikely to be the result of Poisson +noise or modeling artifacts. Simulated catalogs coming +from distributions that do not include these features can +reproduce the height of the ∼ 10 M⊙ peak, but not its +lack of support for zero perturbation. +The ∼ 35 M⊙ +peak is difficult to reproduce from featureless catalogs +in any way. +Our conclusions are consistent with a recent study by +Callister & Farr (in prep.) who fit the BBH mass distri- +bution with an autoregressive model and find that the +primary mass distribution gradually decreases as a func- +tion of mass and exhibits two local maxima but no local +minima. We also find similar results to Edelman et al. +(2022) who construct the mass distribution entirely from +basis splines and find peaks at ∼ 10 M⊙ and ∼ 35 M⊙. +The significance of the peaks near 10 M⊙ and 35 M⊙, as +well as the lack of significance of the dip near 14 M⊙, is +also in agreement with Sadiq et al. (2022) and Wong & +Cranmer (2022). +The dip near ∼ 14 M⊙ may be a large Poisson fluctua- +tion or an artifact of the models used to characterize it. +If it is in fact a feature of the underlying distribution, it +is difficult to resolve with current observations. +The peak centered on ∼ 10 M⊙ is likely an imprint of +the true astrophysical distribution, and additional struc- +ture beyond a power law is needed to explain it. How- +ever, it may either be an additional peak that is dis- +tinct from the one created by the underlying smoothed +power law at ∼ 7 M⊙ (Abbott et al. 2021a; Edelman +et al. 2022; Tiwari 2022) or the sole peak in the re- +gion between ∼ 5 M⊙ and ∼ 20 M⊙ (Edelman et al. +2022). These two possibilities can be seen in Figure 1: +the former scenario is the case where we interpret the +first two peaks in the orange band as distinct from one +another, therefore treating the global maximum inferred +by Power Law + Spline as a different feature from +the global maximum inferred by Power Law + Peak. +If the latter scenario is true, the role of the perturba- +tion function is to shift the global maximum from the +value inferred by the power law component to a slightly +higher value without removing the mass distribution’s +support for 5–10 M⊙ objects. A simple smoothed power +law, such as that employed by the Power Law + Peak +model (see grey and blue bands in Figure 1), may not be +flexible enough to place a global maximum at ∼ 10 M⊙ +while also fitting the correct slope at larger masses and +fitting the correct merger rate below ∼ 10 M⊙, so it +places its global maximum at ∼ 7 M⊙. This scenario, in +which there is a single local maximum below ∼ 12 M⊙, +is consistent with Edelman et al. (2022) and Callister & +Farr (in prep.), both of whom find only one significant +maximum between approximately 3 M⊙ and 12 M⊙ us- +ing fully non-parametric methods. If this interpretation +is correct and the global maximum of the BBH mass +distribution is indeed offset from the minimum mass by +∼ 5 M⊙, the upper edge of the lower mass gap may +not be as morphologically simple as previously assumed +(e.g., Fishbach et al. 2020; Farah et al. 2022; Ezquiaga +& Holz 2022), making it potentially difficult to resolve +with parametric models alone. +The ∼ 10 M⊙ peak could also be indicative of par- +ticular evolutionary processes that are dominant within +formation environments. van Son et al. (2022a) showed +that a global maximum near this value is consistent and +robustly predicted by the stable mass transfer channel in +isolated binary evolution, as stability during mass trans- +fer requires mass ratios between the donor star and ac- +creting compact object to be relatively symmetric, and +stellar companions to ∼ 10 M⊙ BHs must be near this +mass to form compact objects above the minimum BH +mass. This may be an indication that the stable mass +transfer channel operates more efficiently than the tradi- +tional common envelope channel for generating merging +BBHs. Though dynamical formation channels with low +escape velocities, such as globular clusters, struggle to +produce a global maximum at 10 M⊙ (Antonini et al. +2022), dynamical environments with higher escape ve- +locities may more readily produce merging BBHs with +lower masses around 10 M⊙ due to the more prevalent +lower-mass BHs preferentially remaining bound to these +clusters following supernova kicks. +We find that the peak centered on 35 M⊙ is the most +likely to be a feature of the true underlying distribu- +tion. This bodes well for the “spectral siren” method +(Farr et al. 2019; Ezquiaga & Holz 2022) of estimating +cosmological parameters from GW observations, as this +peak happens to be the most informative feature for this +method since it is a well-measured, somewhat sharp fea- +ture in the mass distribution (Abbott et al. 2021). The +astrophysical process that gives rise to this feature is still +a topic of discussion. The key reason for including a flex- +ible bump-like feature in the phenomenology of paramet- +ric models, such as the Power law + Peak model used +by the LVK (Talbot & Thrane 2018), was to accommo- + +12 +Farah, Edelman, et al. +date a potential build-up of BHs with masses just below +the pair instability mass gap, as pulsational pair insta- +bility supernovae are predicted to efficiently shed mate- +rial from high-mass stars with cores in the mass range +of Mcore ∼ 45−65 (Woosley 2017, 2019; Marchant et al. +2019; Renzo et al. 2020). It is difficult to reconcile the +locations of the local maxima found in the BBH primary +mass distribution with predictions of the pair instability +process in the cores of massive stars. The largest uncer- +tainty determining the location of the lower edge of the +pair instability mass gap is the 12C(α, γ)16O reaction +rate, which determines the abundance of oxygen in stel- +lar cores (e.g., Farmer et al. 2019). Higher 12C(α, γ)16O +reaction rates lead to a higher oxygen abundance in the +stellar core, which will ignite explosively during core col- +lapse and lead to (pulsational) pair instability super- +novae occurring at lower core masses. However, even at +3σ deviations above the median measured value of the +12C(α, γ)16O reaction rate, the lower end of the mass +gap only reaches ≈ 38 M⊙ (Farmer et al. 2020). This is +above where the measured overdensity in the observed +mass spectrum occurs. This may be an indication that +the peak at 35 M⊙ is the result of another BBH forma- +tion channel (e.g. globular clusters, see Antonini et al. +2022), or that stellar evolution models are missing par- +ticular ingredients that can shift the location of the pair +instability gap (relaxing the assumption that the explod- +ing stars are hydrogen-free, adjustments to convective +overshooting, see e.g. Iorio et al. 2022). +Additionally, several studies have suggested that the +observed peaks in the BBH mass distribution can be ex- +plained by successive generations of hierarchical mergers +(Tiwari & Fairhurst 2021; Mahapatra et al. 2022; Ti- +wari 2022), though no correlation has been detected in +the spin distribution of BBHs (Biscoveanu et al. 2022a), +which is also necessitated by the hierarchical merger for- +mation scenario (Gerosa & Berti 2017; Fishbach et al. +2017; Rodriguez et al. 2019; Kimball et al. 2020; Doc- +tor et al. 2020, 2021; Gerosa et al. 2021). +Addition- +ally, for these peaks to correspond to hierarchical merg- +ers of the same population, the dominant hierarchical +pairing would have to be the first generation BH with +a third generation BH (Mahapatra et al. 2022; Tiwari +2022), whereas the dominant pairing predicted by Ro- +driguez et al. (2019) is a first generation BH generation +with a second generation BH. While it is certainly pos- +sible that GWTC-3 contains hierarchical mergers (e.g. +Abbott et al. (2020), though also see Fishbach & Holz +(2020b)), the relative fraction of events formed this way +is likely too small to form the structure observed in the +primary mass distribution (Kimball et al. 2021), and +some fine-tuning may be needed to avoid a cluster catas- +trophe (Zevin & Holz 2022). The exact physical reason +for the overdensity at 35 M⊙ therefore remains unclear. +However, we confirm that it is a robust signature in the +observational data; future observing runs will help to +constrain its precise location, width, and possible red- +shift evolution. +5. ACKNOWLEDGEMENTS +The authors gratefully acknowledge Reed Essick for +helpful insights on injections and model systematics, as +well as Thomas Callister and Thomas Dent for useful +comments on the manuscript. A.M.F. is supported by +the National Science Foundation Graduate Research Fel- +lowship Program under Grant No. DGE-1746045. B.E +and B.F are supported in part by the National Science +Foundation under Grant PHY-2146528. +M.Z. is sup- +ported by NASA through the NASA Hubble Fellow- +ship grant HST-HF2-51474.001-A awarded by the Space +Telescope Science Institute, which is operated by the As- +sociation of Universities for Research in Astronomy, Inc., +for NASA, under contract NAS5-26555. J.M.E. is sup- +ported by the European Union’s Horizon 2020 research +and innovation program under the Marie Sklodowska- +Curie grant agreement No. 847523 INTERACTIONS, +by VILLUM FONDEN (grant no. 37766), by the Danish +Research Foundation, and under the European Union’s +H2020 ERC Advanced Grant “Black holes: +gravita- +tional engines of discovery” grant agreement no. Gravi- +tas–101052587. D.E.H is supported by NSF grants AST- +2006645 and PHY-2110507, as well as by the Kavli In- +stitute for Cosmological Physics through an endowment +from the Kavli Foundation and its founder Fred Kavli. +D.E.H also gratefully acknowledges the Marion and Stu- +art Rice Award. This material is based upon work sup- +ported by NSF LIGO Laboratory which is a major fa- +cility fully funded by the National Science Foundation. +This research has made use of data, software and/or web +tools obtained from the Gravitational Wave Open Sci- +ence Center (https://www.gw-openscience.org/), a ser- +vice of LIGO Laboratory, the LIGO Scientific Collabo- +ration and the Virgo Collaboration. +The authors are +grateful for computational resources provided by the +LIGO Laboratory and supported by National Science +Foundation Grants PHY-0757058 and PHY-0823459. +This work benefited from access to the University of +Oregon high performance computer, Talapas. +Software: +gwpopulation (Talbot et al. 2019), +bilby (Ashton et al. 2019; Romero-Shaw et al. 2020), +dynesty (Speagle 2020), numpy (Harris et al. 2020), +xarray (Hoyer & Hamman 2017), matplotlib (Hunter +2007), pandas (pandas development team 2020) + +13 +REFERENCES +Aasi, J., Abbott, B. P., Abbott, R., et al. 2015, Classical +and Quantum Gravity, 32, 074001, +doi: 10.1088/0264-9381/32/7/074001 +Abbott, B. P., et al. 2019, Phys. Rev. X, 9, 031040, +doi: 10.1103/PhysRevX.9.031040 +Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2019, +ApJL, 882, L24, doi: 10.3847/2041-8213/ab3800 +—. 2020, Living Reviews in Relativity, 23, 3, +doi: 10.1007/s41114-020-00026-9 +Abbott, R., Abbott, T. D., Abraham, S., et al. 2020, +Physical Review Letters, 125, 101102, +doi: 10.1103/PhysRevLett.125.101102 +Abbott, R., Abbott, T. D., Acernese, F., et al. 2021, arXiv +e-prints, arXiv:2111.03606. +https://arxiv.org/abs/2111.03606 +Abbott, R., Abe, H., Acernese, F., et al. 2021, Tests of +General Relativity with GWTC-3, arXiv, +doi: 10.48550/arXiv.2112.06861 +Abbott, R., Abe, H., Acernese, F., et al. 2021, arXiv +e-prints, arXiv:2111.03604. +https://arxiv.org/abs/2111.03604 +Abbott, R., Abbott, T. D., Acernese, F., et al. 2021a, The +population of merging compact binaries inferred using +gravitational waves through GWTC-3, arXiv, +doi: 10.48550/ARXIV.2111.03634 +Abbott, R., et al. 2021b, Phys. Rev. X, 11, 021053, +doi: 10.1103/PhysRevX.11.021053 +Abbott, R., Abbott, T. D., Abraham, S., et al. 2021, ApJL, +913, L7, doi: 10.3847/2041-8213/abe949 +Acernese, F., Agathos, M., Agatsuma, K., et al. 2014, +Classical and Quantum Gravity, 32, 024001, +doi: 10.1088/0264-9381/32/2/024001 +Allen, B., Anderson, W. G., Brady, P. R., Brown, D. A., & +Creighton, J. D. E. 2012, PhRvD, 85, 122006, +doi: 10.1103/PhysRevD.85.122006 +Antonini, F., Gieles, M., Dosopoulou, F., & Chattopadhyay, +D. 2022, Coalescing black hole binaries from globular +clusters: mass distributions and comparison to +gravitational wave data from GWTC-3, arXiv, +doi: 10.48550/arXiv.2208.01081 +Ashton, G., H¨ubner, M., Lasky, P. D., et al. 2019, The +Astrophysical Journal Supplement Series, 241, 27, +doi: 10.3847/1538-4365/ab06fc +Barkat, Z., Rakavy, G., & Sack, N. 1967, Physical Review +Letters, 18, 379, doi: 10.1103/PhysRevLett.18.379 +Belczynski, K., Kalogera, V., Rasio, F. A., et al. 2008, The +Astrophysical Journal Supplement Series, 174, 223, +doi: 10.1086/521026 +Belczynski, K., Heger, A., Gladysz, W., et al. 2016, A&A, +594, A97, doi: 10.1051/0004-6361/201628980 +Biscoveanu, S., Callister, T. A., Haster, C.-J., et al. 2022a, +The Astrophysical Journal, 932, L19, +doi: 10.3847/2041-8213/ac71a8 +Biscoveanu, S., Landry, P., & Vitale, S. 2022b, Monthly +Notices of the Royal Astronomical Society, +doi: 10.1093/mnras/stac3052 +Breivik, K., Coughlin, S., Zevin, M., et al. 2020, The +Astrophysical Journal, 898, 71, +doi: 10.3847/1538-4357/ab9d85 +Callister, T., & Farr, W. in prep., Flexible population +inference with autoregressive processes +Chen, H.-Y., Holz, D. E., Miller, J., et al. 2021, Classical +and Quantum Gravity, 38, 055010, +doi: 10.1088/1361-6382/abd594 +Chernoff, D. F., & Finn, L. S. 1993, The Astrophysical +Journal, 411, L5, doi: 10.1086/186898 +Doctor, Z., Farr, B., & Holz, D. E. 2021, The Astrophysical +Journal Letters, 914, L18, doi: 10.3847/2041-8213/ac0334 +Doctor, Z., Wysocki, D., O’Shaughnessy, R., Holz, D. E., & +Farr, B. 2020, The Astrophysical Journal, 893, 35, +doi: 10.3847/1538-4357/ab7fac +Eddington, A. S. 1913, MNRAS, 73, 359, +doi: 10.1093/mnras/73.5.359 +Edelman, B., Doctor, Z., & Farr, B. 2021, ApJL, 913, L23, +doi: 10.3847/2041-8213/abfdb3 +Edelman, B., Doctor, Z., Godfrey, J., & Farr, B. 2022, ApJ, +924, 101, doi: 10.3847/1538-4357/ac3667 +Edelman, B., Farr, B., & Doctor, Z. 2022. +https://ui.adsabs.harvard.edu/abs/2022arXiv221012834E +Essick, R., & Farr, W. 2022, Precision Requirements for +Monte Carlo Sums within Hierarchical Bayesian +Inference, Tech. rep. +https://ui.adsabs.harvard.edu/abs/2022arXiv220400461E +Essick, R., & Fishbach, M. 2022, LIGO-T2200210-v4: On +the consistency of parameter estimation and selection +functions in mock catalogs. +https://dcc.ligo.org/LIGO-T2200210 +Ezquiaga, J. M., & Holz, D. E. 2021, ApJL, 909, L23, +doi: 10.3847/2041-8213/abe638 +Ezquiaga, J. M., & Holz, D. E. 2022, Physical Review +Letters, 129, 061102, +doi: 10.1103/PhysRevLett.129.061102 +Farah, A., Fishbach, M., Essick, R., Holz, D. E., & +Galaudage, S. 2022, The Astrophysical Journal, 931, 108, +doi: 10.3847/1538-4357/ac5f03 + +14 +Farah, Edelman, et al. +Farmer, R., Renzo, M., de Mink, S., Fishbach, M., & +Justham, S. 2020, The Astrophysical Journal, 902, L36, +doi: 10.3847/2041-8213/abbadd +Farmer, R., Renzo, M., de Mink, S. E., Marchant, P., & +Justham, S. 2019, The Astrophysical Journal, 887, 53, +doi: 10.3847/1538-4357/ab518b +Farr, W. M. 2019, Research Notes of the AAS, 3, 66, +doi: 10.3847/2515-5172/ab1d5f +Farr, W. M., Fishbach, M., Ye, J., & Holz, D. E. 2019, The +Astrophysical Journal, 883, L42, +doi: 10.3847/2041-8213/ab4284 +Farr, W. M., Sravan, N., Cantrell, A., et al. 2011, ApJ, 741, +103, doi: 10.1088/0004-637X/741/2/103 +Finn, L. S., & Chernoff, D. F. 1993, PhRvD, 47, 2198, +doi: 10.1103/PhysRevD.47.2198 +Fishbach, M., Essick, R., & Holz, D. E. 2020, The +Astrophysical Journal, 899, L8, +doi: 10.3847/2041-8213/aba7b6 +Fishbach, M., Farr, W. M., & Holz, D. E. 2020, ApJL, 891, +L31, doi: 10.3847/2041-8213/ab77c9 +Fishbach, M., & Holz, D. E. 2017, ApJL, 851, L25, +doi: 10.3847/2041-8213/aa9bf6 +Fishbach, M., & Holz, D. E. 2020a, The Astrophysical +Journal, 891, L27, doi: 10.3847/2041-8213/ab7247 +—. 2020b, The Astrophysical Journal, 904, L26, +doi: 10.3847/2041-8213/abc827 +Fishbach, M., Holz, D. E., & Farr, B. 2017, The +Astrophysical Journal, 840, L24, +doi: 10.3847/2041-8213/aa7045 +Fishbach, M., Holz, D. E., & Farr, W. M. 2018, ApJL, 863, +L41, doi: 10.3847/2041-8213/aad800 +Fishbach, M., Doctor, Z., Callister, T., et al. 2021, The +Astrophysical Journal, 912, 98, +doi: 10.3847/1538-4357/abee11 +Fowler, W. A., & Hoyle, F. 1964, The Astrophysical +Journal Supplement Series, 9, 201, doi: 10.1086/190103 +Fryer, C. L., Belczynski, K., Wiktorowicz, G., et al. 2012, +The Astrophysical Journal, 749, 91, +doi: 10.1088/0004-637X/749/1/91 +Gerosa, D., & Berti, E. 2017, Physical Review D, 95, +124046, doi: 10.1103/PhysRevD.95.124046 +Gerosa, D., Giacobbo, N., & Vecchio, A. 2021, The +Astrophysical Journal, 915, 56, +doi: 10.3847/1538-4357/ac00bb +Harris, C. R., Millman, K. J., van der Walt, S. J., et al. +2020, Nature, 585, 357, doi: 10.1038/s41586-020-2649-2 +Heger, A., Fryer, C. L., Woosley, S. E., Langer, N., & +Hartmann, D. H. 2003, The Astrophysical Journal, 591, +288, doi: 10.1086/375341 +Heger, A., & Woosley, S. E. 2002, The Astrophysical +Journal, 567, 532, doi: 10.1086/338487 +Hoyer, S., & Hamman, J. 2017, Journal of Open Research +Software, 5, 10, doi: 10.5334/jors.148 +Hunter, J. D. 2007, Computing in Science Engineering, 9, +90, doi: 10.1109/MCSE.2007.55 +Iorio, G., Costa, G., Mapelli, M., et al. 2022, Compact +object mergers: exploring uncertainties from stellar and +binary evolution with SEVN, arXiv, +doi: 10.48550/arXiv.2211.11774 +Karathanasis, C., Mukherjee, S., & Mastrogiovanni, S. +2022a, Binary black holes population and cosmology in +new lights: Signature of PISN mass and formation +channel in GWTC-3, arXiv, +doi: 10.48550/arXiv.2204.13495 +Karathanasis, C., Revenu, B., Mukherjee, S., & Stachurski, +F. 2022b, GWSim: A python package to create GW mock +samples for different astrophysical populations and +cosmological models of binary black holes, arXiv, +doi: 10.48550/arXiv.2210.05724 +Kimball, C., Talbot, C., Berry, C. P. L., et al. 2020, The +Astrophysical Journal, 900, 177, +doi: 10.3847/1538-4357/aba518 +Kimball, C., Talbot, C., Berry, C. P. L., et al. 2021, ApJL, +915, L35, doi: 10.3847/2041-8213/ac0aef +Landry, P., & Read, J. S. 2021, The Astrophysical Journal, +921, L25, doi: 10.3847/2041-8213/ac2f3e +Li, A., Miao, Z., Han, S., & Zhang, B. 2021, The +Astrophysical Journal, 913, 27, +doi: 10.3847/1538-4357/abf355 +LIGO Scientific Collaboration, & Virgo Collaboration. +2022, GWTC-2.1: Deep Extended Catalog of Compact +Binary Coalescences Observed by LIGO and Virgo +During the First Half of the Third Observing Run - +Parameter Estimation Data Release, v2, Zenodo, +doi: 10.5281/zenodo.6513631 +LIGO Scientific Collaboration, Virgo Collaboration, & +KAGRA Collaboration. 2021a, GWTC-3: Compact +Binary Coalescences Observed by LIGO and Virgo +During the Second Part of the Third Observing Run — +O3 search sensitivity estimates, Zenodo, +doi: 10.5281/zenodo.5546676 +—. 2021b, GWTC-3: Compact Binary Coalescences +Observed by LIGO and Virgo During the Second Part of +the Third Observing Run — Parameter estimation data +release, Zenodo, doi: 10.5281/zenodo.5546663 +Loredo, T. 2009, 213, 211.04. +https://ui.adsabs.harvard.edu/abs/2009AAS...21321104L + +15 +Mahapatra, P., Gupta, A., Favata, M., Arun, K. G., & +Sathyaprakash, B. S. 2022, Black hole hierarchical growth +efficiency and mass spectrum predictions, arXiv, +doi: 10.48550/arXiv.2209.05766 +Malmquist, K. G. 1922, Meddelanden fran Lunds +Astronomiska Observatorium Serie I, 100, 1. +https://ui.adsabs.harvard.edu/abs/1922MeLuF.100....1M +Malmquist, K. G. 1925, Meddelanden fran Lunds +Astronomiska Observatorium Serie I, 106, 1. +https://ui.adsabs.harvard.edu/abs/1925MeLuF.106....1M +Mandel, I., Farr, W. M., Colonna, A., et al. 2017, Monthly +Notices of the Royal Astronomical Society, 465, 3254, +doi: 10.1093/mnras/stw2883 +Mandel, I., Farr, W. M., & Gair, J. R. 2019, Monthly +Notices of the Royal Astronomical Society, 486, 1086, +doi: 10.1093/mnras/stz896 +Mandel, I., & M¨uller, B. 2020, Monthly Notices of the +Royal Astronomical Society, 499, 3214, +doi: 10.1093/mnras/staa3043 +Marchant, P., Renzo, M., Farmer, R., et al. 2019, The +Astrophysical Journal, 882, 36, +doi: 10.3847/1538-4357/ab3426 +Messenger, C., & Read, J. 2012, Physical Review Letters, +108, 091101, doi: 10.1103/PhysRevLett.108.091101 +¨Ozel, F., Psaltis, D., Narayan, R., & McClintock, J. E. +2010, ApJ, 725, 1918, +doi: 10.1088/0004-637X/725/2/1918 +pandas development team, T. 2020, pandas-dev/pandas: +Pandas, 1.4.3, Zenodo, doi: 10.5281/zenodo.3509134 +Patton, R. A., Sukhbold, T., & Eldridge, J. J. 2022, +Monthly Notices of the Royal Astronomical Society, 511, +903, doi: 10.1093/mnras/stab3797 +Payne, E., & Thrane, E. 2022, Model exploration in +gravitational-wave astronomy with the maximum +population likelihood, arXiv, +doi: 10.48550/ARXIV.2210.11641 +Planck Collaboration, Ade, P. A. R., Aghanim, N., et al. +2016, Astronomy & Astrophysics, 594, A13, +doi: 10.1051/0004-6361/201525830 +Renzo, M., Farmer, R., Justham, S., et al. 2020, Astronomy +& Astrophysics, 640, A56, +doi: 10.1051/0004-6361/202037710 +Riley, J., Agrawal, P., Barrett, J. W., et al. 2022, The +Astrophysical Journal Supplement Series, 258, 34, +doi: 10.3847/1538-4365/ac416c +Rinaldi, S., & Del Pozzo, W. 2022, Monthly Notices of the +Royal Astronomical Society, 509, 5454, +doi: 10.1093/mnras/stab3224 +Rodriguez, C. L., Zevin, M., Amaro-Seoane, P., et al. 2019, +Physical Review D, 100, 043027, +doi: 10.1103/PhysRevD.100.043027 +Romero-Shaw, I. M., Talbot, C., Biscoveanu, S., et al. 2020, +Monthly Notices of the Royal Astronomical Society, 499, +3295–3319, doi: 10.1093/mnras/staa2850 +Sadiq, J., Dent, T., & Wysocki, D. 2022, Physical Review +D, 105, doi: 10.1103/physrevd.105.123014 +Siegel, J. C., Kiato, I., Kalogera, V., et al. 2022, +Investigating the Lower Mass Gap with Low Mass X-ray +Binary Population Synthesis, arXiv, +doi: 10.48550/arXiv.2209.06844 +Speagle, J. S. 2020, Monthly Notices of the Royal +Astronomical Society, 493, 3132, +doi: 10.1093/mnras/staa278 +Stevenson, S., Ohme, F., & Fairhurst, S. 2015, The +Astrophysical Journal, 810, 58, +doi: 10.1088/0004-637X/810/1/58 +Talbot, C., Smith, R., Thrane, E., & Poole, G. B. 2019, +Physical Review D, 100, 043030, +doi: 10.1103/PhysRevD.100.043030 +Talbot, C., & Thrane, E. 2018, The Astrophysical Journal, +856, 173, doi: 10.3847/1538-4357/aab34c +Taylor, S. R., Gair, J. R., & Mandel, I. 2012, PhRvD, 85, +023535, doi: 10.1103/PhysRevD.85.023535 +Thrane, E., & Talbot, C. 2019, Publications of the +Astronomical Society of Australia, 36, e010, +doi: 10.1017/pasa.2019.2 +Tiwari, V. 2022, The Astrophysical Journal, 928, 155, +doi: 10.3847/1538-4357/ac589a +Tiwari, V., & Fairhurst, S. 2021, ApJL, 913, L19, +doi: 10.3847/2041-8213/abfbe7 +van Son, L. A. C., de Mink, S. E., Renzo, M., et al. 2022a, +The Astrophysical Journal, 940, 184, +doi: 10.3847/1538-4357/ac9b0a +van Son, L. A. C., de Mink, S. E., Callister, T., et al. +2022b, The Astrophysical Journal, 931, 17, +doi: 10.3847/1538-4357/ac64a3 +Veitch, J., Raymond, V., Farr, B., et al. 2015, PhRvD, 91, +042003, doi: 10.1103/PhysRevD.91.042003 +Wong, K. W. K., & Cranmer, M. 2022, Automated +discovery of interpretable gravitational-wave population +models, arXiv, doi: 10.48550/arXiv.2207.12409 +Woosley, S. E. 2017, The Astrophysical Journal, 836, 244, +doi: 10.3847/1538-4357/836/2/244 +Woosley, S. E. 2019, The Astrophysical Journal, 878, 49, +doi: 10.3847/1538-4357/ab1b41 +Woosley, S. E., & Heger, A. 2015, 412, 199, +doi: 10.1007/978-3-319-09596-7 7 + +16 +Farah, Edelman, et al. +Wysocki, D., & O’Shaughnessy, R. 2021, PopModels O3a +APS April 2021 Presentation, https: +//gitlab.com/dwysocki/pop-models-o3a-aps-april-2021, +GitLab +Ye, C., & Fishbach, M. 2022, The Astrophysical Journal, +937, 73, doi: 10.3847/1538-4357/ac7f99 +Zevin, M., & Holz, D. E. 2022, ApJL, 935, L20, +doi: 10.3847/2041-8213/ac853d +Zevin, M., Pankow, C., Rodriguez, C. L., et al. 2017, The +Astrophysical Journal, 846, 82, +doi: 10.3847/1538-4357/aa8408 +Zevin, M., Spera, M., Berry, C. P. L., & Kalogera, V. 2020, +The Astrophysical Journal, 899, L1, +doi: 10.3847/2041-8213/aba74e + +17 +APPENDIX +A. GENERATION OF MOCK OBSERVATIONS IN GWMOCKCAT +We describe the process used to simulate gravitational wave event posteriors in mass and redshift, based on the +procedure developed in Fishbach et al. (2020). +This process neglects the generation of spin posteriors as this work only seeks to understand the significance of +features in the mass distribution, and individual-event likelihoods are approximately separable in spin and primary +mass for BBHs, and we do not model any spin populations in this work. However, spin and mass parameters are +not totally uncorrelated for low-mass or high mass ratio events, so future work attempting to validate features seen +in the mass ratio distribution, NSBH, or BNS populations should consider simulating spin parameters as well. A +lightweight, publicly-available python package that can reproduce these mock posteriors and generate similar catalogs +from arbitrary underlying populations and detector sensitivities is available for download and installation4. +The +package is called GWMockCat, and installation instructions, examples, and documentation are available in the git +repository. Several packages exist to draw events from BBH population models (Belczynski et al. 2008; Breivik et al. +2020; Riley et al. 2022, e.g.), some of which also simulate GW detector selection effects (Karathanasis et al. 2022b). +GWMockCat complements these by additionally simulating event-level posteriors without the need to run full parameter +estimation inference, saving significant computational time. +To create realizations of realistic catalogs that would result from a known underlying astrophysical population, +p(m1, m2, z), we first make independent draws of the event parameters, {m1, m2, z}, from that population model. +Each draw corresponds to a potential event in the catalog, although we draw many more potential events than we wish +to keep since not all events generated from the astrophysical distribution will ultimately be detected. We then convert +each event’s redshift z and source-frame component masses to a detector-frame (redshifted) chirp mass, Mdet, and +symmetric mass ratio, η. The symmetric mass ratio and source-frame chirp mass M are related to the source-frame +component masses via +η = +m1m2 +(m1 + m2)2 +(A1) +M = +(m1m2)3/5 +(m1 + m2)1/5 . +(A2) +All detector-frame masses are related to their source-frame values via Mdet = M(1 + z), where M can describe any +parameter with units of mass (e.g. m1, m2, or M). +We then utilize the basic procedure outlined in Fishbach et al. (2018) and Fishbach et al. (2020) to assign “observed” +parameters for each event, using realistic measurement uncertainty and a mock parameter estimation likelihood. +We first calculate an optimally oriented signal-to-noise ratio (SNR) ρopt from the true event parameters using a +characteristic power spectral density (PSD) of the LIGO Livingston detector in O3 (Abbott et al. 2020). ρopt is the +SNR an event would have if it were “optimally oriented” with respect to the detector, that is, directly overhead and +with its angular momentum vector pointed along the line of sight (Chen et al. 2021). In reality, GW sources have +varying sky positions and angular momentum vectors. The effect on the SNR of a source’s deviation from the optimal +orientation can be summarized by a multiplicative constant, Θ, such that +ρ = ρoptΘ, +(A3) +where Θ is between zero and unity. +GW sources are typically assumed to be distributed isotropically in sky position and orientation. +For a single +detector, this yields a corresponding distribution for Θ, described in Finn & Chernoff (1993). Therefore, for each event +i, we assign a true value ˆ +Θi drawn from this distribution and use it to calculate the event’s true single-detector SNR +ˆρ. The set of true parameters for each potential event in the catalog is then ˆθi = { +ˆ +Mdet, ˆη, ˆρ, ˆΘ}i. +4https://git.ligo.org/amanda.farah/mock-PE + +18 +Farah, Edelman, et al. +Given the true parameters, the basic procedure of generating samples from the posterior distribution of each event +is to draw an observation from each event’s likelihood, use that observation as the central value of the posterior +distribution, and then to draw samples from that posterior, assuming a prior. +To obtain observed parameters, θobs +i +, we need the likelihood, Ltotal(θobs +i +|ˆθi). We model each event’s likelihood as +Ltotal(θobs +i +|ˆθi) = LM(Mobs +det,i)Lη(ηobs +i +)LΘ(Θobs +i +)Lρ(ρobs +i +), +(A4) +where +LM +� +ln +� +Mobs +det,i +� +| ln (Mdet,i) +� += N +� +ln (Mobs +det,i)|µ = ln +� +ˆ +Mdet,i +� +, σ = σM +i +� +ρobs +i +�� +Lη +� +ηobs +i +|ηi +� += N +� +ηobs +i +|µ = ˆηi, σ = ση +i +� +ρobs +i +�� +LΘ (Θobs,i|Θi) = N +� +Θobs +i +|µ = ˆΘi, σ = σΘ +i +� +ρobs +i +�� +Lρ +� +ρobs +i +|ρi +� += N +� +ρobsi|µ = ˆρi, σ = σρ +i +� +. +(A5) +Here, N(µ, σ) is the normal distribution with mean µ and standard deviation σ. +The standard deviations are determined by assuming the uncertainties on all parameters except for the SNR scale +inversely with ρobs (Veitch et al. 2015). In stationary, Gaussian noise, we expect the matched-filter SNR in a single +detector to have unit variance (Allen et al. 2012), i.e. σρ +i = 1 for all i. We therefore draw ρobs for each event from +Lρ(ρobs +i +|ρi). This observed SNR will serve as the detection statistic that determines whether each event is observable. +We assume events that pass an SNR threshold of ρobs,i > 8 in a single detector are detected. In this way, we allow +for events near threshold to fluctuate above or below threshold, emulating the actual noise process in the detectors. +Of the events that make it through detection, we randomly select 69 of them to constitute a mock catalog with the +same number of BBHs as were analyzed by Abbott et al. (2021a). The standard deviations for Mdet, η, and Θ of the +detected events are calculated via +σM +i (ρobs +i +) = uM/ρobs +i +ση +i (ρobs +i +) = uη/ρobs +i +σΘ +i (ρobs +i +) = uΘ/ρobs +i +, +(A6) +where we have chosen uM = 0.08 M⊙, uη = 0.022, and uΘ = 0.21 to match uncertainties in these parameters typical +of events observed in O3. +Observed values for all parameters are drawn from Equation A4 with standard deviations defined in Equation A6. +With θobs +i +in hand, we are now ready to construct a posterior distribution. We apply the following priors: +π(Mdet) = U(0 M⊙, 500 M⊙) +π(η) = U(0, 0.25) +π(Θ) = U(0, 1) +π(ρ) = U(0, 300), +(A7) +where U(x1, x2) is the uniform distribution with lower bound x1 and upper bound x2. The bounds on η and Θ are +chosen because those parameters are only physically defined in the domains (0, 0.25] and [0, 1], respectively. M and +ρ are both not defined below zero, but the upper bounds were chosen somewhat arbitrarily: they must only be large +enough that the likelihood has minimal support above them. The posterior distributions for each parameter are then +Gaussians centered on the observed value, with standard deviations defined in Equation A6. They are therefore the +same as the distributions in Equation A5, but with the role of the true and observed values switched. We then simulate +multiple-dimensional posterior samples for each event by drawing 5000 independent samples5 of detector-frame chirp +5We use 5000 samples to optimize the speed of population inference while also ensuring the number of effective samples used for Monte +Carlo sums in the population inference always satisfies the criterion outlined in Farr (2019). That criterion has since been shown to be +insufficient and has been superseded by Essick & Farr (2022), but we utilize the former for consistency with the analysis performed in +Abbott et al. (2021a). However, users of the GWMockCat package can easily modify the number of posterior samples to suit their needs. + +19 +mass, symmetric mass ratio, and Θ from the posterior. Explicitly, +log Mdet,i ∼ N(µ = ln +� +Mobs +det,i +� +, σ = σM +i ) +ηi ∼ N(µ = ηobs +i +, σ = ση +i ) +Θi ∼ N(µ = Θobs +i +, σ = σΘ +i ) +ρi ∼ N(µ = ρobs +i +, σ = σρ +i ). +(A8) +Realistic correlations between other parameters such as component masses and redshift are obtained by transforming +samples in {Mdet, η, Θ, ρ}–space to {m1, m2, z}–space. When necessary, we convert between luminosity distance and +redshift using the cosmological parameters presented in Planck Collaboration et al. (2016) so as to maintain consistency +with the conventions used in (Abbott et al. 2019, 2021b; Abbott et al. 2021). +The induced prior on m1, m2, and z is therefore not uniform in those parameters. This is reasonable, so long as users +appropriately transform the prior when doing population inference on source-frame component masses and redshift. +We therefore provide a module in GWMockCat that performs these transformations. For the case of this analysis, we opt +to re-weigh the samples to a prior that is uniform in detector-frame component mass and proportional to the square +of the luminosity distance in order to mimic the priors used in the standard LVK analysis (Abbott et al. 2019, 2021b; +Abbott et al. 2021). +The fact that Equation A4 is separable up to dependence on ρobs,i means that once ρobs,i is calculated for a +given event, samples for Mdet, ηobs, Θobs, and ρobs can be drawn independently from each other. This approximate +independence is due, in part, to the fact that detector-frame chirp mass, symmetric mass ratio, SNR, and Θ are +the best-measured parameters of any compact binary coalescence signal. This fact saves considerable computational +resources, allowing for many mock event posteriors to be generated quickly on a single CPU6. +We generate sensitivity estimates along with our mock catalogs to ensure that the selection function is calculated +consistently to the event selection criteria (Essick & Fishbach 2022). To do this, we draw 2 × 107 independent samples +in m1, m2, z, and Θ from the following distribution: +p(m1, m2, z, Θ) ∝ mα +1 +�m2 +m1 +�β dVc +dz (1 + z)κ−1p(Θ), +(A9) +where we have chosen α = 2.35, β = 1.70, and κ = 2.7, and p(Θ) is the distribution described in Finn & Chernoff (1993) +which corresponds to isotropically oriented sources that are also isotropically positioned on the sky. We truncate this +distribution below m2 = 1 M⊙, above m1 = 200 M⊙, and above z = 4, and confirm that there are no mock posterior +samples outside of those ranges. We will refer to these draws as “injections.” We then calculate an optimally-oriented +SNR for each injection using the same PSD as was used for the mock observations, and compute a true SNR using +Equation A3. We emulate noise fluctuations in SNR in the same way we do for mock observations, namely by using +Equation A5, so that each injection has a corresponding observed SNR. Injections can then be subject to the same +selection criteria as our mock observations when performing a population inference (in our case, ρobs > 8). +We validate this process by constructing a mock catalog from a known distribution with fixed hyperparameters, and +then fitting the same distribution to our mock catalog, but allowing the hyperparameters to vary. We then verify that +the recovered hyperparameters are consistent with those used to generate the mock catalog. The result is shown in +Appendix B, along with additional validation studies. +B. VALIDATION OF MOCK CATALOGS +In this Appendix, we validate the process of creating mock event posteriors and catalogs from a known underlying +population outlined in Appendix A. For this process, we use the same simulated catalogs utilized in Section 3.3. The +simulated underlying population is described by pmock(m1, m2, z|Λmock), where Λmock is the set of hyperparameters +{α, δ, mmin, mmax, β, κ}, +pmock(m1, m2, z|Λmock) ∝ p(m1|α, δ, mmin, mmax)p(m2|m1, β)p(z|κ), +(B10) +6For example, a catalog of 100 events can be generated in O(10) seconds. + +20 +Farah, Edelman, et al. +Parameter +Description +Value +β +Spectral index for the power law of the mass ratio distribution. +1.70 +α +Negative spectral index for the power law of the primary mass distribution. +3.14 +mmin +Minimum mass of the primary mass distribution. +4.56 M⊙ +mmax +Maximum mass of the primary mass distribution. +81.08 M⊙ +δ +Range of mass tapering at the lower end of the mass distribution. +5.96 M⊙ +κ +Spectral index for the power law factor of the redshift distribution. +2.7 +Table 1. Hyperparameter values for the underlying population of mock catalogs described by Smoothed Power Law (Equa- +tions B10–B13). +and the individual mass and redshift distributions are given by the following: +p(m1|α, δ, mmin, mmax) ∝ +� +� +� +� +� +� +� +� +� +� +� +� +� +0 +if m < mmin +m−α +1 +1 +1+f(m−mmin,δ) +if mmin ≤ m < mmin + δ +m−α +1 +if m ≥ mmin + δ +0 +if m > mmax +, +(B11) +p(m2|m1, β) ∝ +�m2 +m1 +�β +, +(B12) +p(z|κ) ∝ +� +� +� +0 +if (z < 0) ∪ (z > zmax) +dVc +dz (1 + z)κ−1 +otherwise +. +(B13) +This is equivalent to the Power Law + Peak model in Abbott et al. (2021a) and Abbott et al. (2021), with λpeak +set to 0. We will call the population model described by Equations B10–B13 Smoothed Power Law. We generate +catalogs from the model that results from setting Λmock to the values provided in Table 1. These values were chosen +by fitting this population model to GWTC-3 (grey band in Figure 1) and obtaining the median a posteriori value for +each hyperparamter. +We validate the mock catalogs’ generation by fitting them with Smoothed Power Law and allowing the hyperpa- +rameters to be inferred from the mock data. We then determine whether the inferred values of the hyperparameters +are consistent with the values in Table 1. We fit 100 mock catalogs of 69 events each, 10 results of which are shown in +Figure 6. While there is noticeable scatter about the injected value, it is generally consistent with the recovered mass +distributions: the hyperparameters of the underlying mass distribution fall within the inferred mass hyperparameters’ +90% credible intervals 89.6% of the time. We therefore conclude that any biases that the mock posterior generation +process introduces in the mass distribution are subdominant to the statistical uncertainties of the fit. +To further explore systematic differences caused by mock catalog generation that may be subdominant to the +considerable statistical uncertainties resulting from a fit to only 69 events, we fit Smoothed Power Law to a single +catalog of that is five times larger. The result is shown in Figure 7. +The hyperparameters of the underlying distribution +seem to be consistent with the inferred hyperposterior, so we conclude that our mock event posterior generation process +produces biases subdominant to measurement uncertainty typical of 345-event catalogs. We therefore find this method +of generating mock catalogs sufficient to test the significance of features identified in the mass distribution of GWTC-3. + +21 +5 +10 +20 +30 +40 +50 +60 +70 +80 90 +m1 [M⊙] +10−4 +10−3 +10−2 +10−1 +p(m1) [M−1 +⊙ ] +Injected distribution +Inferences on mock +catalogs +−2.5 +0.0 +2.5 +5.0 +7.5 +10.0 +Mass ratio spectral index β +0.0 +0.1 +0.2 +0.3 +0.4 +Density +0 +5 +10 +Redshift spectral index κ +0.0 +0.1 +0.2 +0.3 +Density +Figure 6. Injected (Solid black line) and recovered (colored shaded bands) distributions for 10 mock catalogs. Top: probability +density function of primary masses. Bottom left: hyperposterior distribution for β, the power law spectral index of the mass +ratio distribution. Bottom Right: hyperposterior distribution for κ, the spectral index of the power law factor in the redshift +distribution. + +22 +Farah, Edelman, et al. +Figure 7. A corner plot of the inferred hyperposterior from a fit to a mock catalog with 345 events. The injected values are +shown in orange. The recovered hyperposterior is consistent with the injected population for most of the parameters. + +3.20+0.17 +3.02+1.88 +1.09 +B +4.16+0·48 +mmin +3.0 +83.432·12 +% +mmax +% +m +5 +2.68±0.65 +-0.64 +? +6 +% +3 +6 +9 +3. +? +3.6 +2.8 +1.5 +β +mmin +α +mmax +入 \ No newline at end of file diff --git a/ONAyT4oBgHgl3EQf7Pol/content/tmp_files/load_file.txt b/ONAyT4oBgHgl3EQf7Pol/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f6485906efd1c7357e4e6f9827cbeec6e6a017c0 --- /dev/null +++ b/ONAyT4oBgHgl3EQf7Pol/content/tmp_files/load_file.txt @@ -0,0 +1,1462 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf,len=1461 +page_content='Draft version January 4, 2023 Typeset using LATEX twocolumn style in AASTeX631 Things that might go bump in the night: Assessing structure in the binary black hole mass spectrum Amanda M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Farah ,1 Bruce Edelman ,2 Michael Zevin ,3, 4 Maya Fishbach ,5 Jose Mar´ıa Ezquiaga ,6 Ben Farr ,2 and Daniel E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Holz 1, 3, 4 1Department of Physics, University of Chicago, Chicago, IL 60637, USA 2Institute for Fundamental Science, Department of Physics, University of Oregon, Eugene, OR 97403, USA 3Kavli Institute for Cosmological Physics, The University of Chicago, 5640 South Ellis Avenue, Chicago, Illinois 60637, USA 4Enrico Fermi Institute, The University of Chicago, 933 East 56th Street, Chicago, Illinois 60637, USA 5Canadian Institute for Theoretical Astrophysics, David A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Dunlap Department of Astronomy and Astrophysics, and Department of Physics, 60 St George St, University of Toronto, Toronto, ON M5S 3H8, Canada 6Niels Bohr International Academy, Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen, Denmark ABSTRACT Several features in the mass spectrum of merging binary black holes (BBHs) have been identified using data from the Third Gravitational Wave Transient Catalog (GWTC-3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' These features are of particular interest as they may encode the uncertain mechanism of BBH formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' We assess if the features are statistically significant or the result of Poisson noise due to the finite number of observed events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' We simulate realistic catalogs of BBHs whose underlying distribution does not have the features of interest, apply the analysis previously performed on GWTC-3, and determine how often such features are spuriously found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' We find that two of the features found in GWTC-3, the peaks at ∼ 10 M⊙ and ∼ 35 M⊙, cannot be explained by Poisson noise alone: peaks as significant occur in < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='33% of catalogs generated from a featureless population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' These peaks are therefore likely to be of astrophysical origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' However, additional structure beyond a power law, such as the purported dip at ∼ 14 M⊙, can be explained by Poisson noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' We also provide a publicly-available package, GWMockCat, that creates simulated catalogs of BBH events with realistic measurement uncertainty and selection effects according to user-specified underlying distributions and detector sensitivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' INTRODUCTION Gravitational waves (GWs) from more than 70 merg- ers of compact objects have now been detected in the data of the LIGO (Aasi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2015) and Virgo (Ac- ernese et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2014) detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' A cumulative catalog of these events and their properties has been produced by the LIGO-Virgo-KAGRA (LVK) collaborations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' This collection of all detections to date is called the “Third Gravitational-Wave Transient Catalog” (Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2021, GWTC-3), and has enabled several insights into the nature of gravity (Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2021) , the local expansion of the universe (Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2021), and the population of GW sources (Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The underlying population of GW sources holds infor- mation about the astrophysical processes that give rise to merging binaries of compact objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The mass spec- afarah@uchicago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='edu bedelman@uoregon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='edu trum of binary black holes (BBHs), for example, encodes information about numerous physical processes underly- ing massive-star evolution, supernova physics, compact object formation, and binary interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' For exam- ple, the presence or dearth of black holes with masses between ∼ 2–5 M⊙ (¨Ozel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Farr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Fishbach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Farah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2022) may unveil the maximum neutron star mass, the stability of mass trans- fer, and the timescales relevant for the engines that drive supernova explosions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Fryer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Zevin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Mandel & M¨uller 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' van Son et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2022a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Patton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Siegel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' On the high mass end, a sharp decrease in the mass spectrum for black holes with masses ≳ 50 M⊙ (Fishbach & Holz 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Edelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2021) would be a strong indication that the pair instability process is at play and limiting the core mass of massive stars (Fowler & Hoyle 1964;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Barkat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 1967;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Heger & Woosley 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Heger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Woosley & Heger 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Belczynski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Woosley 2017, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Marchant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Renzo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='00834v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='HE] 2 Jan 2023 ID2 Farah, Edelman, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2020), with the location of the decrease in the differential merger rate acting to constrain relevant nuclear reaction rates (Farmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Other overdensities and un- derdensities in the observed mass distribution (Edelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Tiwari & Fairhurst 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Tiwari 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Edel- man et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2022), as well as the evolution of the mass distribution with redshift (Fishbach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' van Son et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2022b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Karathanasis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2022a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' van Son et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2022a), will further inform the dominant BBH formation channels, binary evolution physics, and the metallicity evolution of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' All of the parameters that are measurable from the signal of a binary merger can provide insight into for- mation mechanisms of merging binaries, especially when used in a population analysis (Stevenson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Zevin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' However, the masses of the objects in the merging system are the best measured and span the largest dynamic range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Additionally, the mass dis- tribution of compact objects can be used to measure cos- mological parameters using the “spectral siren” method, provided there is structure in the distribution beyond a boundless power law (Chernoff & Finn 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Messen- ger & Read 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Taylor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Farr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Ezquiaga & Holz 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Ezquiaga & Holz 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2021), such as edges, gaps, peaks, or changes in the power law slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Multiple features must be present to disentangle redshift evolution of the mass spectrum from cosmology, and more features further aid in break- ing this degeneracy (Ezquiaga & Holz 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' There- fore, considerable effort in the field of GW astronomy has gone towards understanding the mass distribution of GW sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' There are currently many more detected BBH mergers than binary neutron star (BNS) or neu- tron star-black hole (NSBH) mergers, so much of the activity has been on population properties of the BBH distribution, though the mass distribution of BNSs and NSBHs has also been been considered (Fishbach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Landry & Read 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Farah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Ye & Fishbach 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Biscoveanu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The BBH mass distribution is typically parameter- ized by the primary mass m1, the larger of the two component masses in the binary, and the mass ratio q = m2/m1, the ratio of the less massive object’s mass to the primary mass, though other parameterizations are possible and valid (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Farah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Fishbach & Holz 2020a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Tiwari & Fairhurst 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The community has thus far gained a robust understanding of the large- scale features of the BBH mass distribution, and is just beginning to resolve its finer details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' After the release of the First Gravitational-Wave Transient Catalog (Ab- bott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2019, GWTC-1), minimum and maximum masses at ∼ 5 M⊙ and ∼ 40 M⊙ were identified in the 10 30 50 70 90 m1 [MØ] 10°2 100 dR dm1[Gpc°3 yr°1 M°1 Ø ] Smoothed power law Power Law + Peak Power Law + Spline minimum mass maximum mass global maximum additional structure?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Distribution of primary BBH masses inferred using GWTC-3 and three different population models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The smoothed power law model (grey) consists of a single power law slope between a minimum and maximum mass, with the merger rate set to exactly zero outside of those bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' It also includes a smoothing parameter at the low-mass end that allows for an offset between the minimum BH mass and the global maximum of the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The Power Law + Peak model is similar to the smoothed power law, but also includes a Gaussian component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The Power Law + Spline model adds a cubic spline modulation to a smoothed power law to allow for additional substructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' We seek to determine if the perturbations beyond a power law found by Power Law + Spline and other semi-parametric models can be explained by random associations in the data due to a finite number of observations, or if they are features of the true underlying distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' BBH primary mass distribution, but it was not yet pos- sible to distinguish between a uniform distribution and a power law between those two bounds (Fishbach & Holz 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Talbot & Thrane 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The Second Gravitational-Wave Transient Catalog (Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2021b, GWTC-2) brought dozens of additional events, and the BBH mass distribution was found to have a global maximum at ∼ 8 M⊙ and an excess of BHs between ∼ 30 M⊙–40 M⊙ followed by a steep, al- though not infinitely sharp, drop off in the rate at higher masses extending to ∼ 80 M⊙ (instead of sharp cutoff at ∼ 40 M⊙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' At the time, there were not enough observa- tions to determine whether the mass distribution had a local maximum at ∼ 35 M⊙, represented by a Gaussian peak on top of a power law, or whether the steepening towards higher masses was better described as a break in the power law (Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' At the end of the third LIGO–Virgo observing run, the same two features at ∼ 8 M⊙ and ∼ 35 M⊙ remained, and the feature at 35 M⊙ was classified as a peak rather than a break in the power law (Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Additionally, non-parametric (Mandel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Ri- naldi & Del Pozzo 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Sadiq et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Payne & Thrane 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Edelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2022) and semi-parametric (Edelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2022) analyses found robust evidence 3 for an additional peak at ∼ 10 M⊙, the same peak at ∼ 35 M⊙, as well as modest evidence for a paucity of events near ∼ 14 M⊙ (Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' These fea- tures in the primary mass distribution correspond to similar ones in the chirp mass distribution, occurring at ∼ 9 M⊙, ∼ 11 M⊙, and ∼ 26 M⊙, respectively (Tiwari & Fairhurst 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Tiwari 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The current picture of the BBH mass distribution is therefore a decreasing power law from low to high masses, with a global maximum at m1 ∼ 10 M⊙, a potential underdensity at m1 ∼ 14 M⊙, and an overdensity at m1 ∼ 35 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' This can be see in Figure 1, where we plot the results of fitting two pa- rameteric models and one semi-parametric model to the BBHs in GWTC-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' While the existence of this substructure in the current data set appears robust, its interpretation is less clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Plausible explanations for this substructure include (1) Poisson noise, (2) modeling systematics, or (3) astro- physical signatures from one or several formation chan- nels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' We aim to disentangle the first two possibilities from the third using the Power Law + Spline model (Edelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2022), one of the semi-parametric mod- els used to identify the substructure reported in Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' (2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Poisson noise would be caused by the fact that the fiducial BBH analysis in Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' (2021a) includes only 69 events over a mass range that spans more than an order of magnitude, so observations may appear to be clumped at some masses even if the underlying distri- bution is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' We first determine if this explanation accounts for the data by simulating realistic catalogs of BBHs whose underlying distribution does not have the features of interest, applying the analysis previously performed on GWTC-3, and determining how often such features are spuriously found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' We develop several met- rics comparing observations to simulated data in order to assess the statistical significance of the “bumps” in the primary mass distribution found by Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' (2021a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Edelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' All of the metrics de- rived in this work answer the same general question: how often do we infer the existence of a feature when an- alyzing observations of a true population without that feature?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' In this sense, these metrics are analogous to frequentist p-values, as lower values correspond to more significant features in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Readers familiar with gravitational wave data analysis might find it useful to think of these metrics as false alarm rates because they quantify how often noise resembles the observed signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' A similar frequentist analysis on a large number of mock catalogs was performed by Sadiq et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' (2022) on the peak at ∼ 35 M⊙ using an adaptive kernel density es- timator (aKDE) to find features in samples drawn from featureless mass models, as well as from a model with a single peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' They account for selection effects, but not measurement uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' They find that an aKDE is able to identify peaks in the data, and that the peak at ∼ 35 M⊙ found in GWTC-2 is statistically significant within the aKDE model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The second effect mentioned above, model system- atics, could also plausibly cause spurious inference of features beyond a power law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' It is potentially con- cerning that the models considered in Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' (2021a) that find peaks and troughs in the mass dis- tribution are inherently “bumpy”: both Power Law + Peak (Talbot & Thrane 2018) and Multi source employ a smoothed power law with a Gaussian compo- nent (Wysocki & O’Shaughnessy 2021), Flexible Mix- tures is a linear combination of Gaussian components, and Power Law + Spline employs a smoothed power law under a cubic spline modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The question is then whether these “bumpy” models can recover sharp features or if they instead create peaks and troughs that are morphologically dissimilar to the true distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' This is most easily addressed by cross-checking with in- dependent models such as Broken Power Law (Ab- bott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2021a) and the auto- regressive model presented in Callister & Farr (in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Inaccuracies in the selection function are also known to cause systematic biases when inferring the underly- ing population (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Malmquist 1922, 1925).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' These bi- ases could, in principle, also cause an incorrect inference of structure in the astrophysical distribution of BBH masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' However, selection effects in GW detectors are remarkably well-characterized, so we expect this effect to be subdominant to Poisson uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' As the number of events grows, so will our accuracy in the estimation of the selection function (Farr 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Essick & Farr 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' We provide posterior samples from our simulated cata- logs in an accompanying data release, and also provide a publicly-available python package, GWMockCat, to create similar samples according to user-defined populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='1 Section 2 provides a demonstrative example: it fore- goes a full fit to the astrophysical population of sources, and compares the observed distribution of masses to pos- sible observed distributions given an underlying power law in primary mass, (incorrectly) assuming no mea- surement uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' This analysis suggests that the observed peak at ∼ 35 M⊙ is statistically significant, but that all other features beyond a simple power law might be explainable by Poisson noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' This motivates a 1The data release can be found at https://zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='org/record/ 7411991, and GWMockCat can be installed at https://git.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='ligo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='org/ amanda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='farah/mock-PE .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 4 Farah, Edelman, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' thorough study using a full hierarchical Bayesian anal- ysis on simulated event posteriors, which we carry out in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Section 4 summarizes our conclusions and discusses their implications for the astrophysical origin of the gravitational waves observed thus far by the LVK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Readers primarily interested in the significance of fea- tures in the mass distribution may wish to skip to Sec- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='3, whereas those interested in using the package GWMockCat can find details in Appendices A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' MOTIVATION To construct a simple test of feature significance and motivate further study, we first avoid a fit to the mass distribution and instead consider the observed distribu- tion of primary masses and its resemblance to one that would result from a simple power law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The observed population differs significantly from the astrophysical one, as current gravitational wave detectors are subject to selection biases that favor the detection of closer and more massive systems, as well as measurement error that affects each system differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' We construct plausible observed mass distributions that could occur from de- tecting 69 BBHs whose astrophysical distribution is a featureless power law in primary mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' To do this, we use the samples provided by LIGO Scientific Collabora- tion et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' (2021a), which were created for sensitivity es- timation for the LVK’s GWTC-3 analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Each of these samples comes with a probability of being drawn from an assumed underlying distribution and a false alarm rate (FAR) assigned by each search used by the LVK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' We can then re-weight these samples to our desired pop- ulation model (in this case, a power law in m1, q, and z) using the draw probability, and apply the same FAR threshold used in Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' (2021a) to select “found injections.” Of the ∼ 6 × 104 found injections, we re- sample to N = 104 independent sets of 69 draws each to directly compare to observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' We then histogram each set of these found injections, thereby obtaining a distribution of bin heights for our mock populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Using several thousand realizations of found injection sets enables us to construct a null dis- tribution of bin heights and characterize the effect of Poisson noise on the shape of the observed distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' We compare these null distributions with the observed distribution of BBH masses in GWTC-32 by assuming 2For all comparisons to real observations, we use the pub- licly available posterior samples for the GWTC-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='1 and GWTC- 3 data releases (LIGO Scientific Collaboration & Virgo Collabo- ration 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' LIGO Scientific Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2021b, respec- tively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' We use samples generated with the IMRPhenomXPHM waveform and a prior proportional to the square of the luminosity distances (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' the samples were not “cosmologically reweighted”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='06 p(m1) [M−1 ⊙ ] Medians of GWTC-3 BBHs Found Injections, α = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='25 Found Injections, α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='7 10 20 30 40 50 60 70 80 90 m1 [M⊙] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='0 CDF Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Observed source-frame primary mass distribu- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Black solid lines contain the median a posteriori val- ues for the binary black holes in GWTC-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Pink and blue bands indicate the 90% credible interval on the observed dis- tributions predicted from astrophysical distributions that are power laws in primary mass with spectral index α = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='25 and α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='7, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The top panel shows a histogram of observed primary masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' For GWTC-3’s distribution to be consistent with the null distributions, we expect its bin heights in the top panel to be within the 90% credi- ble intervals in 18 out of the 20 bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The uncertainties in these predicted distributions are due only to Poisson noise resulting from a finite number of observations, rather than modeling uncertainty or uncertainty in parameter estima- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Therefore, the cumulative distribution functions in the bottom panel are similar to a conventional posterior predic- tive check, but with only one source of uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The large deviations of the black curve from the shaded bands in some regions indicate the difficulty that a single power law with Poisson shot noise has in fully explaining the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' However, many of the apparent excursions from a power law are well-contained within the predicted bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' To make the most direct comparison with Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' (2021a), 5 the primary masses are measured perfectly and using the median a posteriori values of their primary masses as point estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The result is shown in the top panel of Figure 2, which plots the 90% credible interval on the observed null distributions, along with the distribution of median primary masses of GWTC-3’s BBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' For the null distributions, we consider two power law spectral in- dices as representative examples: α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='7 and α = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' These are chosen to represent a range of plausible values for the BBHs in GWTC-3: a power law fit to GWTC-3 yields α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='98+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='16 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='28, where the bounds represent 1-σ deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' To obtain a more quantitative measure, we compare bin heights from the found injections, hinj, to the bin heights of observed events in GWTC-3, hGWTC-3, ob- taining for each bin i the fraction of simulated bin heights that are lower than those of GWTC-3 BBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Explicitly, ri h = 1 N N � j � � � 1 if hj i,inj < hi,GWTC-3 0 if hj i,inj ≥ hi,GWTC-3 , (1) where the sum is over the N = 104 sets of found injec- tions, and rh is defined for each bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' For example, if rh = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='95 for a given bin, the observed distribution in that bin is larger than would be expected from a feature- less power law 95% of the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' A value of rh approach- ing unity corresponds to a “bump” in the observed mass distribution, and a value of rh approaching zero is in- dicative of a “dip.” Note that the comparison between the null distribu- tions and GWTC-3 are occurring at each bin, rather than across all bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' We do this because the magnitude of Poisson noise depends on the value of m1: since the underlying distribution is not uniform, fewer events are expected at very high m1 and therefore the relative stan- dard deviation is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' This is also a consequence of Eddington bias (Eddington 1913).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Making comparisons at specific points in m1 does not, however, properly cor- rect for the look-elsewhere effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' We will address this effect in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The three most significant values of ri h in the case of α = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='25 are r15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='6 M⊙ h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='033, r27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='9 M⊙ h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='036, r36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='1 M⊙ h > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='999, where the superscripts indicate the centers of the bins at which r was calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' This means that less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='1% of mock populations had more events near 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='1 M⊙ than GWTC-3 does, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='3% of mock populations had fewer events near 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='6 M⊙ than we keep events with secondary mass larger than 3 M⊙ and FAR less than 1 yr−1, resulting in 69 events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' GWTC-3, and at 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='9 M⊙, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='6% of mock populations had fewer events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Repeating the exercise for α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='7, we find the three most significant values of ri hto be r40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='2 M⊙ h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='935, r27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='9 M⊙ h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='020, r36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='1 M⊙ h > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The locations of the significant features differ when the assumed under- lying distribution changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' In either case, the bump at ∼ 35 M⊙ is unlikely to be due to Poisson noise, but other features may be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' To avoid the need to arbitrarily choose bins, we ad- ditionally construct a cumulative distribution function (CDF) of the primary masses and compare it to the CDFs of the null distributions, shown in the bottom panel of Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' This comparison is akin to a poste- rior predictive check in that it can highlight where the model fails to predict the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Importantly, though, it differs from the conventional posterior predictive check because we have purposefully left out the effects of mod- eling uncertainty and measurement uncertainty in order to isolate the effects of Poisson noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The prior distri- butions are therefore also not included, since each event is assumed to be measured with perfect accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' If α = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='25, the null distributions are consistent with the data below ∼ 18 M⊙ and above ∼ 35 M⊙, but not between them, meaning that the ∼ 10 M⊙ and ∼ 35 M⊙ peaks can be explained by Poisson noise, but the under- density between them could not be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' On the other hand, if α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='7, the null distributions are consistent with the data everywhere except for above ∼ 40 M⊙, suggesting that under this scenario, Poisson noise can explain all features except for the ∼ 35 M⊙ peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' For both spectral indices considered, two of the three features found by Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' (2021a) can be explained by Poisson noise from a finite number of observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' However, this does not mean that exactly two of the fea- tures are the result of Poisson noise, just that no more than two can be caused by the phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Addition- ally, it is not clear which of the features are more likely to have physical origin, as this method offers no quan- titative way to determine which power law slope is pre- ferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Importantly, this methodology does not account for the effects of measurement error, which can cause signif- icant biases near the edges of sharp distributions when not properly accounted for (Fishbach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' We therefore turn to a full hierarchical Bayesian analysis of simulated catalogs, which will allow us to fit for popula- tion model parameters, take measurement uncertainty into account, and directly compare to metrics used in Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' (2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' FULL ANALYSIS 6 Farah, Edelman, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' We determine how often the features inferred in the mass distribution of BBHs would be spuriously found in data whose underlying distribution does not have those features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' To do this, we construct a null distri- bution by simulating BBH observations that would oc- cur if the underlying astrophysical distribution was a single power law with no substructure in a finite range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The procedure for creating synthetic BBH observations is described in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Mock observations are com- bined with corresponding sensitivity estimates in a hier- archical Bayesian analysis, described in Loredo (2009);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Mandel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Thrane & Talbot (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' We an- alyze these simulations in the same way as the BBHs in GWTC-3 to determine how often the features observed in GWTC-3 would be found from an underlying distri- bution without those features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Power Law + Spline Mass Model We use the Power Law + Spline semi-parametric primary mass model as a flexible model that is easily capable of finding peaks and valleys in the mass distri- bution (Edelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' This model parameterizes perturbations or deviations from a simpler underlying distribution with flexible cubic spline functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Specifically, given an underlying hyper-prior for primary mass, p(m1|Λ), the Power Law + Spline model describes the primary mass distribution as: pspline(m1|Λ, {mi}, {fi}) ∝ p(m1|Λ) exp(f(m1|{mi}, {fi})) (2) where f(m1|{mi}, {fi}) is the function describing the perturbations, which we model with a cubic spline function interpolated by introduced hyper-parameters, {mi}, the locations of spline knots in mass space, and {fi}, the height of the perturbation function at each knot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' This describes a semi-parametric model as it in- cludes a simple “parametric” component (the underly- ing distribution) in addition to a non-parametric com- ponent that models the perturbation around the simple description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' For this study we use the simplest primary mass model for the underlying description, which is the Truncated model, describing a power law with sharp cutoffs at the lower and upper mass bounds (Fishbach & Holz 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Edelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' While this model has been shown to insufficiently describe the primary mass distribution, it captures the majority of the broadest features (Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' To assess the significance of peaks or valleys found with the Power Law + Spline model one can look at the posterior distribution of the perturbation heights as a function of mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' This tells us how far “off” the simple power law description is from accounting for the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Specifically we can find what percentile f(m1) = 0 falls in the posterior distribution as a function of mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' For data exactly distributed as a power law (the un- derlying population), the inferred perturbation function should be symmetric about 0 with widths determined by the prior distributions on the knot heights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' At masses where the percentile of zero perturbation approaches 100% (0%) we can say there is an over (under) density of events at these masses, compared to the underlying power law distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' This is identical to the analysis done by Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' (2021a), who use the percentile at which the perturbation function excludes zero at a given location as a metric for how significant a feature is at that location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Metrics of Feature Significance As described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='1, the Power Law + Spline model makes use of a perturbation function con- structed from cubic splines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The height of the pertur- bation function, f(m1), at a point in primary mass, m1, is then a direct measure of the deviation from a power law at that point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' We can determine how often one would find spurious evidence for substructure by simu- lating catalogs from a power law, fitting them with the Power Law + Spline model, and examining the re- sulting perturbation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' If the mock catalogs produce perturbation functions with similar amplitudes to those seen for GWTC-3, the structure in the GWTC-3 fit might be described by Pois- son noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' On the other hand, if the perturbation func- tions produced by fits to the mock catalogs are always lower in amplitude to that of the GWTC-3 fit, the struc- ture in the GWTC-3 data is likely to be present in the underlying distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' For a given mock catalog, we find the m1 value where the perturbation function is maximal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' We obtain the posterior distribution of perturbation function ampli- tudes at that location, g(fmax).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' We repeat this for all mock catalogs, obtaining a set of maximal perturbation function distributions, {gj(fmax)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' These are plotted in light grey on the left panels of Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The locations of the three maximal perturbation function amplitudes in the GWTC-3 fit are, from least to most significant, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='5 M⊙, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='2 M⊙, and 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='6 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The posterior distri- butions of perturbation function heights at these loca- tions are gGWTC-3(f(13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='5 M⊙)), gGWTC-3(f(10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='2 M⊙)), and gGWTC-3(f(34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='6 M⊙)), and are plotted in orange in the left panels of Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The amplitude of the per- turbation function at 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='5 M⊙ is negative (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' it is a dip rather than a bump), so we flip its distribution about zero for more direct comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The same is done for 7 all g(fmax) whose medians are negative, as the pertur- bation function’s prior is symmetric about zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Simulation Study To determine whether the features in the mass spec- trum of GWTC-3 BBHs are the result of Poisson noise of a finite number of observations drawn from a feature- less power law, we compare Power Law + Spline fits using the GWTC-3 catalog and 300 mock catalogs gen- erated from a “featureless” power law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The mock cat- alogs considered in this section are all generated from the same underlying distribution: a truncated power law in primary mass, mass ratio, and redshift, with a smoothing at low component masses to ensure the peak of the mass distribution is not in the same location as the minimum mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The explicit form of the mock cat- alogs’ population model, including values of all of its hyperparameters, can be found in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Despite knowing the parameters of the underlying population for the mock catalogs, we allow all hyperparameters to vary when fitting Power Law + Spline to the mock catalogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The resulting perturbation functions are shown in Fig- ure 3 for 10 randomly chosen mock catalogs and GWTC- 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The perturbation functions deviate from their prior distribution in the mass range where detections exist (above ∼ 5 M⊙ and below ∼ 85 M⊙), even in the case of mock catalogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' This means that the perturbation functions are informed by the mock data despite the mock data not inherently requiring a deviation from a power law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The question still remains whether the perturbation function heights inferred from mock cata- logs with no substructure are larger than those inferred from GWTC-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' While nonzero values of the perturba- tion function are common in the 10 mock catalog fits shown in Figure 3, only a few amplitudes appear com- parable in height to the three largest amplitudes of the GWTC-3 perturbation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' To verify this, we isolate the largest amplitude per- turbations for all 300 mock catalog fits and compare them to the three largest amplitude perturbations for the GWTC-3 fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' These are plotted in the leftmost pan- els of Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The light grey curves are the poste- rior distributions of largest perturbation function am- plitudes {gj(fmax)} for each simulated catalog j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' These appear to have the same general shape as one another, though with noticeable scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The orange curves in each panel are the posterior distributions of GWTC-3’s perturbation function gGWTC-3(f(m1)) at its three max- imal locations: m1 = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='5 M⊙, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='2 M⊙, and 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='6 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The distribution for the ∼ 14 M⊙ dip appears qual- itatively similar to that of the simulated catalogs, the ∼ 10 M⊙ peak appears to be slightly shifted with respect to most of the simulated catalogs but still within their range, and the ∼ 35 M⊙ peak is noticeably shifted to- wards higher values relative to the bulk of the simulated catalog distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' This suggests that the ∼ 35 M⊙ peak is unlikely to be the result of Poisson noise or mod- eling systematics, while other features could plausibly be explained by those effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Maximum Perturbation Amplitude To obtain a more quantitative measure, we derive sev- eral metrics from the distributions of maximal pertur- bation function amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The first uses the Kol- mogorov–Smirnov (KS) test: we compute the KS di- vergence D between each of the {gj(fmax)} values to obtain a null distribution of KS divergences, shown in the solid black curve in the middle panels of Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' We then perform a KS test between the {gj(fmax)} val- ues and gGWTC-3(f(m1)) and obtain the orange curves in the middle panels of Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' From this, we find that the KS divergences for GWTC-3 are larger than those of the mock catalogs 15%, 10%, and 4% of the time for the 14 M⊙, 10 M⊙, and 35 M⊙ features, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' This means, for example, that mock catalogs can produce perturbation function posteriors as tall as the one inferred from GWTC-3 at ∼ 35 M⊙ only 4% of the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' In terms of g(f), gGWTC-3(f(14 M⊙)) ̸= gj(fmax) to 16%, gGWTC-3(f(10 M⊙)) ̸= gj(fmax) to 9%, and gGWTC-3(f(35 M⊙)) ̸= gj(fmax) to 3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Though none of these percentages are convincingly small, this indi- cates that the orange histograms are more statistically distinct from the black histograms in the case of the ∼ 35 M⊙ peak than they are in the cases of the features at 10 M⊙ and 14 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The second metric is obtained by quantifying the shift of gGWTC-3(f(m1)) relative to the set of {gj(fmax)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' For each point in gGWTC-3(f(m1)), we calculate the per- centile in which it lies in each of the {gj(fmax)}, ob- taining the orange bands in the rightmost panels of Fig- ure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' For comparison, we do the same for each of the {gj(fmax)} relative to each other, constructing the grey bands in the rightmost panels of Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' We then take the mean of the set of light orange bands and light black bands to obtain the solid orange and solid black curves, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The black bands serve as null distribu- tions, so large deviations from those indicate significant shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' We observe a large deviation for the ∼ 35 M⊙ peak, a moderate deviation for the ∼ 10 M⊙ peak, and only a slight deviation for the ∼ 14 M⊙ dip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Quan- titatively, gGWTC-3(f(35 M⊙)) ≥ gj(fmax) to 94+6 −80% (90% credible interval), meaning that the ∼ 35 M⊙ peak lies in the 94+6 −80th percentile of the mock cata- 8 Farah, Edelman, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' −3 −2 −1 0 1 2 Perturbation Function f(m1) Mock Catalogs GWTC-3 5 10 20 30 40 50 60 70 80 90 m1 [M⊙] −3 −2 −1 0 1 2 Perturbation Function f(m1) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Median (top panel) and 90% credible interval (bottom panel) of the perturbation function resulting from the Power Law + Spline fit to the primary masses in GWTC-3 (orange) and in 10 mock catalogs (grey).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The perturbation function multiplies a smoothed power law in primary mass to add modulations to an otherwise monotonic distribution, making it a direct measure of deviations from a power law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' It is a cubic spline with knots fixed at the locations indicated by the black vertical tick marks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The prior on the perturbation heights is the unit normal distribution, as can be seen below ∼ 5 M⊙ where there are no detections to constrain the likelihood and the posterior reverts to the prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The perturbation function corresponding to GWTC-3 events appears large in amplitude in three locations: ∼ 10 M⊙, ∼ 14 M⊙, and ∼ 35 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' While the medians of the perturbation function at these distributions are comparable in amplitude, the posterior distribution at ∼ 35 M⊙ (∼ 14 M⊙) is the most (least) tightly constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' logs’ largest perturbation heights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' For the other fea- tures, gGWTC-3(f(10 M⊙)) ≥ gj(fmax) to 78+22 −69% and gGWTC-3(f(14 M⊙)) ≥ gj(fmax) to 68+31 −61%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' In compari- son, the corresponding statistic for the null distributions is gj(fmax) ≥ gi(fmax) to 50+47 −46%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' It is not possible to draw firm conclusions from these large uncertainties, especially since all features are con- sistent with being in both the 100th and 50th percentiles of mock catalog perturbation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' However, the central values indicate that the ∼ 35 M⊙ peak is notice- ably shifted relative to the mock catalogs’ perturbation functions, while the other features are not shifted as sig- nificantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Inconsistency With a Power Law The final metric we consider is inspired by the statis- tic presented in Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' (2021a), which states that “the inferred perturbation f(m1) strongly disfavors zero at both the 10 M⊙ and 35 M⊙ peak.” We therefore turn from considering the full distribution of perturbation function heights at a given location to the percentile at which it excludes zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' A perturbation function am- plitude of zero is a useful reference point for several rea- sons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The most intuitive is that it causes the population 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='0 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='0 Cumulative Density 0 5 Largest Perturbation f(m1)|max 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='0 Density 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='6 M⊙ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='6 KS divergence D 0 1 2 3 4 Density 0 50 100 Percentile 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='0 Cumulative Density Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Three largest deviations from a power law observed in GWTC-3 compared to mock catalogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Left column: The posterior distribution of perturbation function heights at the location where the posterior distribution is maximal for mock catalogs (light grey) and GWTC-3 (solid orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Middle column: Null distribution (black) and GWTC-3 distribution (orange) of Kolmogorov–Smirnov (KS) divergences between the individual distributions in the left column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Smaller values of the KS divergence indicate more similar distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Right column: Null distribution (black) and GWTC-3 distribution (orange) of percentiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Large deviations from the diagonal indicate a more significant rightward shift of the GWTC-3 distribution relative to the mock catalogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Each row corresponds to a different local extremum for GWTC-3: m1 = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='5 M⊙ (top), m1 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='2 M⊙ (middle), and m1 = 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='6 M⊙ (bottom), while the global extrema for each mock catalog are shown in all rows, along with the aggregated distribution across all mock catalogs (solid black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The ∼ 35 M⊙ peak is an outlier with respect to both the KS and percentile statistics, but the other two features are more ambiguous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 10 Farah, Edelman, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='5 1 2 5 10 20 Zero-exclusion Percentile [%] 0 20 40 60 Density 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='5 M⊙ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='2 M⊙ 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='6 M⊙ Mock Catalogs Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Percentile at which the posterior distribution of the perturbation function excludes zero for GWTC-3 (or- ange vertical lines) and catalogs drawn from a featureless distribution (black histogram).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' For GWTC-3, we evaluate the perturbation function’s posterior distribution at primary mass (m1) values of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='2 M⊙ (dashed), 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='5 M⊙ (dotted) and 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='6 M⊙ (solid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' For mock catalogs, we find the primary mass value at which the perturbation function is maximal and evaluate its posterior distribution there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The values re- ported here are the percentage of the posterior distribution that is greater than zero at those values in m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='5 M⊙ feature excludes zero to a level comparable to some of the mock catalogs, but the other two features exclude zero to a level not reproducible by any mock catalogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' model to behave like a featureless power law, so a pos- terior that excludes zero to high credibility indicates an inconsistency with a power law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Zero is also the mean of the prior predictive distribution for the perturbation function: the prior allows for equal upwards and down- wards fluctuations, symmetric about zero perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Similarly, a vanishing perturbation function amplitude is the state to which we expect the posterior predictive distribution to asymptote in the limit of infinite detec- tions from an underlying power law distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' We therefore plot the percentile at which each mock catalog excludes zero perturbation in Figure 5 We calculate how often a simulated catalog’s pertur- bation function excludes zero to the same credibility as that of GWTC-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' None of the 300 {gj(fmax)} exclude zero to the same percentile as gGWTC-3(f(35 M⊙)) or gGWTC-3(f(10 M⊙)), and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='3% of the {gj(fmax)} exclude zero to the same percentile as gGWTC-3(f(14 M⊙))3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 3The fact that gGWTC-3(f(14 M⊙)) < 0 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='51% but 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='3% of mock catalogs have a similar or smaller statistic is due to the differ- ence between Bayesian credible intervals and frequentist p-values, and because our metric corrects for the look-elsewhere effect by comparing GWTC-3’s perturbation function at specific locations to all possible locations in the mock catalogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' This, combined with the metrics presented in Sec- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='1, lead us to conclude that the peaks at ∼ 10 M⊙ and ∼ 35 M⊙ are difficult to reproduce with featureless catalogs, but it is possible that the dip at ∼ 14 M⊙ is a large fluctuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' In summary, even though featureless catalogs can pro- duce perturbations as tall as the ∼ 10 M⊙ peak, they cannot create perturbations constrained away from zero with the same confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' This means that the ampli- tude of the ∼ 10 M⊙ peak can be reproduced by Pois- son fluctuations, but its inconsistency with a power law cannot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The dip at ∼ 14 M⊙ could be a Poisson fluc- tuation because fits to featureless catalogs can easily produce perturbations as large, and can sometimes pro- duce fluctuations as confidently constrained away from zero perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The peak at ∼ 35 M⊙ cannot be re- produced by mock catalogs in any way: its perturbation amplitude is too large and too confidently constrained away from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The fact that we find one of the features explainable by Poisson noise is consistent with Section 2, which suggests that up to two of the excursions from a power law can be explained by Poisson fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Our conclusions are also in broad agreement with those presented in Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' (2021a), as they report confident detections for the two largest peaks in the mass distribution but only modest evidence for the dip at ∼ 14 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' DISCUSSION Previous analyses of the BBH mass spectrum by the LVK and others have found evidence for structure be- yond a simple power law (Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' There has been considerable work ex- ploring possible astrophysical causes of these identified features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Our aim is instead to determine, from a statis- tical viewpoint, whether astrophysical arguments need be invoked at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' We first demonstrate that it is only possible for up to two of the three deviations from a power law to be explained by Poisson noise about a single power law dis- tribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Therefore, at least one feature must be added on top of a power law to describe the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' We then perform a more thorough analysis, simulat- ing thousands of BBHs with realistic measurement un- certainty, selection effects, and a known underlying dis- tribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' We fit the Power Law + Spline model to the resulting catalogs and find that the data is incon- sistent with a single power law, agreeing with the LVK result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' However, we find that one of the previously iden- tified features, an underdensity at ∼ 14 M⊙, may not be present in the true astrophysical distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Instead, it may have been the result of a Poisson fluctuation around 11 a simple power law, or an artifact of the models used to fit the mass spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The metrics constructed in this work differ from those previously used to assess the sig- nificance of features in the mass distribution because, by virtue of comparing to several simulated catalogs, they correct for the look-elsewhere effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' This is only in mild tension with the conclusions reached by Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' (2021a), as they report “modest evidence” in favor of a dip at 14 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' We find the other two previously identified peaks, at ∼ 10 M⊙ and ∼ 35 M⊙, unlikely to be the result of Poisson noise or modeling artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Simulated catalogs coming from distributions that do not include these features can reproduce the height of the ∼ 10 M⊙ peak, but not its lack of support for zero perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The ∼ 35 M⊙ peak is difficult to reproduce from featureless catalogs in any way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Our conclusions are consistent with a recent study by Callister & Farr (in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=') who fit the BBH mass distri- bution with an autoregressive model and find that the primary mass distribution gradually decreases as a func- tion of mass and exhibits two local maxima but no local minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' We also find similar results to Edelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' (2022) who construct the mass distribution entirely from basis splines and find peaks at ∼ 10 M⊙ and ∼ 35 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The significance of the peaks near 10 M⊙ and 35 M⊙, as well as the lack of significance of the dip near 14 M⊙, is also in agreement with Sadiq et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' (2022) and Wong & Cranmer (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The dip near ∼ 14 M⊙ may be a large Poisson fluctua- tion or an artifact of the models used to characterize it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' If it is in fact a feature of the underlying distribution, it is difficult to resolve with current observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The peak centered on ∼ 10 M⊙ is likely an imprint of the true astrophysical distribution, and additional struc- ture beyond a power law is needed to explain it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' How- ever, it may either be an additional peak that is dis- tinct from the one created by the underlying smoothed power law at ∼ 7 M⊙ (Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Edelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Tiwari 2022) or the sole peak in the re- gion between ∼ 5 M⊙ and ∼ 20 M⊙ (Edelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' These two possibilities can be seen in Figure 1: the former scenario is the case where we interpret the first two peaks in the orange band as distinct from one another, therefore treating the global maximum inferred by Power Law + Spline as a different feature from the global maximum inferred by Power Law + Peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' If the latter scenario is true, the role of the perturba- tion function is to shift the global maximum from the value inferred by the power law component to a slightly higher value without removing the mass distribution’s support for 5–10 M⊙ objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' A simple smoothed power law, such as that employed by the Power Law + Peak model (see grey and blue bands in Figure 1), may not be flexible enough to place a global maximum at ∼ 10 M⊙ while also fitting the correct slope at larger masses and fitting the correct merger rate below ∼ 10 M⊙, so it places its global maximum at ∼ 7 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' This scenario, in which there is a single local maximum below ∼ 12 M⊙, is consistent with Edelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' (2022) and Callister & Farr (in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' ), both of whom find only one significant maximum between approximately 3 M⊙ and 12 M⊙ us- ing fully non-parametric methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' If this interpretation is correct and the global maximum of the BBH mass distribution is indeed offset from the minimum mass by ∼ 5 M⊙, the upper edge of the lower mass gap may not be as morphologically simple as previously assumed (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Fishbach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Farah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Ezquiaga & Holz 2022), making it potentially difficult to resolve with parametric models alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The ∼ 10 M⊙ peak could also be indicative of par- ticular evolutionary processes that are dominant within formation environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' van Son et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' (2022a) showed that a global maximum near this value is consistent and robustly predicted by the stable mass transfer channel in isolated binary evolution, as stability during mass trans- fer requires mass ratios between the donor star and ac- creting compact object to be relatively symmetric, and stellar companions to ∼ 10 M⊙ BHs must be near this mass to form compact objects above the minimum BH mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' This may be an indication that the stable mass transfer channel operates more efficiently than the tradi- tional common envelope channel for generating merging BBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Though dynamical formation channels with low escape velocities, such as globular clusters, struggle to produce a global maximum at 10 M⊙ (Antonini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2022), dynamical environments with higher escape ve- locities may more readily produce merging BBHs with lower masses around 10 M⊙ due to the more prevalent lower-mass BHs preferentially remaining bound to these clusters following supernova kicks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' We find that the peak centered on 35 M⊙ is the most likely to be a feature of the true underlying distribu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' This bodes well for the “spectral siren” method (Farr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Ezquiaga & Holz 2022) of estimating cosmological parameters from GW observations, as this peak happens to be the most informative feature for this method since it is a well-measured, somewhat sharp fea- ture in the mass distribution (Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The astrophysical process that gives rise to this feature is still a topic of discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The key reason for including a flex- ible bump-like feature in the phenomenology of paramet- ric models, such as the Power law + Peak model used by the LVK (Talbot & Thrane 2018), was to accommo- 12 Farah, Edelman, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' date a potential build-up of BHs with masses just below the pair instability mass gap, as pulsational pair insta- bility supernovae are predicted to efficiently shed mate- rial from high-mass stars with cores in the mass range of Mcore ∼ 45−65 (Woosley 2017, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Marchant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Renzo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' It is difficult to reconcile the locations of the local maxima found in the BBH primary mass distribution with predictions of the pair instability process in the cores of massive stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The largest uncer- tainty determining the location of the lower edge of the pair instability mass gap is the 12C(α, γ)16O reaction rate, which determines the abundance of oxygen in stel- lar cores (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Farmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Higher 12C(α, γ)16O reaction rates lead to a higher oxygen abundance in the stellar core, which will ignite explosively during core col- lapse and lead to (pulsational) pair instability super- novae occurring at lower core masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' However, even at 3σ deviations above the median measured value of the 12C(α, γ)16O reaction rate, the lower end of the mass gap only reaches ≈ 38 M⊙ (Farmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' This is above where the measured overdensity in the observed mass spectrum occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' This may be an indication that the peak at 35 M⊙ is the result of another BBH forma- tion channel (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' globular clusters, see Antonini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2022), or that stellar evolution models are missing par- ticular ingredients that can shift the location of the pair instability gap (relaxing the assumption that the explod- ing stars are hydrogen-free, adjustments to convective overshooting, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Iorio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Additionally, several studies have suggested that the observed peaks in the BBH mass distribution can be ex- plained by successive generations of hierarchical mergers (Tiwari & Fairhurst 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Mahapatra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Ti- wari 2022), though no correlation has been detected in the spin distribution of BBHs (Biscoveanu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2022a), which is also necessitated by the hierarchical merger for- mation scenario (Gerosa & Berti 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Fishbach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Rodriguez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Kimball et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Doc- tor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2020, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Gerosa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Addition- ally, for these peaks to correspond to hierarchical merg- ers of the same population, the dominant hierarchical pairing would have to be the first generation BH with a third generation BH (Mahapatra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Tiwari 2022), whereas the dominant pairing predicted by Ro- driguez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' (2019) is a first generation BH generation with a second generation BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' While it is certainly pos- sible that GWTC-3 contains hierarchical mergers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' (2020), though also see Fishbach & Holz (2020b)), the relative fraction of events formed this way is likely too small to form the structure observed in the primary mass distribution (Kimball et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2021), and some fine-tuning may be needed to avoid a cluster catas- trophe (Zevin & Holz 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The exact physical reason for the overdensity at 35 M⊙ therefore remains unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' However, we confirm that it is a robust signature in the observational data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' future observing runs will help to constrain its precise location, width, and possible red- shift evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' ACKNOWLEDGEMENTS The authors gratefully acknowledge Reed Essick for helpful insights on injections and model systematics, as well as Thomas Callister and Thomas Dent for useful comments on the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' is supported by the National Science Foundation Graduate Research Fel- lowship Program under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' DGE-1746045.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='E and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='F are supported in part by the National Science Foundation under Grant PHY-2146528.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' is sup- ported by NASA through the NASA Hubble Fellow- ship grant HST-HF2-51474.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='001-A awarded by the Space Telescope Science Institute, which is operated by the As- sociation of Universities for Research in Astronomy, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', for NASA, under contract NAS5-26555.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' is sup- ported by the European Union’s Horizon 2020 research and innovation program under the Marie Sklodowska- Curie grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 847523 INTERACTIONS, by VILLUM FONDEN (grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 37766), by the Danish Research Foundation, and under the European Union’s H2020 ERC Advanced Grant “Black holes: gravita- tional engines of discovery” grant agreement no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Gravi- tas–101052587.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='H is supported by NSF grants AST- 2006645 and PHY-2110507, as well as by the Kavli In- stitute for Cosmological Physics through an endowment from the Kavli Foundation and its founder Fred Kavli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='H also gratefully acknowledges the Marion and Stu- art Rice Award.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' This material is based upon work sup- ported by NSF LIGO Laboratory which is a major fa- cility fully funded by the National Science Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' This research has made use of data, software and/or web tools obtained from the Gravitational Wave Open Sci- ence Center (https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='gw-openscience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='org/), a ser- vice of LIGO Laboratory, the LIGO Scientific Collabo- ration and the Virgo Collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The authors are grateful for computational resources provided by the LIGO Laboratory and supported by National Science Foundation Grants PHY-0757058 and PHY-0823459.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' This work benefited from access to the University of Oregon high performance computer, Talapas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Software: gwpopulation (Talbot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2019), bilby (Ashton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Romero-Shaw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2020), dynesty (Speagle 2020), numpy (Harris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2020), xarray (Hoyer & Hamman 2017), matplotlib (Hunter 2007), pandas (pandas development team 2020) 13 REFERENCES Aasi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Abbott, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Abbott, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2015, Classical and Quantum Gravity, 32, 074001, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='1088/0264-9381/32/7/074001 Abbott, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2019, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' X, 9, 031040, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='1103/PhysRevX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='031040 Abbott, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Abbott, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Abbott, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2019, ApJL, 882, L24, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='3847/2041-8213/ab3800 —.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2020, Living Reviews in Relativity, 23, 3, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='1007/s41114-020-00026-9 Abbott, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Abbott, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Abraham, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2020, Physical Review Letters, 125, 101102, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='101102 Abbott, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Abbott, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Acernese, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2021, arXiv e-prints, arXiv:2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='03606.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='org/abs/2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='03606 Abbott, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Abe, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Acernese, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2021, Tests of General Relativity with GWTC-3, arXiv, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='48550/arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='06861 Abbott, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Abe, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Acernese, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2021, arXiv e-prints, arXiv:2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='03604.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='org/abs/2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='03604 Abbott, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Abbott, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Acernese, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2021a, The population of merging compact binaries inferred using gravitational waves through GWTC-3, arXiv, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='03634 Abbott, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2021b, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' X, 11, 021053, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='1103/PhysRevX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='021053 Abbott, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Abbott, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Abraham, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2021, ApJL, 913, L7, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='3847/2041-8213/abe949 Acernese, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Agathos, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Agatsuma, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2014, Classical and Quantum Gravity, 32, 024001, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='1088/0264-9381/32/2/024001 Allen, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Anderson, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Brady, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Brown, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', & Creighton, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2012, PhRvD, 85, 122006, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='122006 Antonini, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Gieles, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Dosopoulou, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', & Chattopadhyay, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2022, Coalescing black hole binaries from globular clusters: mass distributions and comparison to gravitational wave data from GWTC-3, arXiv, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='48550/arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='01081 Ashton, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', H¨ubner, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Lasky, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2019, The Astrophysical Journal Supplement Series, 241, 27, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='3847/1538-4365/ab06fc Barkat, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Rakavy, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', & Sack, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 1967, Physical Review Letters, 18, 379, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='379 Belczynski, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Kalogera, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Rasio, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2008, The Astrophysical Journal Supplement Series, 174, 223, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='1086/521026 Belczynski, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Heger, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Gladysz, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2016, A&A, 594, A97, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='1051/0004-6361/201628980 Biscoveanu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Callister, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Haster, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2022a, The Astrophysical Journal, 932, L19, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='3847/2041-8213/ac71a8 Biscoveanu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Landry, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', & Vitale, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2022b, Monthly Notices of the Royal Astronomical Society, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='1093/mnras/stac3052 Breivik, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Coughlin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Zevin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2020, The Astrophysical Journal, 898, 71, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='3847/1538-4357/ab9d85 Callister, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', & Farr, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Flexible population inference with autoregressive processes Chen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Holz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Miller, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2021, Classical and Quantum Gravity, 38, 055010, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='1088/1361-6382/abd594 Chernoff, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', & Finn, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 1993, The Astrophysical Journal, 411, L5, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='1086/186898 Doctor, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Farr, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', & Holz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2021, The Astrophysical Journal Letters, 914, L18, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='3847/2041-8213/ac0334 Doctor, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Wysocki, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', O’Shaughnessy, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Holz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', & Farr, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2020, The Astrophysical Journal, 893, 35, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='3847/1538-4357/ab7fac Eddington, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 1913, MNRAS, 73, 359, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='1093/mnras/73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='359 Edelman, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Doctor, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', & Farr, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2021, ApJL, 913, L23, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='3847/2041-8213/abfdb3 Edelman, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Doctor, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Godfrey, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', & Farr, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2022, ApJ, 924, 101, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='3847/1538-4357/ac3667 Edelman, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Farr, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', & Doctor, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' https://ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='adsabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='harvard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='edu/abs/2022arXiv221012834E Essick, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', & Farr, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2022, Precision Requirements for Monte Carlo Sums within Hierarchical Bayesian Inference, Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' https://ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='adsabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='harvard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='edu/abs/2022arXiv220400461E Essick, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', & Fishbach, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2022, LIGO-T2200210-v4: On the consistency of parameter estimation and selection functions in mock catalogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' https://dcc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='ligo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='org/LIGO-T2200210 Ezquiaga, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', & Holz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2021, ApJL, 909, L23, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='3847/2041-8213/abe638 Ezquiaga, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', & Holz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2022, Physical Review Letters, 129, 061102, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='061102 Farah, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Fishbach, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Essick, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Holz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', & Galaudage, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2022, The Astrophysical Journal, 931, 108, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='3847/1538-4357/ac5f03 14 Farah, Edelman, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Farmer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Renzo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', de Mink, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Fishbach, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', & Justham, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2020, The Astrophysical Journal, 902, L36, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='3847/2041-8213/abbadd Farmer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Renzo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', de Mink, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Marchant, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', & Justham, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2019, The Astrophysical Journal, 887, 53, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='3847/1538-4357/ab518b Farr, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2019, Research Notes of the AAS, 3, 66, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='3847/2515-5172/ab1d5f Farr, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Fishbach, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Ye, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', & Holz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2019, The Astrophysical Journal, 883, L42, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='3847/2041-8213/ab4284 Farr, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Sravan, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Cantrell, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2011, ApJ, 741, 103, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='1088/0004-637X/741/2/103 Finn, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', & Chernoff, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 1993, PhRvD, 47, 2198, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='2198 Fishbach, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Essick, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', & Holz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2020, The Astrophysical Journal, 899, L8, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='3847/2041-8213/aba7b6 Fishbach, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Farr, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', & Holz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2020, ApJL, 891, L31, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='3847/2041-8213/ab77c9 Fishbach, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', & Holz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2017, ApJL, 851, L25, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='3847/2041-8213/aa9bf6 Fishbach, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', & Holz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2020a, The Astrophysical Journal, 891, L27, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='3847/2041-8213/ab7247 —.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2020b, The Astrophysical Journal, 904, L26, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='3847/2041-8213/abc827 Fishbach, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Holz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', & Farr, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2017, The Astrophysical Journal, 840, L24, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='3847/2041-8213/aa7045 Fishbach, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Holz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', & Farr, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2018, ApJL, 863, L41, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='3847/2041-8213/aad800 Fishbach, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Doctor, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Callister, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2021, The Astrophysical Journal, 912, 98, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='3847/1538-4357/abee11 Fowler, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', & Hoyle, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 1964, The Astrophysical Journal Supplement Series, 9, 201, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='1086/190103 Fryer, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Belczynski, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Wiktorowicz, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2012, The Astrophysical Journal, 749, 91, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='1088/0004-637X/749/1/91 Gerosa, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', & Berti, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2017, Physical Review D, 95, 124046, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='124046 Gerosa, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Giacobbo, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', & Vecchio, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2021, The Astrophysical Journal, 915, 56, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='3847/1538-4357/ac00bb Harris, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Millman, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', van der Walt, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2020, Nature, 585, 357, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='1038/s41586-020-2649-2 Heger, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Fryer, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Woosley, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Langer, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', & Hartmann, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2003, The Astrophysical Journal, 591, 288, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='1086/375341 Heger, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', & Woosley, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2002, The Astrophysical Journal, 567, 532, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='1086/338487 Hoyer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', & Hamman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2017, Journal of Open Research Software, 5, 10, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='5334/jors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='148 Hunter, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2007, Computing in Science Engineering, 9, 90, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='1109/MCSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='55 Iorio, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Costa, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Mapelli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2022, Compact object mergers: exploring uncertainties from stellar and binary evolution with SEVN, arXiv, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='48550/arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='11774 Karathanasis, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Mukherjee, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', & Mastrogiovanni, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2022a, Binary black holes population and cosmology in new lights: Signature of PISN mass and formation channel in GWTC-3, arXiv, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='48550/arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='13495 Karathanasis, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Revenu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Mukherjee, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', & Stachurski, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2022b, GWSim: A python package to create GW mock samples for different astrophysical populations and cosmological models of binary black holes, arXiv, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='48550/arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='05724 Kimball, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Talbot, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Berry, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2020, The Astrophysical Journal, 900, 177, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='3847/1538-4357/aba518 Kimball, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Talbot, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Berry, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2021, ApJL, 915, L35, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='3847/2041-8213/ac0aef Landry, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', & Read, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2021, The Astrophysical Journal, 921, L25, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='3847/2041-8213/ac2f3e Li, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Miao, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Han, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', & Zhang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2021, The Astrophysical Journal, 913, 27, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='3847/1538-4357/abf355 LIGO Scientific Collaboration, & Virgo Collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2022, GWTC-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='1: Deep Extended Catalog of Compact Binary Coalescences Observed by LIGO and Virgo During the First Half of the Third Observing Run - Parameter Estimation Data Release, v2, Zenodo, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='6513631 LIGO Scientific Collaboration, Virgo Collaboration, & KAGRA Collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2021a, GWTC-3: Compact Binary Coalescences Observed by LIGO and Virgo During the Second Part of the Third Observing Run — O3 search sensitivity estimates, Zenodo, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='5546676 —.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2021b, GWTC-3: Compact Binary Coalescences Observed by LIGO and Virgo During the Second Part of the Third Observing Run — Parameter estimation data release, Zenodo, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='5546663 Loredo, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2009, 213, 211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' https://ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='adsabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='harvard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='edu/abs/2009AAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='21321104L 15 Mahapatra, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Gupta, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Favata, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Arun, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', & Sathyaprakash, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2022, Black hole hierarchical growth efficiency and mass spectrum predictions, arXiv, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='48550/arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='05766 Malmquist, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 1922, Meddelanden fran Lunds Astronomiska Observatorium Serie I, 100, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' https://ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='adsabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='harvard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='edu/abs/1922MeLuF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='.1M Malmquist, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 1925, Meddelanden fran Lunds Astronomiska Observatorium Serie I, 106, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' https://ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='adsabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='harvard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='edu/abs/1925MeLuF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='.1M Mandel, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Farr, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Colonna, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2017, Monthly Notices of the Royal Astronomical Society, 465, 3254, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='1093/mnras/stw2883 Mandel, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Farr, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', & Gair, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2019, Monthly Notices of the Royal Astronomical Society, 486, 1086, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='1093/mnras/stz896 Mandel, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', & M¨uller, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2020, Monthly Notices of the Royal Astronomical Society, 499, 3214, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='1093/mnras/staa3043 Marchant, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Renzo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Farmer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2019, The Astrophysical Journal, 882, 36, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='3847/1538-4357/ab3426 Messenger, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', & Read, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2012, Physical Review Letters, 108, 091101, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='091101 ¨Ozel, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Psaltis, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Narayan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', & McClintock, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2010, ApJ, 725, 1918, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='1088/0004-637X/725/2/1918 pandas development team, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2020, pandas-dev/pandas: Pandas, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='3, Zenodo, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='3509134 Patton, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Sukhbold, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', & Eldridge, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2022, Monthly Notices of the Royal Astronomical Society, 511, 903, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='1093/mnras/stab3797 Payne, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', & Thrane, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2022, Model exploration in gravitational-wave astronomy with the maximum population likelihood, arXiv, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='11641 Planck Collaboration, Ade, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Aghanim, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2016, Astronomy & Astrophysics, 594, A13, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='1051/0004-6361/201525830 Renzo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Farmer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Justham, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2020, Astronomy & Astrophysics, 640, A56, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='1051/0004-6361/202037710 Riley, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Agrawal, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Barrett, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2022, The Astrophysical Journal Supplement Series, 258, 34, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='3847/1538-4365/ac416c Rinaldi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', & Del Pozzo, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2022, Monthly Notices of the Royal Astronomical Society, 509, 5454, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='1093/mnras/stab3224 Rodriguez, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Zevin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Amaro-Seoane, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2019, Physical Review D, 100, 043027, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='043027 Romero-Shaw, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Talbot, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Biscoveanu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2020, Monthly Notices of the Royal Astronomical Society, 499, 3295–3319, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='1093/mnras/staa2850 Sadiq, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Dent, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', & Wysocki, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2022, Physical Review D, 105, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='1103/physrevd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='123014 Siegel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Kiato, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Kalogera, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2022, Investigating the Lower Mass Gap with Low Mass X-ray Binary Population Synthesis, arXiv, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='48550/arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='06844 Speagle, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2020, Monthly Notices of the Royal Astronomical Society, 493, 3132, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='1093/mnras/staa278 Stevenson, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Ohme, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', & Fairhurst, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2015, The Astrophysical Journal, 810, 58, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='1088/0004-637X/810/1/58 Talbot, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Smith, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Thrane, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', & Poole, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2019, Physical Review D, 100, 043030, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='043030 Talbot, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', & Thrane, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2018, The Astrophysical Journal, 856, 173, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='3847/1538-4357/aab34c Taylor, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Gair, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', & Mandel, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2012, PhRvD, 85, 023535, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='023535 Thrane, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', & Talbot, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2019, Publications of the Astronomical Society of Australia, 36, e010, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='1017/pasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='2 Tiwari, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2022, The Astrophysical Journal, 928, 155, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='3847/1538-4357/ac589a Tiwari, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', & Fairhurst, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2021, ApJL, 913, L19, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='3847/2041-8213/abfbe7 van Son, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', de Mink, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Renzo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2022a, The Astrophysical Journal, 940, 184, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='3847/1538-4357/ac9b0a van Son, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', de Mink, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Callister, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2022b, The Astrophysical Journal, 931, 17, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='3847/1538-4357/ac64a3 Veitch, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Raymond, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Farr, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2015, PhRvD, 91, 042003, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='042003 Wong, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', & Cranmer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2022, Automated discovery of interpretable gravitational-wave population models, arXiv, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='48550/arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='12409 Woosley, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2017, The Astrophysical Journal, 836, 244, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='3847/1538-4357/836/2/244 Woosley, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2019, The Astrophysical Journal, 878, 49, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='3847/1538-4357/ab1b41 Woosley, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', & Heger, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2015, 412, 199, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='1007/978-3-319-09596-7 7 16 Farah, Edelman, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Wysocki, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', & O’Shaughnessy, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2021, PopModels O3a APS April 2021 Presentation, https: //gitlab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='com/dwysocki/pop-models-o3a-aps-april-2021, GitLab Ye, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', & Fishbach, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2022, The Astrophysical Journal, 937, 73, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='3847/1538-4357/ac7f99 Zevin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', & Holz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2022, ApJL, 935, L20, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='3847/2041-8213/ac853d Zevin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Pankow, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Rodriguez, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2017, The Astrophysical Journal, 846, 82, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='3847/1538-4357/aa8408 Zevin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Spera, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', Berry, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=', & Kalogera, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2020, The Astrophysical Journal, 899, L1, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='3847/2041-8213/aba74e 17 APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' GENERATION OF MOCK OBSERVATIONS IN GWMOCKCAT We describe the process used to simulate gravitational wave event posteriors in mass and redshift, based on the procedure developed in Fishbach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' This process neglects the generation of spin posteriors as this work only seeks to understand the significance of features in the mass distribution, and individual-event likelihoods are approximately separable in spin and primary mass for BBHs, and we do not model any spin populations in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' However, spin and mass parameters are not totally uncorrelated for low-mass or high mass ratio events, so future work attempting to validate features seen in the mass ratio distribution, NSBH, or BNS populations should consider simulating spin parameters as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' A lightweight, publicly-available python package that can reproduce these mock posteriors and generate similar catalogs from arbitrary underlying populations and detector sensitivities is available for download and installation4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The package is called GWMockCat, and installation instructions, examples, and documentation are available in the git repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Several packages exist to draw events from BBH population models (Belczynski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Breivik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Riley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2022, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' ), some of which also simulate GW detector selection effects (Karathanasis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' GWMockCat complements these by additionally simulating event-level posteriors without the need to run full parameter estimation inference, saving significant computational time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' To create realizations of realistic catalogs that would result from a known underlying astrophysical population, p(m1, m2, z), we first make independent draws of the event parameters, {m1, m2, z}, from that population model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Each draw corresponds to a potential event in the catalog, although we draw many more potential events than we wish to keep since not all events generated from the astrophysical distribution will ultimately be detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' We then convert each event’s redshift z and source-frame component masses to a detector-frame (redshifted) chirp mass, Mdet, and symmetric mass ratio, η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The symmetric mass ratio and source-frame chirp mass M are related to the source-frame component masses via η = m1m2 (m1 + m2)2 (A1) M = (m1m2)3/5 (m1 + m2)1/5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' (A2) All detector-frame masses are related to their source-frame values via Mdet = M(1 + z), where M can describe any parameter with units of mass (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' m1, m2, or M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' We then utilize the basic procedure outlined in Fishbach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' (2018) and Fishbach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' (2020) to assign “observed” parameters for each event, using realistic measurement uncertainty and a mock parameter estimation likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' We first calculate an optimally oriented signal-to-noise ratio (SNR) ρopt from the true event parameters using a characteristic power spectral density (PSD) of the LIGO Livingston detector in O3 (Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' ρopt is the SNR an event would have if it were “optimally oriented” with respect to the detector, that is, directly overhead and with its angular momentum vector pointed along the line of sight (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' In reality, GW sources have varying sky positions and angular momentum vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The effect on the SNR of a source’s deviation from the optimal orientation can be summarized by a multiplicative constant, Θ, such that ρ = ρoptΘ, (A3) where Θ is between zero and unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' GW sources are typically assumed to be distributed isotropically in sky position and orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' For a single detector, this yields a corresponding distribution for Θ, described in Finn & Chernoff (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Therefore, for each event i, we assign a true value ˆ Θi drawn from this distribution and use it to calculate the event’s true single-detector SNR ˆρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The set of true parameters for each potential event in the catalog is then ˆθi = { ˆ Mdet, ˆη, ˆρ, ˆΘ}i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 4https://git.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='ligo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='org/amanda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='farah/mock-PE 18 Farah, Edelman, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Given the true parameters, the basic procedure of generating samples from the posterior distribution of each event is to draw an observation from each event’s likelihood, use that observation as the central value of the posterior distribution, and then to draw samples from that posterior, assuming a prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' To obtain observed parameters, θobs i , we need the likelihood, Ltotal(θobs i |ˆθi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' We model each event’s likelihood as Ltotal(θobs i |ˆθi) = LM(Mobs det,i)Lη(ηobs i )LΘ(Θobs i )Lρ(ρobs i ), (A4) where LM � ln � Mobs det,i � | ln (Mdet,i) � = N � ln (Mobs det,i)|µ = ln � ˆ Mdet,i � , σ = σM i � ρobs i �� Lη � ηobs i |ηi � = N � ηobs i |µ = ˆηi, σ = ση i � ρobs i �� LΘ (Θobs,i|Θi) = N � Θobs i |µ = ˆΘi, σ = σΘ i � ρobs i �� Lρ � ρobs i |ρi � = N � ρobsi|µ = ˆρi, σ = σρ i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' (A5) Here, N(µ, σ) is the normal distribution with mean µ and standard deviation σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The standard deviations are determined by assuming the uncertainties on all parameters except for the SNR scale inversely with ρobs (Veitch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' In stationary, Gaussian noise, we expect the matched-filter SNR in a single detector to have unit variance (Allen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2012), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' σρ i = 1 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' We therefore draw ρobs for each event from Lρ(ρobs i |ρi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' This observed SNR will serve as the detection statistic that determines whether each event is observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' We assume events that pass an SNR threshold of ρobs,i > 8 in a single detector are detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' In this way, we allow for events near threshold to fluctuate above or below threshold, emulating the actual noise process in the detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Of the events that make it through detection, we randomly select 69 of them to constitute a mock catalog with the same number of BBHs as were analyzed by Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' (2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The standard deviations for Mdet, η, and Θ of the detected events are calculated via σM i (ρobs i ) = uM/ρobs i ση i (ρobs i ) = uη/ρobs i σΘ i (ρobs i ) = uΘ/ρobs i , (A6) where we have chosen uM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='08 M⊙, uη = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='022, and uΘ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='21 to match uncertainties in these parameters typical of events observed in O3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Observed values for all parameters are drawn from Equation A4 with standard deviations defined in Equation A6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' With θobs i in hand, we are now ready to construct a posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' We apply the following priors: π(Mdet) = U(0 M⊙, 500 M⊙) π(η) = U(0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='25) π(Θ) = U(0, 1) π(ρ) = U(0, 300), (A7) where U(x1, x2) is the uniform distribution with lower bound x1 and upper bound x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The bounds on η and Θ are chosen because those parameters are only physically defined in the domains (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='25] and [0, 1], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' M and ρ are both not defined below zero, but the upper bounds were chosen somewhat arbitrarily: they must only be large enough that the likelihood has minimal support above them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The posterior distributions for each parameter are then Gaussians centered on the observed value, with standard deviations defined in Equation A6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' They are therefore the same as the distributions in Equation A5, but with the role of the true and observed values switched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' We then simulate multiple-dimensional posterior samples for each event by drawing 5000 independent samples5 of detector-frame chirp 5We use 5000 samples to optimize the speed of population inference while also ensuring the number of effective samples used for Monte Carlo sums in the population inference always satisfies the criterion outlined in Farr (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' That criterion has since been shown to be insufficient and has been superseded by Essick & Farr (2022), but we utilize the former for consistency with the analysis performed in Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' (2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' However, users of the GWMockCat package can easily modify the number of posterior samples to suit their needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 19 mass, symmetric mass ratio, and Θ from the posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Explicitly, log Mdet,i ∼ N(µ = ln � Mobs det,i � , σ = σM i ) ηi ∼ N(µ = ηobs i , σ = ση i ) Θi ∼ N(µ = Θobs i , σ = σΘ i ) ρi ∼ N(µ = ρobs i , σ = σρ i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' (A8) Realistic correlations between other parameters such as component masses and redshift are obtained by transforming samples in {Mdet, η, Θ, ρ}–space to {m1, m2, z}–space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' When necessary, we convert between luminosity distance and redshift using the cosmological parameters presented in Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' (2016) so as to maintain consistency with the conventions used in (Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2019, 2021b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The induced prior on m1, m2, and z is therefore not uniform in those parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' This is reasonable, so long as users appropriately transform the prior when doing population inference on source-frame component masses and redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' We therefore provide a module in GWMockCat that performs these transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' For the case of this analysis, we opt to re-weigh the samples to a prior that is uniform in detector-frame component mass and proportional to the square of the luminosity distance in order to mimic the priors used in the standard LVK analysis (Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2019, 2021b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The fact that Equation A4 is separable up to dependence on ρobs,i means that once ρobs,i is calculated for a given event, samples for Mdet, ηobs, Θobs, and ρobs can be drawn independently from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' This approximate independence is due, in part, to the fact that detector-frame chirp mass, symmetric mass ratio, SNR, and Θ are the best-measured parameters of any compact binary coalescence signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' This fact saves considerable computational resources, allowing for many mock event posteriors to be generated quickly on a single CPU6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' We generate sensitivity estimates along with our mock catalogs to ensure that the selection function is calculated consistently to the event selection criteria (Essick & Fishbach 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' To do this, we draw 2 × 107 independent samples in m1, m2, z, and Θ from the following distribution: p(m1, m2, z, Θ) ∝ mα 1 �m2 m1 �β dVc dz (1 + z)κ−1p(Θ), (A9) where we have chosen α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='35, β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='70, and κ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='7, and p(Θ) is the distribution described in Finn & Chernoff (1993) which corresponds to isotropically oriented sources that are also isotropically positioned on the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' We truncate this distribution below m2 = 1 M⊙, above m1 = 200 M⊙, and above z = 4, and confirm that there are no mock posterior samples outside of those ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' We will refer to these draws as “injections.” We then calculate an optimally-oriented SNR for each injection using the same PSD as was used for the mock observations, and compute a true SNR using Equation A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' We emulate noise fluctuations in SNR in the same way we do for mock observations, namely by using Equation A5, so that each injection has a corresponding observed SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Injections can then be subject to the same selection criteria as our mock observations when performing a population inference (in our case, ρobs > 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' We validate this process by constructing a mock catalog from a known distribution with fixed hyperparameters, and then fitting the same distribution to our mock catalog, but allowing the hyperparameters to vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' We then verify that the recovered hyperparameters are consistent with those used to generate the mock catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The result is shown in Appendix B, along with additional validation studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' VALIDATION OF MOCK CATALOGS In this Appendix, we validate the process of creating mock event posteriors and catalogs from a known underlying population outlined in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' For this process, we use the same simulated catalogs utilized in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The simulated underlying population is described by pmock(m1, m2, z|Λmock), where Λmock is the set of hyperparameters {α, δ, mmin, mmax, β, κ}, pmock(m1, m2, z|Λmock) ∝ p(m1|α, δ, mmin, mmax)p(m2|m1, β)p(z|κ), (B10) 6For example, a catalog of 100 events can be generated in O(10) seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 20 Farah, Edelman, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Parameter Description Value β Spectral index for the power law of the mass ratio distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='70 α Negative spectral index for the power law of the primary mass distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='14 mmin Minimum mass of the primary mass distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='56 M⊙ mmax Maximum mass of the primary mass distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='08 M⊙ δ Range of mass tapering at the lower end of the mass distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='96 M⊙ κ Spectral index for the power law factor of the redshift distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='7 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Hyperparameter values for the underlying population of mock catalogs described by Smoothed Power Law (Equa- tions B10–B13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' and the individual mass and redshift distributions are given by the following: p(m1|α, δ, mmin, mmax) ∝ � � � � � � � � � � � � � 0 if m < mmin m−α 1 1 1+f(m−mmin,δ) if mmin ≤ m < mmin + δ m−α 1 if m ≥ mmin + δ 0 if m > mmax , (B11) p(m2|m1, β) ∝ �m2 m1 �β , (B12) p(z|κ) ∝ � � � 0 if (z < 0) ∪ (z > zmax) dVc dz (1 + z)κ−1 otherwise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' (B13) This is equivalent to the Power Law + Peak model in Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' (2021a) and Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' (2021), with λpeak set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' We will call the population model described by Equations B10–B13 Smoothed Power Law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' We generate catalogs from the model that results from setting Λmock to the values provided in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' These values were chosen by fitting this population model to GWTC-3 (grey band in Figure 1) and obtaining the median a posteriori value for each hyperparamter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' We validate the mock catalogs’ generation by fitting them with Smoothed Power Law and allowing the hyperpa- rameters to be inferred from the mock data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' We then determine whether the inferred values of the hyperparameters are consistent with the values in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' We fit 100 mock catalogs of 69 events each, 10 results of which are shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' While there is noticeable scatter about the injected value, it is generally consistent with the recovered mass distributions: the hyperparameters of the underlying mass distribution fall within the inferred mass hyperparameters’ 90% credible intervals 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='6% of the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' We therefore conclude that any biases that the mock posterior generation process introduces in the mass distribution are subdominant to the statistical uncertainties of the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' To further explore systematic differences caused by mock catalog generation that may be subdominant to the considerable statistical uncertainties resulting from a fit to only 69 events, we fit Smoothed Power Law to a single catalog of that is five times larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The result is shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The hyperparameters of the underlying distribution seem to be consistent with the inferred hyperposterior, so we conclude that our mock event posterior generation process produces biases subdominant to measurement uncertainty typical of 345-event catalogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' We therefore find this method of generating mock catalogs sufficient to test the significance of features identified in the mass distribution of GWTC-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 21 5 10 20 30 40 50 60 70 80 90 m1 [M⊙] 10−4 10−3 10−2 10−1 p(m1) [M−1 ⊙ ] Injected distribution Inferences on mock catalogs −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='0 Mass ratio spectral index β 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='4 Density 0 5 10 Redshift spectral index κ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content='3 Density Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Injected (Solid black line) and recovered (colored shaded bands) distributions for 10 mock catalogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Top: probability density function of primary masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Bottom left: hyperposterior distribution for β, the power law spectral index of the mass ratio distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Bottom Right: hyperposterior distribution for κ, the spectral index of the power law factor in the redshift distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' 22 Farah, Edelman, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' A corner plot of the inferred hyperposterior from a fit to a mock catalog with 345 events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The injected values are shown in orange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQf7Pol/content/2301.00834v1.pdf'} +page_content=' The recovered hyperposterior is consistent with the injected population for most of the 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Rieger𝑎,𝑏 +𝑎Institute for Theoretical Physics, Heidelberg University, Philosophenweg 12, 69120 Heidelberg, Germany +𝑏Max-Planck-Institute for Nuclear Physics (MPIK), P.O. Box 103980, 69029 Heidelberg, Germany +E-mail: f.rieger@uni-heidelberg.de +The last decade has seen tremendous developments in gamma-ray astronomy with the extragalactic +sky becoming highly populated by Active Galactic Nuclei (AGNs). This brief review highlights +some of the progress in AGN research achieved over the years, and discusses exemplary advances in +the theory and physics of gamma-ray loud AGNs, including black-hole magnetospheric processes, +the physics of pc-scales jets, as well as particle acceleration and high-energy emission in the +large-scale jets of AGNs. +7th Heidelberg International Symposium on High-Energy Gamma-Ray Astronomy (Gamma2022) +4-8 July 2022 +Barcelona, Spain +© Copyright owned by the author(s) under the terms of the Creative Commons +Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). +https://pos.sissa.it/ +arXiv:2301.13494v1 [astro-ph.HE] 31 Jan 2023 + +Theory of Gamma-Ray Loud AGNs +1. +Introduction +Radio-loud or jetted AGNs [1], i.e. AGNs with strong relativistic jets, are the most persistent, +powerful sources in the Universe. Their detection at gamma-ray energies reveals that a significant +amount of their power is deposited a highest energies, and demonstrates them to be exceptional +cosmic particle accelerators [2]. All types of jetted AGNs have been seen at gamma-ray energies, +from more misaligned radio galaxies to blazar-type sources where the jets are seen face-on [3–5]. +Gamma-ray astrophysics is thus expected to play a fundamental role in resolving key issues in AGN +physics, such as: (i) How are relativistic jets being formed? Are they preferentially ergospheric or +disk-driven? If the former, what is their plasma source; if the latter, what is their accretion-disk +connection? (ii) What makes jets radiate? What is the dominant gamma-ray radiation process and +particle acceleration mechanism? Where is it located? What is the plasma composition? (iii) How +are small and large scales connected? How is the energy transported from the black hole (BH) to +the outer lobes scales? How are jets confined? What kind of instabilities are important? +2. +On the Extragalactic Gamma-Ray Sky +The extragalactic sky has become bright at gamma-ray energies. The Fermi-LAT 12 year point +source catalog (4FGL-DR3) for example, reports the detection of 6658 sources at high energies +(HE) > 50 MeV, with more than 3740 identified as belonging to the blazar class of AGNs, cf. Fig. 1. +About 70 sources have been identified as non-blazar AGNs, out of which 45 are radio galaxies +Figure 1: High-energy sky map (4FGL) based on 8 yr of Fermi-LAT data. All AGNs (all classes) are plotted +with the same blue symbol. From ref. [6]. ©AAS. Reproduced with permission. +(among them 22 Fanaroff-Riley (FR) I and 14 FR II) and 8 are Narrow Line Seyfert 1 (NLSy 1) +galaxies [6, 7]. At very high energies (VHE; ≥ 100 GeV), about 85 AGNs are listed in the TeVCat +catalog, with redshifts up to 𝑧 ≃ 1. While most of the TeV sources are BL Lac objects (55 HBL, +10 IBL, 2 LBL), a few prominent radio galaxies (e.g., Cen A, M87 and NGC 1275) are present as +well. The latter sources have raised considerable interest by allowing insights into the immediate +vicinity of the supermassive BH environment in AGNs (cf. Sec. 4.1 below). For a recent review of +2 + +No association +回 +PossibleassociationwithSNRorPwN +AGN +Pulsar +Globular cluster +Starburst Galaxy +PWN +A +Binary +Galaxy +SNR ++ +Nova +Star-forming region +回 +UnclassifiedsourceTheory of Gamma-Ray Loud AGNs +the physics case of gamma-ray emitting, non-blazar AGNs, the reader is referred to ref. [8]. +Spectral and timing capabilities at gamma-ray energies have significantly developed over time, +allowing, in some sources, to probe rapid variability down to a few minutes and unusual spectral +features that could signal new physical processes, see e.g., Fig. 2. +Figure 2: Left: The VHE SED of Mkn 501 (𝑧 = 0.034) during an elevated state in 2014 July 19 as measured +by MAGIC, revealing an unusual feature at ∼ 3 TeV. From ref. [9], reproduced with permission © ESO. +Right: The VHE light curve of PKS 2155-304 (𝑧 = 0.116) during its famous outburst in July 2006 as seen +by H.E.S.S., revealing substantial flux changes down to ∼ 3 min. From ref. [10]. +The detection of NLSy 1 galaxies at HE energies has raised several issues. Radio-quiet (non- +jetted) NLSy 1 are commonly thought to be high-Eddington sources, with moderate BH masses +(∼ 106−8𝑀⊙), which are hosted by spiral galaxies [11]. Since HE-emitting (radio-loud) NLSy 1 +appear to be of the jetted AGN type, showing some blazar-like properties such as one-sided jets and +superluminal motion, this has triggered discussion about the BH mass limit and accretion/merger +history conducive for the origin and formation of relativistic jets. Relevant to this discussion is the +question, whether jetted NLSy 1 might belong to a special subclass harbouring BHs with larger +masses and hosted by elliptical galaxies instead. There are growing indications, however, that many +of them are better modelled as disk-like galaxies [12]. For a discussion of these topics and related +literature, the reader is referred to the recent overviews in refs. [13, 14]. +3. +On Challenges and Progress in (jetted) AGN Physics +3.1 Challenges in AGN Physics +Understanding the physics of AGN jets and the role played by its supermassive BH is a +challenging, multi-scale problem. Phenomenologically, the observed scale separation covers a +range of almost ten orders (as in, e.g., Cen A) of magnitude, see e.g. Fig. 3. This fact translates +into a fundamental physics and modelling challenge of how to consistently bridge these different +scales, e.g., of how to connect the global, source dynamical scale, the radiation (e.g., synchrotron, +inverse Compton) scale and the dissipation (e.g., shock, reconnection, turbulence) scale. While +solid progress has been achieved over the years, no complete picture is existing up to now. In +principle, this also applies to numerical simulations (cf. [16, 17] for related reviews). +While +essential to inform and advance our understanding, they are not without caveats, i.e., in general, +AGN physics is also accompanied by a methodological (computational) challenge. In the context +of jet formation for example, conventionally employed general-relativistic/magneto-hydrodynamic +3 + +I(>200 GeV) [10′ cm2 s'] +10-10 +3.5 +SED [TeV cm-2 s-1] +2.5 +2 +1.5 +LP fit +LP+EP fit +10-11, +MJD 56857.98 Observed +0.5 +102 +103 +40 +60 +80 +100 +20 +Time - MJD53944.0 [min] +E [GeV]Theory of Gamma-Ray Loud AGNs +Figure 3: The radio galaxy M87 as seen from large, radio-halo (VLA) down to black-hole (EHT) scales, +corresponding to an "observed scale separation" of ∼ 108. Adapted based on ref. [15]. +(GR/MHD) simulations usually rely on an ambiguous numerical floor model; methodologically, +it may seem rather surprising that a single fluid description is able to provide important insights +(as it has) in cases where we expect the plasma to be collisionless; further, in ideal MHD there is +little physical understanding of reconnection, particle acceleration and radiation, and so forth. On +the other hand, first-principle particle-in-cell (PIC) simulations in astrophysics mostly deal with +highly idealized setups, often in reduced dimensionality and for quite limited duration, along with +artificially large gyro-radii. In fact, in the case of AGNs, the relevant scale separation (system size +𝑟 vs plasma skin depth 𝑙𝑝) can be as high as 𝑟/𝑙𝑝 ∼ 106−8, cf. ref. [18], and it remains unclear to +which extent features seen in these simulations might persist up to the physical scale of interest. +Notwithstanding these limitations, numerical simulations have become an indispensable tool for +progress. +3.2 Progress in AGN Physics +The last years have seen both, significant progress and consolidation of knowledge in our +understanding of the physics of AGNs. The following provides a short selection of exemplary +results with relevance to gamma-ray emitting AGNs.1 +3.2.1 Supermassive Black Holes in the Center of AGNs +The presence of supermassive BHs in galactic nuclei has been suggested for more than half +a century based on strong theoretical arguments. Early experimental efforts related to dynamical +searches have been described ∼ 25 yr ago [19]. +Most recently, theoretical and experimental +progress in BH research has eventually been documented by the 2020 Nobel prize for Physics to +Roger Penrose, Reinhard Genzel and Andrea Ghez. While the benchmark BH Sgr A* is not (yet) +an established gamma-ray emitter [20], the event horizon telescope (EHT) radio image of the BH +shadow in M87 [21] provides important information relevant to VHE research. The size ≃ 2.5𝑟𝑠 of +the photon ring in M87 (where 𝑟𝑠 := 2𝑟𝑔 := 2𝐺𝑀BH/𝑐2), implies a BH mass of 𝑀BH ≃ 6.5×109𝑀⊙ +and constrains horizon scale (minimum) variability to 𝑡𝐻 ≃ 0.38 d, close to what can be probed +1For recent results on hadronic processes and neutrino emission, the reader is referred to the related reviews by +Paolo Padovani and Elisa Resconi. +4 + +VLA +VLBA +VLBI +25 kpc +800 pc +20 pc +0.5 pc +0.05 pc +0.005 pcTheory of Gamma-Ray Loud AGNs +with VHE instruments [8]. Further research along these lines (by, e.g., the next-generation EHT) +will allow to probe deeper into the central engine in AGNs (BH - disk - jet, its connection and +dynamics), and tighten the constraints on the current jet power (e.g., BH spin) relevant to VHE +emission models. +3.2.2 Convergence of theoretical, numerical & observational evidence for jet stratification +There is consolidation of knowledge that the jets in AGN are multi-layered, revealing a lateral +stratification similar to the one induced by a fast BH-driven (Blandford-Znajek: BZ, [22]) jet sur- +rounded (and possibly confined) by a slower moving disk-driven outflow, cf. [23]. Resolved (lateral) +emission structures such as limb-brightening and linear polarisation signatures, e.g., refs. [24, 25], +substantiate early two-flow and spine-sheath type non-thermal emission models [26, 27]. In the +case of M87, for example, significant structural patterns across the jet on sub-pc-scale have been +detected, indicating the presence of both slow (∼ 0.5𝑐) and fast (∼ 0.92𝑐) components [28]. Pro- +nounced edge-brightened features have now been seen in both, M87 and Cen A [29]. In the high +energy context, characterising internal jet stratification is potentially important to, e.g., address the +(putative) Doppler factor "crisis" in TeV (HBL) blazars [30], to develop more advanced acceleration +(cf. Sec. 4.3) and emission models, and to probe into AGN unification scenarios (e.g., relative power +of inner vs outer jet?). +3.2.3 Acceleration and collimation of relativistic jets +Detailed radio studies now provide insights into the jet collimation profile in more and more +nearby sources [31, 32]. In the case of M87 for example, the jet width profile is initially (semi- +)parabolic and then transitions (at around the Bondi radius 𝑟𝐵 ∼ 5 × 105𝑟𝑔 ∼ 150 pc) to a conical +shape, see Fig. 4. There is evidence that the radio flow is initially slow (on scales ∼ 0.03 pc) +and gradually accelerates with distance, reaching Γ𝛽 ∼ 3 on scales of 𝑟𝐵 [33, 34]. This can be +Figure 4: The jet collimation profile for the radio galaxy M87 from sub-parsec to kilo-parsec scales. From +ref. [33]. ©AAS. Reproduced with permission. +5 + +de-projected distance (pc) +10-4 +10-3 +10-2 +10-1 +1 +101 +102 +103 +104 +106 +●VLBA15GHz(AN12.H13) +conical z. α R +MERLIN 1.8 GHz (AN12) +EVN1.6GHz(AN12) +●VLBA22GHz(H13) +VLBA 2.3GHz (H13) +VLBA43GHz(AN12.H13) +105 +VLBA 5 GHz (H13) +● HSA 86 GHz (H16) + VLBA 8.4 GHz (H13) +VLBACore43&86GHz(NA13) +104 +■VLBACore5,8,15,24,43,&86GHz(H13),HSACore86GHz(H16) +△EHTCore230GHz(D12.A15) +(Area)FFEparabolicjet(NMF07,TMN08) +(Area)FFEgenuineparabolic jet(NMFO7,TMNO8) +103 +a = 0.5 (upper edge) - 0.99 (lower edge) +UUT +HST-1 +parabolic +102 +UTT +2 α R1.6 +101 +radius +UTT +Bondi +1 +T +1 +101 +102 +103 +104 +105 +106 +107 +108 +Jet axial distance (de-projected): z (r.)Theory of Gamma-Ray Loud AGNs +understood in terms of a jet that is magnetically dominated at its base and whose magnetic energy +is gradually converted into kinetic energy while experiencing some external confinement [35, 36]. +These results are instructive for improving our understanding of the non-thermal emission in low- +luminosity FR I type sources, and of the dynamical role (confinement) played by a disk-driven wind. +While consequences of flow deceleration on high-energy SED modelling have been explored in the +past [37], consideration of flow acceleration seems an important desideratum. +3.2.4 Convergent evidence for relativistic motion in large-scale AGN jets +While jets experience deceleration on larger scales, they can still be substantially relativistic +on kilo-parsec scales, even for moderately powerful sources. The nearby (𝑑 ∼ 95 Mpc) FR I +radio galaxy 3C 264 is a prototype example (with estimated jet power 𝐿 𝑗 ∼ 5 × 1043 erg/s) [38], +exhibiting superluminal motion and collision of knots (internal shocks) in HST observations on +scales of ∼ (0.5 − 1) kpc, corresponding to a bulk flow Lorentz factor Γ ∼ 7 and a Doppler factor +𝐷 ∼ 7 for an inclination 𝜃 ∼ 8◦ [39]. These findings are of interest not only because 3C264 is a +recently detected VHE emitter [40], but also by offering general insights into particle acceleration +at shocks, and the role of nearby large-scale jets as putative sites of ultra-high energy cosmic ray +(UHECR) acceleration. +3.2.5 Extreme TeV blazars +A sizeable fraction of VHE-emitting AGNs (∼ 1/4 of all HBL) are now composed of BL +Lac objects with hard intrinsic VHE spectra, i.e. with de-absorbed power-law (PL) photon indices +ΓVHE ≲ 1.5 − 1.9, implying an SED bump peaking above 1 TeV, see ref. [4] for a review. The +prototypical source of these extreme highly peaked BL Lacs (EHBLs) is 1ES 0229+200 (𝑧 = 0.14) +with an SED peak (de-absorbed) above 4 TeV. The SED modelling of these sources is challenging, +and in the common synchrotron-self Compton (SSC) approach requires unusual (if not extreme) +conditions such as an electron distribution with a very high minimum Lorentz factor 𝛾min ≳ 104, +or one following a relativistic Maxwellian [41]. The narrow pile-up feature seen in Mkn 501 [9], +cf. Fig. 1 (left), might be reminiscent of the latter [42]. These extreme TeV blazars are particularly +interesting as they indicate emitting regions significantly away from equipartition (𝑢𝐵/𝑢𝑒 ≲ 10−4 +in SSC). The implied low magnetisation 𝜎 would seem to disfavour reconnection and instead point +to stochastic or multiple shocks as possible acceleration mechanism (see also Sec. 4.2). VHE +variability studies will enable to probe into this as only modest (≥ month-type) intrinsic variability +would be expected (unless geometrical effects would interfere). Unusual spectral shapes at VHE +energies will complicate EBL (Extragalactic background light) studies using simple extrapolation +functions (for a recent review of developments, see e.g. ref. [43]). +3.2.6 Gamma-Ray Astrophysics in the Time Domain +Within the last decade, gamma-ray astronomy has successfully entered the time domain [44]. +Key examples include the detection and characterisation of ultra-fast VHE variability (down to a +few minutes) [4, 45] as well as evidence for year-type quasi-periodic oscillations (QPOs) in the +Fermi-LAT light curves of blazars [46, 47]. A prominent example of the latter is the HBL object +PG 1553+113 (𝑧 ∼ 0.5) with an (observed) HE period of ≃ 2.2 yr, possibly caused by a close +supermassive binary BH system (SBBHs) [48, 49]. Application of more advanced time series +6 + +Theory of Gamma-Ray Loud AGNs +techniques, exploring power-law noise characteristics and log-normality has been steadily growing +[50]. Theoretical efforts are now gaining momentum, cf. [44] for a review, and refs. [51, 52] +for selected contributions. +Properly understanding the variability characteristics has important +implications for the origin of the emission (e.g., location and driving process) and the overall source +dynamics and evolution (e.g., SBBHs, disk physics). +4. +Selected Theoretical Advances +The following subsections aim to provide exemplary and more detailed insights into three +different fields where theoretical advances in AGN physics, whether conceptual or methodological, +have been significantly influenced by gamma-ray observations. +4.1 Magnetospheric Processes in AGNs +The detection of VHE variability on BH horizon crossing time scales (∼ 𝑟𝑔/𝑐), as in e.g. M87, +has been a significant driving force for the development of magnetospheric particle acceleration +and emission models, see e.g. ref. [8] for a review. Magnetic fields that are brought in and dragged +into rotation (Ω𝐹) by the BH can induce an electric field component (parallel to the magnetic +field), that, if not screened, can facilitate efficient particle acceleration. This could potentially occur +either around the so-called null surface across which the Goldreich-Julian [53] charge density, +𝜌GJ ∼ (Ω𝐹 − 𝜔)𝐵, required to screen the electric field, changes sign, or at the stagnation surface +that separates MHD in- and outflows, see Figure 5. The maximum available voltage drop for a BH +Figure 5: Left: Illustration of possible locations of charge-deficient regions (gaps) around a rotating BH. The +red line denotes the null surface across which 𝜌GJ changes sign, while the blue line delineates the stagnation +surface from which stationary MHD flows start. Right: Exemplary gap evolution for a Kerr BH of 109𝑀⊙ +accreting in a radiatively inefficient mode. The curves show the radial distribution of the parallel electric field +around the null surface for different accretion rates �𝑚. The gap width and voltage drop increase as �𝑚 drops +because pair creation of energetic photons with the soft photon field becomes less efficient. From ref. [54]. +gap of width ℎ is of the order of [55] +ΔΦ = 1 +𝑐Ω𝐹𝑟2 +𝐻 𝐵𝐻 (ℎ/𝑟𝑔)2 ≃ 2 × 1021 �𝑚1/2𝑀1/2 +9 +(ℎ/𝑟𝑔)2 [𝑉] , +(1) +for a horizon-threading field 𝐵𝐻 ≃ 2 × 105 �𝑚1/2𝑀−1/2 +9 +G, where 𝑀9 ≡ 𝑀BH/109𝑀⊙, Ω𝐹 = +Ω𝐻/2 = 𝑐/4𝑟𝑔, and �𝑚 ≡ +�𝑀/ �𝑀Edd. Electrons that get accelerated in these gaps produce 𝛾-rays +7 + +0 +Stagnation +Surface +0.05 +E +Parallel electric field, +-0.1 +IInN +PGJ <0 +Surface +-0.15 +PG >0 +magnetic field +accretion rate: 10-5.0 +-0.2 +accretion rate: 10-6.0 +BH +Ergo + sphere +1.2 +1.4 +1.6 +1.8 +2 +2.2 +2.4 +2.6 +rH +s=r/rTheory of Gamma-Ray Loud AGNs +via curvature radiation and inverse Compton (IC) up-scattering of ambient soft photons, producing +an observable and variable VHE signal [56]. Interaction (𝛾𝛾-absorption) of energetic 𝛾-rays with +low-energy soft photons can trigger a pair cascade that generates plasma and ensures closure of the +gap [8]. On conceptual grounds, magnetospheric gaps thus provide a self-consistent means for the +continuous plasma supply ("numerical floor") needed to activate a force-free (BZ) MHD outflow +[57]. Correspondingly, unless accretion rates are sufficiently low (i.e., well below �𝑚 ∼ 10−4), +efficient pair production will occur (if non-negligible gaps form at all, cf. [56, 58]), leading to gap +sizes ℎ ≪ 𝑟𝑔, thereby significantly reducing the accessible gap potential, cf. eq. [1]. Typically, +achievable (radiation-limited) electron Lorentz factors are constrained to 𝛾𝑒 ∼ 108−10 [55]. To +improve insights into characteristic electric field strengths and achievable voltage drops, one can +seek for self-consistent steady gap solutions [54, 58] by solving the system of relevant partial +differential equations (e.g., Gauss’ law, equation of particle motion, continuity equation), see e.g. +Figure 5 (right). For example, taking an accretion rate �𝑚 = 10−5.75, which is close to the mean +MAD value used in EHT-GRMHD simulations for M87 [59] and which corresponds to jet powers +of a few times 1043 erg/s, the gap properties for M87 (assuming the inner disk to be ADAF-type) +can be evaluated [54]. The resultant gap sizes are of the order of ∼ 0.8𝑟𝑔, suggesting that the VHE +emission in M87 might be variable down to timescales of ∼ 0.4 days. The inferred gap power of +∼ 5×1041 erg/s would make it in principle possible to accommodate the VHE emission seen during +its high states [60]. The achievable voltage drop would be of the order of ∼ 1018 V, suggesting that +the BH in M87 is not an efficient UHECR proton accelerator. The appreciation that gap operation is +expected to be intermittent, has in recent times triggered a variety of time-dependent investigations +using PIC simulations, see e.g. refs. [61–64]. Relative comparison is often not straightforward as +different setups and complexities have been employed (e.g., with regard to box sizes, cell numbers, +run times, 1d/2d, radiation reaction, choice of soft photon field). Generically, the outcome is highly +sensitive to the assumed ambient soft photon field (i.e., minimum photon energy 𝜖min and spectral +shape, e.g., PL index), making an adequate description an important prerequisite as it determines +the efficiency of pair cascades. There are indications for a periodic opening (timescale ∼ 𝑟𝑔/𝑐) of +macroscopic gaps (with ℎ ∼ [0.1−1] 𝑟𝑔), cf. Fig. 6. Issues that deserve further studies concern e.g., +the degree to which detected oscillation periods could be dependent on the simulation setup (e.g., +box size), and the extent to which the emerging particle distributions might be bimodal. Insights +gained will help to better characterise the link between gap activity, VHE flaring and jet formation. +4.2 Modelling Parsec-Scale Jet Emission in Blazars +The spectral energy distributions (SEDs) of gamma-ray blazars have for long been modelled +by leptonic (SSC) and/or hadronic (e.g., 𝑝𝛾) radiation processes, see e.g. [65, 66] for a review. +Quite often their SEDs can be satisfactorily reproduced with different sets of assumptions, which +significantly limits inferences on the source physics. Incorporating microphysical insights from +kinetic plasma simulations into SED modelling can allow an important step forward. One exem- +plary context relates to the requirement of high minimum electron Lorentz factor 𝛾min,e and low +magnetisation 𝜎 in SSC models for extreme TeV blazars (cf. Sec. 3.2.5). A recently proposed +framework considers co-acceleration (diffusive shock) of electrons and protons at mildly relativistic +(e.g., Γ𝑠 ∼ 3), weakly magnetised (internal or recollimation) shocks in blazar jets [67]. Shocks +are known to convert a significant fraction of the kinetic energy of the incoming plasma flow +8 + +Theory of Gamma-Ray Loud AGNs +Figure 6: Exemplary gap cycle seen in 1d PIC simulations (spit monopole field) for a steep ambient photon +field (𝑛𝑠) with 𝜖min = 0.0005 eV and fiducial Thomson depth 𝜏0 := 𝑛𝑠𝜎Th𝑟𝑔 = 100 (yielding a pair creation +length ∼ 0.3𝑟𝑔). Top panel: Evolution of the normalized densities of electrons, positrons and photons. +Bottom panel: Evolution of the electric field. Vertical dashed line denotes the null surface. From ref. [63]. +©AAS. Reproduced with permission. +to thermal energy, represented by a Maxwellian particle distribution 𝑛(𝛾) ∝ 𝛾2 exp[−𝛾/Θ] with +Θ ≡ 𝑘𝐵𝑇/𝑚𝑐2. Assuming an efficient energy transfer (heating/thermal coupling) from protons to +electrons in the shock transition layer, one can write 𝑘𝐵𝑇𝑃 ∼ Γ𝑠𝑚 𝑝𝑐2 and 𝑇𝑒 = 𝜉𝑇𝑝 where 𝜉 < 1, +with Γ𝑠 the Lorentz factor of the shock. If the peak in the Maxwellian electron distribution is +identified with 𝛾min,e, then roughly 𝛾min,e ∼ 𝑘𝐵𝑇𝑒/(𝑚𝑒𝑐2), i.e. +𝛾min,e ∼ +�𝑚 𝑝 +𝑚𝑒 +� +Γ𝑠𝜉 ≃ 1800 Γ𝑠𝜉 +(2) +Hence, given suitable conditions, a high 𝛾min,e is naturally expected. +In PIC simulations of +highly relativistic shocks, efficient Fermi-type particle acceleration has only been seen for weakly +magnetized (e.g., 𝜎 <∼ 10−3) shocks, in line with considerations in the above framework. Since +these shocks will also generate magnetic (micro)turbulence, the effective magnetization (𝜎 := +𝐵2/(4𝜋⟨𝛾⟩𝑛𝑚𝑐2) could in principle be higher than the pre-shock one, yet decaying downstream +with distance. Accordingly, the magnetization in the radiation zone, 𝜎rad, may be in the range +𝜎 <∼ 𝜎rad +<∼ 0.01. +For high (Alfvenic) Mach numbers, significant thermal coupling can occur. +In large 2d PIC +simulations of quasi-parallel, mildly relativistic (𝛽𝑠 ≃ 0.8), weakly magnetized (𝜎 = 0.007, +defined downstream) electron-ion shocks (with reduced mass ration 𝑚 𝑝/𝑚𝑒 = 64), for example, +𝑇𝑒 ∼ (0.2−0.3) 𝑇𝑝 and 𝑘𝐵𝑇𝑝 ∼ 0.2𝑚 𝑝𝑐2 has been seen, cf. Fig. 7 [68]. For smaller magnetizations +and larger shock speeds, electron preheating can be further increased [69, 70]. Figure 8 shows an +exemplary reproduction of the SED of the prototypical EHBL 1ES 0229+200 motivated by such +a framework. In this case, re-acceleration at multiple (𝑛 ≥ 2) shocks is needed to account for +the hard particle spectra required for spectral fitting. This leads to a modification (hardening and +power-law deviation) of the input, single shock electron spectrum (of power law momentum index +𝑠 = 2.2), and also increases the effective injection Lorentz factor to 𝛾min,eff ∼ Γ2𝑛/3 +𝑠 +𝛾min,e [67]. +9 + +t/(r +t/(r +=191.55 +t/(rg +/c) +=191.97 +t/(ra +/C) +=192.29 +101 +100 +△n/△n +10 +:01X ( +0 +-5 +-10 +15 +2 +3 +4 +2 +3 +4 +2 +3 +4 +2 +3 +4 +YTheory of Gamma-Ray Loud AGNs +Figure 7: 2d PIC simulations of a weakly magnetized, mildly relativistic, high Mach number (𝑀𝐴 = 15) +shock with Γ𝑠 ≃ 1.7. Right: Shock evolution and magnetic field generation (initially Weibel-, later Bell- +mediated) with time. Right: Downstream particle distributions 𝑝4 𝑓 (𝑝) at different times, with the formation +of a non-thermal power-law tail on top of a thermalized particle distribution. At late times, the downstream +electron temperature is 20 − 30% of the ion temperature, indicating significant coupling. Based on ref. [68]. +Figure 8: Reproduction of the SED of the EHBL source 1ES 0229+200 in a multiple shock scenario with +input parameters for the initial electron distribution as shown in the table to the right. From ref. [67], +reproduced with permission © ESO. +The noted framework represents an exemplary approach to utilise microphysical insights for blazar +SED modelling. One central assumption is that the emission arises in a jet region that is kinetically +dominated by an electron-proton composition (with protonic emission expected to be suppressed) +rather than an 𝑒+𝑒− one. Similar indications have been found in the case of another prominent 𝛾-ray +blazar, PKS 2155-304, based on a statistical analysis of its X-ray emission [71], though there is also +other evidence (based on optical circular polarization) suggesting that the (non-thermal) positron +fraction cannot be too low [72], cf. also ref. [73]. A possible caveat for the noted approach relates +to the model requirement of a region significantly out of equipartition and characterized by a very +low magnetic field. In principle, variability studies could allow to probe into these assumptions. +A related framework considers the electrons in the jet to be (initially) diffusively accelerated +at a recollimation-type shock, but then further energized through stochastic (2nd order Fermi) +10 + +downstream spectrum +25 +y [clopi] +Ions +n/no +10-2 +6 +io1 +3375 +y [clOpi], +100 +10-4 +0 +-101 +50 +75 +100 +125 +150 +175 +200 +225 +250 +x [clopi] +10-6 +50 +12 +Electrons +10-2 +Wpit = 4148 +n/no +6 +0 +Te~0.23T, +50 +iol +f(p) αp-4. 2 +10- +-4 +100 +25 +n(E) α E-2.2 +0 +-101 +150 +200 +250 +300 +350 +400 +450 +500 +x [clopi] +10-6 +.0 +10-1 +100 +101 +10-2 +p/(yoβomic)-9 +1 ES 0229+200 +Modelling parameters +-10 +(multiple shocks) +50 +Rsrc [1016cm] +1.3 +B[mG] +4.4 +12 +1.0 × 10-4 +O rad +270 +rad/o +.13 +1.8 × 103 +Ye,min +Ye,max +2.9 × 106 +14 +ne [cm3] +0.13 +12 +14 +16 +18 +20 +22 +24 +26 +28 +log (v [Hz])Theory of Gamma-Ray Loud AGNs +acceleration in the downstream flow [74]. +Hence, instead of acceleration at multiple shocks, +recollimation in a weakly magnetized flow is presumed to result in strong turbulence. While the +synchrotron spectrum can be satisfactorily reproduced in a SSC model where synchrotron cooling +is comparable to particle escape (cf. also [41] for another variant), the Compton SED component +appears somewhat too hard. Since EHBL SED models are in general far from equipartition (i.e., +𝜎 ≪ 1), characteristic Alfven speeds are sub-relativistic (𝑣 𝐴 ∝ √𝜎 ≪ 𝑐) making stochastic +acceleration less efficient (𝑡 ∝ 1/𝑣2 +𝐴), so that it remains to be explored to what extent a self- +consistent SED fit can be achieved. Nevertheless, these approaches represent instructive examples +of how to fruitfully combine insights from jet simulation, particle acceleration physics and SED +modelling. +4.3 Understanding the large-scale jet emission of AGNs +The origin of extended X-ray emission along the large-scale jets in AGNs has for long been a +matter of debate, with electron-synchrotron and IC-CMB scenarios as the leading contenders, see +e.g. [75, 76] for review. Within more recent years, however, IC-CMB models have been disfavoured +in an increasing number of sources as they tend to, e.g., over-predict Fermi-LAT gamma-ray flux +limits [77, 78] or are not supported by detailed X-ray spectral information [79]. +An electron +synchrotron origin, however, requires to maintain electrons with Lorentz factors 𝛾𝑒 ∼ 108 all along +the jet. The short cooling timescales 𝑡cool ∝ 1/𝛾 of these electrons, and correspondingly short +cooling lengths 𝑐𝑡cool ≪ 1 kpc, calls for the operation of a "distributed" acceleration mechanism +along the jet, beyond "localised" shock acceleration. +The recent detection of extended VHE +emission along the kpc-scale jet of Cen A [80] has provided an important validation to this picture, +see Fig. 9, by directly tracing the electrons and removing the degeneracy (𝐵, 𝛾) inherent in an +X-ray synchrotron model. IC up-scattering of dust by ultra-relativistic (synchrotron X-ray emitting) +electrons with 𝛾𝑒 = 108 accounts for the observed VHE emission and confirms the X-ray synchrotron +interpretation. The results imply a jet that is only weakly magnetized (𝜎 < 0.5) on kiloparsec scales. +Stochastic (2nd order Fermi) and shear particle acceleration processes are among the most promising +mechanisms facilitating continuous re-acceleration in kinetically dominated jets [82–85]. Particle +acceleration in the latter framework essentially draws on the jet flow velocity difference that a +charged particles experiences as it moves across the jet shear (cf. Sec. 3.2.2), see e.g. ref. [86] for +an accessible introduction. It can be viewed as a second-order (Δ𝐸 ∝ 𝑢2 +sc𝐸) Fermi-type acceleration +process, in which the common scattering center speed, 𝑢sc, is replaced by an effective velocity, ¯𝑢, +determined by the shear flow profile, i.e., by the characteristic flow velocity change sampled over +the mean free path 𝜆 ≃ 𝜏𝑐 of the particle. Hence, for instance ¯𝑢 = (𝜕𝑢𝑧/𝜕𝑟) 𝜆 for a continuous +shear profile �𝑢 = 𝑢𝑧(𝑟) �𝑒𝑧. This results in a characteristic acceleration timescale +𝑡acc ≃ +𝐸 +𝑑𝐸/𝑑𝑡 ≃ +𝜏 +⟨Δ𝐸/𝐸⟩ ∝ 𝜏 +¯𝑢2 ∝ 1 +𝜆 , +(3) +scaling inversely with the particle mean free path 𝜆 or scattering time 𝜏. While this makes shear +flows also interesting for UHE cosmic-ray acceleration in the jets of AGNs [87], it typically requires +some energetic seed injection for electron acceleration to proceed efficiently. In principle, this +could be achieved by shock or classical 2nd order Fermi acceleration (for which 𝑡acc ∝ 𝜆) [82], +or possibly, by kinetic-scale reconnection processes [88]. Phenomenologically, this could result +11 + +Theory of Gamma-Ray Loud AGNs +Figure 9: Extended VHE emission along the kpc-scale jet of Cen A as seen by H.E.S.S. The SED of the +kpc-scale jet can be reproduced by means of an (exponential-cutoff) broken-power law electron distribution +with 𝛾𝑏 ≃ 106 and cut-off energy 𝛾𝑐 ≃ 108, for a characteristic jet magnetic field strength of 𝐵 ≃ 20𝜇G. +Interestingly, IC emission by the kpc-scale jet may be behind the unusual spectral hardening [81] seen at +GeV energies. Based on ref. [80]. +in multi-component particle distributions, e.g., broken power-laws with changes in spectral indices +different from a classical cooling break. A recent application shows that in the presence of a +Kolmogorov-type 𝜆 ∝ 𝛾1/3, the SED of the kpc-scale jet in Cen A can be successfully reproduced +within a linear shear model for spine velocities of 𝑣 𝑗 ≃ 0.7c and shear layer widths of Δ𝑟 ≃ 100 +pc [84]. In general, efficient shear acceleration requires relativistic flow speeds [89, 90] and this +confines its applicability to astrophysical objects with fast outflows. There is currently a promising +trend to incorporate particle acceleration and non-thermal emission recipes in large-scale MHD +jet simulations, e.g. [91]. While such a hybrid approach is vital, it should be kept in mind that +it still involves significant extrapolations (cf. Sec. 3.1.) that may need to be further explored. +The prospects are good, though, for achieving further progress soon as several new observational +facilities are coming online [15]. +5. +Conclusions +By providing new and often unexpected insights, multi-wavelengh and multi-messenger obser- +vations have become an indispensable tool for progress in many areas of astronomy. The field of +AGNs, and in particular jetted AGNs, to which most extragalactic gamma-ray sources belong, is no +exception in this regard [92]. In the present review, I have highlighted some of the developments that +are of particular relevance for our understanding of the nature of these sources, and/or directly driven +and inspired by gamma-ray observations, ranging from new observational findings to conceptual +and methodological advances. I believe that we have seen true progress in our understanding of the +physics of gamma-ray loud AGNs. As for jetted AGNs in general and for gamma-ray loud AGNs +in particular: "The observational prospects for securing our understanding of AGN jets are bright." +(Blandford et al., [15]). In particular, given available (from e.g. Fermi-LAT and the current IACTs, +HESS, MAGIC and VERITAS to HAWC, LHAASO) and upcoming capabilities (CTA) at highest +photon energies. +12 + +10-10 +radio convolved with H.E.S.S. PSF +core +-42°56'00.0" +10-11 + synchrotron +IC +58'00.0" +E2 dN/dE [erg s-1 cm-2] +2.2 kpc +10-12 +optical +H.E.S.S. +(2000) +kpc jet +radio + X-rays +-43°00'00.0" +10-13 +Dec +02'00.0" +10-14 +IC (CMB) +IC (starlight) +IC (dust) +10-15, +IC (SSC) +04'00.0" +IC (total) +Sync +10-16 +H.E.S.S. +10-8 +10-6 +10-4 +10-2 +100 +102 +104 +106 +108 +1010 +10121014 +06'00.0" +Photon energy [eV] +26m00.00s +48.00s +36.00s +24.00s +12.00s +13h25m0iTheory of Gamma-Ray Loud AGNs +Acknowledgments +I would like to thank Josep Maria Parades, Valenti Bosch-Ramon, Pol Bordas, Marc Ribo and +the whole Barcelona LOC team for a truly pleasant and inspiring conference. I am grateful to +Felix Aharonian for fruitful collaboration over the years. Funding by a DFG Fellowship under +RI 1187/8-1 is gratefully acknowledged. +References +[1] P. Padovani, On the two main classes of active galactic nuclei, Nature Astronomy 1 (2017) +0194 [1707.08069]. +[2] J. Biteau, E. Prandini, L. Costamante, M. Lemoine, P. Padovani, E. Pueschel et al., Progress +in unveiling extreme particle acceleration in persistent astrophysical jets, Nature Astronomy +4 (2020) 124 [2001.09222]. +[3] C.M. Urry and P. Padovani, Unified Schemes for Radio-Loud Active Galactic Nuclei, PASP +107 (1995) 803 [astro-ph/9506063]. +[4] L. Costamante, Blazars - an updated review, in Multifrequency Behaviour of High Energy +Cosmic Sources - XIII. 3-8 June 2019. Palermo, p. 35, Dec., 2020, DOI. +[5] C. Rulten, Radio Galaxies at TeV Energies, Galaxies 10 (2022) 61. +[6] S. Abdollahi, F. Acero, M. Ackermann, M. Ajello, W.B. Atwood, M. Axelsson et al., Fermi +Large Area Telescope Fourth Source Catalog, ApJS 247 (2020) 33 [1902.10045]. +[7] S. Abdollahi, F. Acero, L. Baldini, J. Ballet, D. Bastieri, R. Bellazzini et al., Incremental +Fermi Large Area Telescope Fourth Source Catalog, ApJS 260 (2022) 53 [2201.11184]. +[8] F. Rieger and A. Levinson, Radio Galaxies at VHE Energies, Galaxies 6 (2018) 116 +[1810.05409]. +[9] MAGIC Collaboration, V.A. Acciari, S. Ansoldi, L.A. Antonelli, A. Babić, B. Banerjee +et al., Study of the variable broadband emission of Markarian 501 during the most extreme +Swift X-ray activity, A&A 637 (2020) A86 [2001.07729]. +[10] F. Aharonian, A.G. Akhperjanian, A.R. Bazer-Bachi, B. Behera, M. Beilicke, W. Benbow +et al., An Exceptional Very High Energy Gamma-Ray Flare of PKS 2155-304, ApJL 664 +(2007) L71 [0706.0797]. +[11] S. Komossa, Narrow-line Seyfert 1 Galaxies, in Revista Mexicana de Astronomia y +Astrofisica Conference Series, vol. 32 of Revista Mexicana de Astronomia y Astrofisica +Conference Series, pp. 86–92, Apr., 2008, DOI [0710.3326]. +[12] I. Varglund, E. Järvelä, A. Lähteenmäki, M. Berton, S. Ciroi and E. Congiu, Jetted +narrow-line Seyfert 1 galaxies breaking the jet paradigm – a comprehensive study of host +galaxy morphologies, arXiv e-prints (A&A accepted) (2022) arXiv:2209.09270 +[2209.09270]. +13 + +Theory of Gamma-Ray Loud AGNs +[13] F. D’Ammando, Relativistic Jets in Gamma-Ray-Emitting Narrow-Line Seyfert 1 Galaxies, +Galaxies 7 (2019) 87 [1911.03500]. +[14] L. Foschini, Jetted Narrow-Line Seyfert 1 Galaxies & Co.: Where Do We Stand?, Universe 6 +(2020) 136 [2008.13383]. +[15] R. Blandford, D. Meier and A. Readhead, Relativistic Jets from Active Galactic Nuclei, +ARA&A 57 (2019) 467 [1812.06025]. +[16] O. Porth, K. Chatterjee, R. Narayan, C.F. Gammie, Y. Mizuno, P. Anninos et al., The Event +Horizon General Relativistic Magnetohydrodynamic Code Comparison Project, ApJS 243 +(2019) 26 [1904.04923]. +[17] K. Nishikawa, I. Duţan, C. Köhn and Y. Mizuno, PIC methods in astrophysics: simulations +of relativistic jets and kinetic physics in astrophysical systems, Living Reviews in +Computational Astrophysics 7 (2021) 1 [2008.02105]. +[18] H. Ji and W. Daughton, Phase diagram for magnetic reconnection in heliophysical, +astrophysical, and laboratory plasmas, Physics of Plasmas 18 (2011) 111207 [1109.0756]. +[19] J. Kormendy and D. Richstone, Inward Bound—The Search For Supermassive Black Holes +In Galactic Nuclei, ARA&A 33 (1995) 581. +[20] G. Katsoulakos, F.M. Rieger and B. Reville, Constraining Cosmic-Ray Acceleration in the +Magnetospheric Gaps of Sgr A*, ApJL 899 (2020) L7 [2007.12189]. +[21] Event Horizon Telescope Collaboration, K. Akiyama, A. Alberdi, W. Alef, K. Asada, +R. Azulay et al., First M87 Event Horizon Telescope Results. I. The Shadow of the +Supermassive Black Hole, ApJL 875 (2019) L1 [1906.11238]. +[22] R.D. Blandford and R.L. Znajek, Electromagnetic extraction of energy from Kerr black +holes., MNRAS 179 (1977) 433. +[23] Y. Mizuno, GRMHD Simulations and Modeling for Jet Formation and Acceleration Region +in AGNs, Universe 8 (2022) 85 [2201.12608]. +[24] B. Boccardi, T.P. Krichbaum, E. Ros and J.A. Zensus, Radio observations of active galactic +nuclei with mm-VLBI, A&ARv 25 (2017) 4 [1711.07548]. +[25] J.Y. Kim, T.P. Krichbaum, R.S. Lu, E. Ros, U. Bach, M. Bremer et al., The limb-brightened +jet of M87 down to the 7 Schwarzschild radii scale, A&A 616 (2018) A188 [1805.02478]. +[26] H. Sol, G. Pelletier and E. Asseo, Two-flow model for extragalactic radio jets., MNRAS 237 +(1989) 411. +[27] G. Ghisellini, F. Tavecchio and M. Chiaberge, Structured jets in TeV BL Lac objects and +radiogalaxies. Implications for the observed properties, A&A 432 (2005) 401 +[astro-ph/0406093]. +14 + +Theory of Gamma-Ray Loud AGNs +[28] F. Mertens, A.P. Lobanov, R.C. Walker and P.E. Hardee, Kinematics of the jet in M 87 on +scales of 100-1000 Schwarzschild radii, A&A 595 (2016) A54 [1608.05063]. +[29] M. Janssen, H. Falcke, M. Kadler, E. Ros, M. Wielgus, K. Akiyama et al., Event Horizon +Telescope observations of the jet launching and collimation in Centaurus A, Nature +Astronomy 5 (2021) 1017 [2111.03356]. +[30] G. Henri and L. Saugé, The Bulk Lorentz Factor Crisis of TeV Blazars: Evidence for an +Inhomogeneous Pileup Energy Distribution?, ApJ 640 (2006) 185 [astro-ph/0511610]. +[31] K. Hada, Relativistic Jets from AGN Viewed at Highest Angular Resolution, Galaxies 8 +(2019) 1. +[32] Y.Y. Kovalev, A.B. Pushkarev, E.E. Nokhrina, A.V. Plavin, V.S. Beskin, A.V. Chernoglazov +et al., A transition from parabolic to conical shape as a common effect in nearby AGN jets, +MNRAS 495 (2020) 3576 [1907.01485]. +[33] M. Nakamura, K. Asada, K. Hada, H.-Y. Pu, S. Noble, C. Tseng et al., Parabolic Jets from +the Spinning Black Hole in M87, ApJ 868 (2018) 146 [1810.09963]. +[34] J. Park, K. Hada, M. Kino, M. Nakamura, J. Hodgson, H. Ro et al., Kinematics of the M87 +Jet in the Collimation Zone: Gradual Acceleration and Velocity Stratification, ApJ 887 +(2019) 147 [1911.02279]. +[35] V.S. Beskin and E.E. Nokhrina, The effective acceleration of plasma outflow in the +paraboloidal magnetic field, MNRAS 367 (2006) 375. +[36] S.S. Komissarov, M.V. Barkov, N. Vlahakis and A. Königl, Magnetic acceleration of +relativistic active galactic nucleus jets, MNRAS 380 (2007) 51 [astro-ph/0703146]. +[37] M. Georganopoulos, E.S. Perlman and D. Kazanas, Is the Core of M87 the Source of Its TeV +Emission? Implications for Unified Schemes, ApJL 634 (2005) L33 [astro-ph/0510783]. +[38] E.T. Meyer, G. Fossati, M. Georganopoulos and M.L. Lister, From the Blazar Sequence to +the Blazar Envelope: Revisiting the Relativistic Jet Dichotomy in Radio-loud Active Galactic +Nuclei, ApJ 740 (2011) 98 [1107.5105]. +[39] E.T. Meyer, M. Georganopoulos, W.B. Sparks, E. Perlman, R.P. van der Marel, J. Anderson +et al., A kiloparsec-scale internal shock collision in the jet of a nearby radio galaxy, Nature +521 (2015) 495. +[40] A. Archer, W. Benbow, R. Bird, A. Brill, M. Buchovecky, J.H. Buckley et al., VERITAS +Discovery of VHE Emission from the Radio Galaxy 3C 264: A Multiwavelength Study, ApJ +896 (2020) 41 [2005.03110]. +[41] E. Lefa, F.M. Rieger and F. Aharonian, Formation of Very Hard Gamma-Ray Spectra of +Blazars in Leptonic Models, ApJ 740 (2011) 64 [1106.4201]. +15 + +Theory of Gamma-Ray Loud AGNs +[42] E. Lefa, F.A. Aharonian and F.M. Rieger, “Leading Blob” Model in a Stochastic Acceleration +Scenario: The Case of the 2009 Flare of Mkn 501, ApJL 743 (2011) L19 [1108.4568]. +[43] J. Biteau and M. Meyer, Gamma-Ray Cosmology and Tests of Fundamental Physics, +Galaxies 10 (2022) 39 [2202.00523]. +[44] F. Rieger, Gamma-Ray Astrophysics in the Time Domain, Galaxies 7 (2019) 28 +[1901.10216]. +[45] F.A. Aharonian, M.V. Barkov and D. Khangulyan, Scenarios for Ultrafast Gamma-Ray +Variability in AGN, ApJ 841 (2017) 61 [1704.08148]. +[46] F. Ait Benkhali, W. Hofmann, F.M. Rieger and N. Chakraborty, Evaluating quasi-periodic +variations in the 𝛾-ray light curves of Fermi-LAT blazars, A&A 634 (2020) A120 +[1901.10246]. +[47] P. Peñil, M. Ajello, S. Buson, A. Domínguez, J.R. Westernacher-Schneider and J. Zrake, +Evidence of Periodic Variability in Gamma-ray Emitting Blazars with Fermi-LAT, arXiv +e-prints (2022) arXiv:2211.01894 [2211.01894]. +[48] E. Sobacchi, M.C. Sormani and A. Stamerra, A model for periodic blazars, MNRAS 465 +(2017) 161 [1610.04709]. +[49] M. Tavani, A. Cavaliere, P. Munar-Adrover and A. Argan, The Blazar PG 1553+113 as a +Binary System of Supermassive Black Holes, ApJ 854 (2018) 11 [1801.03335]. +[50] H. E. S. S. Collaboration, H. Abdalla, A. Abramowski, F. Aharonian, F. Ait Benkhali, +A.G. Akhperjanian et al., Characterizing the 𝛾-ray long-term variability of PKS 2155-304 +with H.E.S.S. and Fermi-LAT, A&A 598 (2017) A39 [1610.03311]. +[51] M. Polkas, M. Petropoulou, G. Vasilopoulos, A. Mastichiadis, C.M. Urry, P. Coppi et al., A +numerical study of long-term multiwavelength blazar variability, MNRAS 505 (2021) 6103 +[2105.07030]. +[52] A. Dmytriiev, H. Sol and A. Zech, Connecting steady emission and very high energy flaring +states in blazars: the case of Mrk 421, MNRAS 505 (2021) 2712 [2105.12480]. +[53] P. Goldreich and W.H. Julian, Pulsar Electrodynamics, ApJ 157 (1969) 869. +[54] G. Katsoulakos and F.M. Rieger, Gap-type Particle Acceleration in the Magnetospheres of +Rotating Supermassive Black Holes, ApJ 895 (2020) 99 [2005.05076]. +[55] G. Katsoulakos and F.M. Rieger, Magnetospheric Gamma-Ray Emission in Active Galactic +Nuclei, ApJ 852 (2018) 112 [1712.04203]. +[56] A. Levinson and F. Rieger, Variable TeV Emission as a Manifestation of Jet Formation in +M87?, ApJ 730 (2011) 123 [1011.5319]. +16 + +Theory of Gamma-Ray Loud AGNs +[57] I. Contopoulos, The immediate environment of an astrophysical black hole, MNRAS 473 +(2018) L146 [1711.00302]. +[58] K. Hirotani, H.-Y. Pu, L.C.-C. Lin, H.-K. Chang, M. Inoue, A.K.H. Kong et al., Lepton +Acceleration in the Vicinity of the Event Horizon: High-energy and Very-high-energy +Emissions from Rotating Black Holes with Various Masses, ApJ 833 (2016) 142 +[1610.07819]. +[59] Event Horizon Telescope Collaboration, K. Akiyama, A. Alberdi, W. Alef, K. Asada, +R. Azulay et al., First M87 Event Horizon Telescope Results. V. Physical Origin of the +Asymmetric Ring, ApJL 875 (2019) L5 [1906.11242]. +[60] F. Ait Benkhali, N. Chakraborty and F.M. Rieger, Complex gamma-ray behavior of the radio +galaxy M 87, A&A 623 (2019) A2 [1802.03103]. +[61] A.Y. Chen and Y. Yuan, Physics of Pair Producing Gaps in Black Hole Magnetospheres. II. +General Relativity, ApJ 895 (2020) 121 [1908.06919]. +[62] B. Crinquand, B. Cerutti, A. Philippov, K. Parfrey and G. Dubus, Multidimensional +Simulations of Ergospheric Pair Discharges around Black Holes, Phys Rev Lett. 124 (2020) +145101 [2003.03548]. +[63] S. Kisaka, A. Levinson and K. Toma, Comprehensive Analysis of Magnetospheric Gaps +around Kerr Black Holes Using 1D GRPIC Simulations, ApJ 902 (2020) 80 [2007.02838]. +[64] S. Kisaka, A. Levinson, K. Toma and I. Niv, The Response of Black Hole Spark Gaps to +External Changes: A Production Mechanism of Rapid TeV Flares?, ApJ 924 (2022) 28 +[2108.02971]. +[65] M. Cerruti, Leptonic and Hadronic Radiative Processes in Supermassive-Black-Hole Jets, +Galaxies 8 (2020) 72 [2012.13302]. +[66] H. Sol and A. Zech, Blazars at Very High Energies: Emission Modelling, Galaxies 10 (2022) +105 [2211.03580]. +[67] A. Zech and M. Lemoine, Electron-proton co-acceleration on relativistic shocks in +extreme-TeV blazars, A&A 654 (2021) A96 [2108.12271]. +[68] P. Crumley, D. Caprioli, S. Markoff and A. Spitkovsky, Kinetic simulations of mildly +relativistic shocks - I. Particle acceleration in high Mach number shocks, MNRAS 485 (2019) +5105 [1809.10809]. +[69] L. Sironi, A. Spitkovsky and J. Arons, The Maximum Energy of Accelerated Particles in +Relativistic Collisionless Shocks, ApJ 771 (2013) 54 [1301.5333]. +[70] A. Vanthieghem, M. Lemoine, I. Plotnikov, A. Grassi, M. Grech, L. Gremillet et al., Physics +and Phenomenology of Weakly Magnetized, Relativistic Astrophysical Shock Waves, +Galaxies 8 (2020) 33 [2002.01141]. +17 + +Theory of Gamma-Ray Loud AGNs +[71] S.K. Jagan, S. Sahayanathan, F.M. Rieger and C.D. Ravikumar, Convex X-ray spectra of PKS +2155-304 and constraints on the minimum electron energy, MNRAS 506 (2021) 3996 +[2107.04534]. +[72] I. Liodakis, D. Blinov, S.B. Potter and F.M. Rieger, Constraints on magnetic field and +particle content in blazar jets through optical circular polarization, MNRAS 509 (2022) L21 +[2110.11434]. +[73] G.E. Romero, The content of astrophysical jets, Astronomische Nachrichten 342 (2021) 727 +[2106.14346]. +[74] F. Tavecchio, A. Costa and A. Sciaccaluga, Extreme blazars: the result of unstable +recollimated jets?, MNRAS 517 (2022) L16 [2207.12766]. +[75] D.E. Harris and H. Krawczynski, X-Ray Emission from Extragalactic Jets, ARA&A 44 (2006) +463 [astro-ph/0607228]. +[76] M. Georganopoulos, E. Meyer and E. Perlman, Recent Progress in Understanding the Large +Scale Jets of Powerful Quasars, Galaxies 4 (2016) 65. +[77] P. Breiding, E.T. Meyer, M. Georganopoulos, M.E. Keenan, N.S. DeNigris and J. Hewitt, +Fermi Non-detections of Four X-Ray Jet Sources and Implications for the IC/CMB +Mechanism, ApJ 849 (2017) 95 [1710.04250]. +[78] P. Breiding, E.T. Meyer, M. Georganopoulos, K. Reddy, K.E. Kollmann and +A. Roychowdhury, A multiwavelength study of multiple spectral component jets in AGN: +testing the IC/CMB model for the large-scale-jet X-ray emission, MNRAS 518 (2023) 3222 +[2210.13104]. +[79] X.-N. Sun, R.-Z. Yang, F.M. Rieger, R.-Y. Liu and F. Aharonian, Energy distribution of +relativistic electrons in the kiloparsec scale jet of M 87 with Chandra, A&A 612 (2018) A106 +[1712.06390]. +[80] H. E. S. S. Collaboration, H. Abdalla, R. Adam, F. Aharonian, F. Ait Benkhali, E.O. Angüner +et al., Resolving acceleration to very high energies along the jet of Centaurus A, Nature 582 +(2020) 356 [2007.04823]. +[81] H. E. S. S. Collaboration, H. Abdalla, A. Abramowski, F. Aharonian, F. Ait Benkhali, +E.O. Angüner et al., The 𝛾-ray spectrum of the core of Centaurus A as observed with +H.E.S.S. and Fermi-LAT, A&A 619 (2018) A71 [1807.07375]. +[82] R.-Y. Liu, F.M. Rieger and F.A. Aharonian, Particle Acceleration in Mildly Relativistic +Shearing Flows: The Interplay of Systematic and Stochastic Effects, and the Origin of the +Extended High-energy Emission in AGN Jets, ApJ 842 (2017) 39 [1706.01054]. +[83] F. Tavecchio, Constraining the shear acceleration model for the X-ray emission of +large-scale extragalactic jets, MNRAS 501 (2021) 6199 [2011.03264]. +18 + +Theory of Gamma-Ray Loud AGNs +[84] J.-S. Wang, B. Reville, R.-Y. Liu, F.M. Rieger and F.A. Aharonian, Particle acceleration in +shearing flows: the case for large-scale jets, MNRAS 505 (2021) 1334 [2105.08600]. +[85] J.-S. Wang, B. Reville, Y. Mizuno, F.M. Rieger and F.A. Aharonian, Particle acceleration in +shearing flows: the self-generation of turbulent spine-sheath structures in relativistic MHD +jet simulations, arXiv e-prints (2022) arXiv:2212.03226 [2212.03226]. +[86] F.M. Rieger, An Introduction to Particle Acceleration in Shearing Flows, Galaxies 7 (2019) +78 [1909.07237]. +[87] F.M. Rieger, Active Galactic Nuclei as Potential Sources of Ultra-High Energy Cosmic Rays, +Universe 8 (2022) 607 [2211.12202]. +[88] L. Sironi, M.E. Rowan and R. Narayan, Reconnection-driven Particle Acceleration in +Relativistic Shear Flows, ApJL 907 (2021) L44 [2009.11877]. +[89] G.M. Webb, A.F. Barghouty, Q. Hu and J.A. le Roux, Particle Acceleration Due to +Cosmic-ray Viscosity and Fluid Shear in Astrophysical Jets, ApJ 855 (2018) 31. +[90] F.M. Rieger and P. Duffy, Particle Acceleration in Relativistic Shearing Flows: Energy +Spectrum, ApJ 933 (2022) 149 [2206.13098]. +[91] D. Mukherjee, G. Bodo, P. Rossi, A. Mignone and B. Vaidya, Simulating the dynamics and +synchrotron emission from relativistic jets - II. Evolution of non-thermal electrons, MNRAS +505 (2021) 2267 [2105.02836]. +[92] M. Böttcher, Progress in Multi-wavelength and Multi-Messenger Observations of Blazars +and Theoretical Challenges, Galaxies 7 (2019) 20 [1901.04178]. +19 + diff --git a/PtFRT4oBgHgl3EQfJDc4/content/tmp_files/load_file.txt b/PtFRT4oBgHgl3EQfJDc4/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6f0cc85fc80584b90366a9f305e877f9c311c045 --- /dev/null +++ b/PtFRT4oBgHgl3EQfJDc4/content/tmp_files/load_file.txt @@ -0,0 +1,1138 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf,len=1137 +page_content='Theory of Gamma-Ray Loud AGNs Frank M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Rieger𝑎,𝑏 𝑎Institute for Theoretical Physics, Heidelberg University, Philosophenweg 12, 69120 Heidelberg, Germany 𝑏Max-Planck-Institute for Nuclear Physics (MPIK), P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Box 103980, 69029 Heidelberg, Germany E-mail: f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='rieger@uni-heidelberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='de The last decade has seen tremendous developments in gamma-ray astronomy with the extragalactic sky becoming highly populated by Active Galactic Nuclei (AGNs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' This brief review highlights some of the progress in AGN research achieved over the years, and discusses exemplary advances in the theory and physics of gamma-ray loud AGNs, including black-hole magnetospheric processes, the physics of pc-scales jets, as well as particle acceleration and high-energy emission in the large-scale jets of AGNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' 7th Heidelberg International Symposium on High-Energy Gamma-Ray Astronomy (Gamma2022) 4-8 July 2022 Barcelona, Spain © Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='0 International License (CC BY-NC-ND 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' https://pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='sissa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='it/ arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='13494v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='HE] 31 Jan 2023 Theory of Gamma-Ray Loud AGNs 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Introduction Radio-loud or jetted AGNs [1], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' AGNs with strong relativistic jets, are the most persistent, powerful sources in the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Their detection at gamma-ray energies reveals that a significant amount of their power is deposited a highest energies, and demonstrates them to be exceptional cosmic particle accelerators [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' All types of jetted AGNs have been seen at gamma-ray energies, from more misaligned radio galaxies to blazar-type sources where the jets are seen face-on [3–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Gamma-ray astrophysics is thus expected to play a fundamental role in resolving key issues in AGN physics, such as: (i) How are relativistic jets being formed?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Are they preferentially ergospheric or disk-driven?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' If the former, what is their plasma source;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' if the latter, what is their accretion-disk connection?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' (ii) What makes jets radiate?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' What is the dominant gamma-ray radiation process and particle acceleration mechanism?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Where is it located?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' What is the plasma composition?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' (iii) How are small and large scales connected?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' How is the energy transported from the black hole (BH) to the outer lobes scales?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' How are jets confined?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' What kind of instabilities are important?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' On the Extragalactic Gamma-Ray Sky The extragalactic sky has become bright at gamma-ray energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' The Fermi-LAT 12 year point source catalog (4FGL-DR3) for example, reports the detection of 6658 sources at high energies (HE) > 50 MeV, with more than 3740 identified as belonging to the blazar class of AGNs, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' About 70 sources have been identified as non-blazar AGNs, out of which 45 are radio galaxies Figure 1: High-energy sky map (4FGL) based on 8 yr of Fermi-LAT data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' All AGNs (all classes) are plotted with the same blue symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' From ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' ©AAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Reproduced with permission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' (among them 22 Fanaroff-Riley (FR) I and 14 FR II) and 8 are Narrow Line Seyfert 1 (NLSy 1) galaxies [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' At very high energies (VHE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' ≥ 100 GeV), about 85 AGNs are listed in the TeVCat catalog, with redshifts up to 𝑧 ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' While most of the TeV sources are BL Lac objects (55 HBL, 10 IBL, 2 LBL), a few prominent radio galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', Cen A, M87 and NGC 1275) are present as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' The latter sources have raised considerable interest by allowing insights into the immediate vicinity of the supermassive BH environment in AGNs (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='1 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' For a recent review of 2 No association 回 PossibleassociationwithSNRorPwN AGN Pulsar Globular cluster Starburst Galaxy PWN A Binary Galaxy SNR + Nova Star-forming region 回 UnclassifiedsourceTheory of Gamma-Ray Loud AGNs the physics case of gamma-ray emitting, non-blazar AGNs, the reader is referred to ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Spectral and timing capabilities at gamma-ray energies have significantly developed over time, allowing, in some sources, to probe rapid variability down to a few minutes and unusual spectral features that could signal new physical processes, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Figure 2: Left: The VHE SED of Mkn 501 (𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='034) during an elevated state in 2014 July 19 as measured by MAGIC, revealing an unusual feature at ∼ 3 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' From ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [9], reproduced with permission © ESO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Right: The VHE light curve of PKS 2155-304 (𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='116) during its famous outburst in July 2006 as seen by H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', revealing substantial flux changes down to ∼ 3 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' From ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' The detection of NLSy 1 galaxies at HE energies has raised several issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Radio-quiet (non- jetted) NLSy 1 are commonly thought to be high-Eddington sources, with moderate BH masses (∼ 106−8𝑀⊙), which are hosted by spiral galaxies [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Since HE-emitting (radio-loud) NLSy 1 appear to be of the jetted AGN type, showing some blazar-like properties such as one-sided jets and superluminal motion, this has triggered discussion about the BH mass limit and accretion/merger history conducive for the origin and formation of relativistic jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Relevant to this discussion is the question, whether jetted NLSy 1 might belong to a special subclass harbouring BHs with larger masses and hosted by elliptical galaxies instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' There are growing indications, however, that many of them are better modelled as disk-like galaxies [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' For a discussion of these topics and related literature, the reader is referred to the recent overviews in refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' On Challenges and Progress in (jetted) AGN Physics 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='1 Challenges in AGN Physics Understanding the physics of AGN jets and the role played by its supermassive BH is a challenging, multi-scale problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Phenomenologically, the observed scale separation covers a range of almost ten orders (as in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', Cen A) of magnitude, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' This fact translates into a fundamental physics and modelling challenge of how to consistently bridge these different scales, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', of how to connect the global, source dynamical scale, the radiation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', synchrotron, inverse Compton) scale and the dissipation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', shock, reconnection, turbulence) scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' While solid progress has been achieved over the years, no complete picture is existing up to now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' In principle, this also applies to numerical simulations (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [16, 17] for related reviews).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' While essential to inform and advance our understanding, they are not without caveats, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', in general, AGN physics is also accompanied by a methodological (computational) challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=" In the context of jet formation for example, conventionally employed general-relativistic/magneto-hydrodynamic 3 I(>200 GeV) [10′ cm2 s'] 10-10 3." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='5 SED [TeV cm-2 s-1] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='5 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='5 LP fit LP+EP fit 10-11, MJD 56857.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='98 Observed 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='5 102 103 40 60 80 100 20 Time - MJD53944.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='0 [min] E [GeV]Theory of Gamma-Ray Loud AGNs Figure 3: The radio galaxy M87 as seen from large, radio-halo (VLA) down to black-hole (EHT) scales, corresponding to an "observed scale separation" of ∼ 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Adapted based on ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' (GR/MHD) simulations usually rely on an ambiguous numerical floor model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' methodologically, it may seem rather surprising that a single fluid description is able to provide important insights (as it has) in cases where we expect the plasma to be collisionless;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' further, in ideal MHD there is little physical understanding of reconnection, particle acceleration and radiation, and so forth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' On the other hand, first-principle particle-in-cell (PIC) simulations in astrophysics mostly deal with highly idealized setups, often in reduced dimensionality and for quite limited duration, along with artificially large gyro-radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' In fact, in the case of AGNs, the relevant scale separation (system size 𝑟 vs plasma skin depth 𝑙𝑝) can be as high as 𝑟/𝑙𝑝 ∼ 106−8, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [18], and it remains unclear to which extent features seen in these simulations might persist up to the physical scale of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Notwithstanding these limitations, numerical simulations have become an indispensable tool for progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='2 Progress in AGN Physics The last years have seen both, significant progress and consolidation of knowledge in our understanding of the physics of AGNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' The following provides a short selection of exemplary results with relevance to gamma-ray emitting AGNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='1 Supermassive Black Holes in the Center of AGNs The presence of supermassive BHs in galactic nuclei has been suggested for more than half a century based on strong theoretical arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Early experimental efforts related to dynamical searches have been described ∼ 25 yr ago [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Most recently, theoretical and experimental progress in BH research has eventually been documented by the 2020 Nobel prize for Physics to Roger Penrose, Reinhard Genzel and Andrea Ghez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' While the benchmark BH Sgr A* is not (yet) an established gamma-ray emitter [20], the event horizon telescope (EHT) radio image of the BH shadow in M87 [21] provides important information relevant to VHE research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' The size ≃ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='5𝑟𝑠 of the photon ring in M87 (where 𝑟𝑠 := 2𝑟𝑔 := 2𝐺𝑀BH/𝑐2), implies a BH mass of 𝑀BH ≃ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='5×109𝑀⊙ and constrains horizon scale (minimum) variability to 𝑡𝐻 ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='38 d, close to what can be probed 1For recent results on hadronic processes and neutrino emission, the reader is referred to the related reviews by Paolo Padovani and Elisa Resconi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' 4 VLA VLBA VLBI 25 kpc 800 pc 20 pc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='5 pc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='05 pc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='005 pcTheory of Gamma-Ray Loud AGNs with VHE instruments [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Further research along these lines (by, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', the next-generation EHT) will allow to probe deeper into the central engine in AGNs (BH - disk - jet, its connection and dynamics), and tighten the constraints on the current jet power (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', BH spin) relevant to VHE emission models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='2 Convergence of theoretical, numerical & observational evidence for jet stratification There is consolidation of knowledge that the jets in AGN are multi-layered, revealing a lateral stratification similar to the one induced by a fast BH-driven (Blandford-Znajek: BZ, [22]) jet sur- rounded (and possibly confined) by a slower moving disk-driven outflow, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Resolved (lateral) emission structures such as limb-brightening and linear polarisation signatures, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [24, 25], substantiate early two-flow and spine-sheath type non-thermal emission models [26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' In the case of M87, for example, significant structural patterns across the jet on sub-pc-scale have been detected, indicating the presence of both slow (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='5𝑐) and fast (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='92𝑐) components [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Pro- nounced edge-brightened features have now been seen in both, M87 and Cen A [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' In the high energy context, characterising internal jet stratification is potentially important to, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', address the (putative) Doppler factor "crisis" in TeV (HBL) blazars [30], to develop more advanced acceleration (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='3) and emission models, and to probe into AGN unification scenarios (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', relative power of inner vs outer jet?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='3 Acceleration and collimation of relativistic jets Detailed radio studies now provide insights into the jet collimation profile in more and more nearby sources [31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' In the case of M87 for example, the jet width profile is initially (semi- )parabolic and then transitions (at around the Bondi radius 𝑟𝐵 ∼ 5 × 105𝑟𝑔 ∼ 150 pc) to a conical shape, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' There is evidence that the radio flow is initially slow (on scales ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='03 pc) and gradually accelerates with distance, reaching Γ𝛽 ∼ 3 on scales of 𝑟𝐵 [33, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' This can be Figure 4: The jet collimation profile for the radio galaxy M87 from sub-parsec to kilo-parsec scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' From ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' ©AAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Reproduced with permission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' 5 de-projected distance (pc) 10-4 10-3 10-2 10-1 1 101 102 103 104 106 VLBA15GHz(AN12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='H13) conical z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' α R MERLIN 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='8 GHz (AN12) EVN1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='6GHz(AN12) VLBA22GHz(H13) VLBA 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='3GHz (H13) VLBA43GHz(AN12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='H13) 105 VLBA 5 GHz (H13) HSA 86 GHz (H16) VLBA 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='4 GHz (H13) VLBACore43&86GHz(NA13) 104 ■VLBACore5,8,15,24,43,&86GHz(H13),HSACore86GHz(H16) △EHTCore230GHz(D12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='A15) (Area)FFEparabolicjet(NMF07,TMN08) (Area)FFEgenuineparabolic jet(NMFO7,TMNO8) 103 a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='5 (upper edge) - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='99 (lower edge) UUT HST-1 parabolic 102 UTT 2 α R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='6 101 radius UTT Bondi 1 T 1 101 102 103 104 105 106 107 108 Jet axial distance (de-projected): z (r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=')Theory of Gamma-Ray Loud AGNs understood in terms of a jet that is magnetically dominated at its base and whose magnetic energy is gradually converted into kinetic energy while experiencing some external confinement [35, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' These results are instructive for improving our understanding of the non-thermal emission in low- luminosity FR I type sources, and of the dynamical role (confinement) played by a disk-driven wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' While consequences of flow deceleration on high-energy SED modelling have been explored in the past [37], consideration of flow acceleration seems an important desideratum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='4 Convergent evidence for relativistic motion in large-scale AGN jets While jets experience deceleration on larger scales, they can still be substantially relativistic on kilo-parsec scales, even for moderately powerful sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' The nearby (𝑑 ∼ 95 Mpc) FR I radio galaxy 3C 264 is a prototype example (with estimated jet power 𝐿 𝑗 ∼ 5 × 1043 erg/s) [38], exhibiting superluminal motion and collision of knots (internal shocks) in HST observations on scales of ∼ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='5 − 1) kpc, corresponding to a bulk flow Lorentz factor Γ ∼ 7 and a Doppler factor 𝐷 ∼ 7 for an inclination 𝜃 ∼ 8◦ [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' These findings are of interest not only because 3C264 is a recently detected VHE emitter [40], but also by offering general insights into particle acceleration at shocks, and the role of nearby large-scale jets as putative sites of ultra-high energy cosmic ray (UHECR) acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='5 Extreme TeV blazars A sizeable fraction of VHE-emitting AGNs (∼ 1/4 of all HBL) are now composed of BL Lac objects with hard intrinsic VHE spectra, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' with de-absorbed power-law (PL) photon indices ΓVHE ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='5 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='9, implying an SED bump peaking above 1 TeV, see ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [4] for a review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' The prototypical source of these extreme highly peaked BL Lacs (EHBLs) is 1ES 0229+200 (𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='14) with an SED peak (de-absorbed) above 4 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' The SED modelling of these sources is challenging, and in the common synchrotron-self Compton (SSC) approach requires unusual (if not extreme) conditions such as an electron distribution with a very high minimum Lorentz factor 𝛾min ≳ 104, or one following a relativistic Maxwellian [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' The narrow pile-up feature seen in Mkn 501 [9], cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' 1 (left), might be reminiscent of the latter [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' These extreme TeV blazars are particularly interesting as they indicate emitting regions significantly away from equipartition (𝑢𝐵/𝑢𝑒 ≲ 10−4 in SSC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' The implied low magnetisation 𝜎 would seem to disfavour reconnection and instead point to stochastic or multiple shocks as possible acceleration mechanism (see also Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' VHE variability studies will enable to probe into this as only modest (≥ month-type) intrinsic variability would be expected (unless geometrical effects would interfere).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Unusual spectral shapes at VHE energies will complicate EBL (Extragalactic background light) studies using simple extrapolation functions (for a recent review of developments, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [43]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='6 Gamma-Ray Astrophysics in the Time Domain Within the last decade, gamma-ray astronomy has successfully entered the time domain [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Key examples include the detection and characterisation of ultra-fast VHE variability (down to a few minutes) [4, 45] as well as evidence for year-type quasi-periodic oscillations (QPOs) in the Fermi-LAT light curves of blazars [46, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' A prominent example of the latter is the HBL object PG 1553+113 (𝑧 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='5) with an (observed) HE period of ≃ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='2 yr, possibly caused by a close supermassive binary BH system (SBBHs) [48, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Application of more advanced time series 6 Theory of Gamma-Ray Loud AGNs techniques, exploring power-law noise characteristics and log-normality has been steadily growing [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Theoretical efforts are now gaining momentum, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [44] for a review, and refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [51, 52] for selected contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Properly understanding the variability characteristics has important implications for the origin of the emission (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', location and driving process) and the overall source dynamics and evolution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', SBBHs, disk physics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Selected Theoretical Advances The following subsections aim to provide exemplary and more detailed insights into three different fields where theoretical advances in AGN physics, whether conceptual or methodological, have been significantly influenced by gamma-ray observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='1 Magnetospheric Processes in AGNs The detection of VHE variability on BH horizon crossing time scales (∼ 𝑟𝑔/𝑐), as in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' M87, has been a significant driving force for the development of magnetospheric particle acceleration and emission models, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [8] for a review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Magnetic fields that are brought in and dragged into rotation (Ω𝐹) by the BH can induce an electric field component (parallel to the magnetic field), that, if not screened, can facilitate efficient particle acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' This could potentially occur either around the so-called null surface across which the Goldreich-Julian [53] charge density, 𝜌GJ ∼ (Ω𝐹 − 𝜔)𝐵, required to screen the electric field, changes sign, or at the stagnation surface that separates MHD in- and outflows, see Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' The maximum available voltage drop for a BH Figure 5: Left: Illustration of possible locations of charge-deficient regions (gaps) around a rotating BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' The red line denotes the null surface across which 𝜌GJ changes sign, while the blue line delineates the stagnation surface from which stationary MHD flows start.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Right: Exemplary gap evolution for a Kerr BH of 109𝑀⊙ accreting in a radiatively inefficient mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' The curves show the radial distribution of the parallel electric field around the null surface for different accretion rates �𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' The gap width and voltage drop increase as �𝑚 drops because pair creation of energetic photons with the soft photon field becomes less efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' From ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' gap of width ℎ is of the order of [55] ΔΦ = 1 𝑐Ω𝐹𝑟2 𝐻 𝐵𝐻 (ℎ/𝑟𝑔)2 ≃ 2 × 1021 �𝑚1/2𝑀1/2 9 (ℎ/𝑟𝑔)2 [𝑉] , (1) for a horizon-threading field 𝐵𝐻 ≃ 2 × 105 �𝑚1/2𝑀−1/2 9 G, where 𝑀9 ≡ 𝑀BH/109𝑀⊙, Ω𝐹 = Ω𝐻/2 = 𝑐/4𝑟𝑔, and �𝑚 ≡ �𝑀/ �𝑀Edd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Electrons that get accelerated in these gaps produce 𝛾-rays 7 0 Stagnation Surface 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='05 E Parallel electric field, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='1 IInN PGJ <0 Surface 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='15 PG >0 magnetic field accretion rate: 10-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='2 accretion rate: 10-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='0 BH Ergo sphere 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='6 rH s=r/rTheory of Gamma-Ray Loud AGNs via curvature radiation and inverse Compton (IC) up-scattering of ambient soft photons, producing an observable and variable VHE signal [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Interaction (𝛾𝛾-absorption) of energetic 𝛾-rays with low-energy soft photons can trigger a pair cascade that generates plasma and ensures closure of the gap [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' On conceptual grounds, magnetospheric gaps thus provide a self-consistent means for the continuous plasma supply ("numerical floor") needed to activate a force-free (BZ) MHD outflow [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Correspondingly, unless accretion rates are sufficiently low (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', well below �𝑚 ∼ 10−4), efficient pair production will occur (if non-negligible gaps form at all, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [56, 58]), leading to gap sizes ℎ ≪ 𝑟𝑔, thereby significantly reducing the accessible gap potential, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Typically, achievable (radiation-limited) electron Lorentz factors are constrained to 𝛾𝑒 ∼ 108−10 [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' To improve insights into characteristic electric field strengths and achievable voltage drops, one can seek for self-consistent steady gap solutions [54, 58] by solving the system of relevant partial differential equations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', Gauss’ law, equation of particle motion, continuity equation), see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Figure 5 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' For example, taking an accretion rate �𝑚 = 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='75, which is close to the mean MAD value used in EHT-GRMHD simulations for M87 [59] and which corresponds to jet powers of a few times 1043 erg/s, the gap properties for M87 (assuming the inner disk to be ADAF-type) can be evaluated [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' The resultant gap sizes are of the order of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='8𝑟𝑔, suggesting that the VHE emission in M87 might be variable down to timescales of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='4 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' The inferred gap power of ∼ 5×1041 erg/s would make it in principle possible to accommodate the VHE emission seen during its high states [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' The achievable voltage drop would be of the order of ∼ 1018 V, suggesting that the BH in M87 is not an efficient UHECR proton accelerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' The appreciation that gap operation is expected to be intermittent, has in recent times triggered a variety of time-dependent investigations using PIC simulations, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [61–64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Relative comparison is often not straightforward as different setups and complexities have been employed (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', with regard to box sizes, cell numbers, run times, 1d/2d, radiation reaction, choice of soft photon field).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Generically, the outcome is highly sensitive to the assumed ambient soft photon field (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', minimum photon energy 𝜖min and spectral shape, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', PL index), making an adequate description an important prerequisite as it determines the efficiency of pair cascades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' There are indications for a periodic opening (timescale ∼ 𝑟𝑔/𝑐) of macroscopic gaps (with ℎ ∼ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='1−1] 𝑟𝑔), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Issues that deserve further studies concern e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', the degree to which detected oscillation periods could be dependent on the simulation setup (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', box size), and the extent to which the emerging particle distributions might be bimodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Insights gained will help to better characterise the link between gap activity, VHE flaring and jet formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='2 Modelling Parsec-Scale Jet Emission in Blazars The spectral energy distributions (SEDs) of gamma-ray blazars have for long been modelled by leptonic (SSC) and/or hadronic (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', 𝑝𝛾) radiation processes, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [65, 66] for a review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Quite often their SEDs can be satisfactorily reproduced with different sets of assumptions, which significantly limits inferences on the source physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Incorporating microphysical insights from kinetic plasma simulations into SED modelling can allow an important step forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' One exem- plary context relates to the requirement of high minimum electron Lorentz factor 𝛾min,e and low magnetisation 𝜎 in SSC models for extreme TeV blazars (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' A recently proposed framework considers co-acceleration (diffusive shock) of electrons and protons at mildly relativistic (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', Γ𝑠 ∼ 3), weakly magnetised (internal or recollimation) shocks in blazar jets [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Shocks are known to convert a significant fraction of the kinetic energy of the incoming plasma flow 8 Theory of Gamma-Ray Loud AGNs Figure 6: Exemplary gap cycle seen in 1d PIC simulations (spit monopole field) for a steep ambient photon field (𝑛𝑠) with 𝜖min = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='0005 eV and fiducial Thomson depth 𝜏0 := 𝑛𝑠𝜎Th𝑟𝑔 = 100 (yielding a pair creation length ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='3𝑟𝑔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Top panel: Evolution of the normalized densities of electrons, positrons and photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Bottom panel: Evolution of the electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Vertical dashed line denotes the null surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' From ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' ©AAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Reproduced with permission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' to thermal energy, represented by a Maxwellian particle distribution 𝑛(𝛾) ∝ 𝛾2 exp[−𝛾/Θ] with Θ ≡ 𝑘𝐵𝑇/𝑚𝑐2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Assuming an efficient energy transfer (heating/thermal coupling) from protons to electrons in the shock transition layer, one can write 𝑘𝐵𝑇𝑃 ∼ Γ𝑠𝑚 𝑝𝑐2 and 𝑇𝑒 = 𝜉𝑇𝑝 where 𝜉 < 1, with Γ𝑠 the Lorentz factor of the shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' If the peak in the Maxwellian electron distribution is identified with 𝛾min,e, then roughly 𝛾min,e ∼ 𝑘𝐵𝑇𝑒/(𝑚𝑒𝑐2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' 𝛾min,e ∼ �𝑚 𝑝 𝑚𝑒 � Γ𝑠𝜉 ≃ 1800 Γ𝑠𝜉 (2) Hence, given suitable conditions, a high 𝛾min,e is naturally expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' In PIC simulations of highly relativistic shocks, efficient Fermi-type particle acceleration has only been seen for weakly magnetized (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', 𝜎 <∼ 10−3) shocks, in line with considerations in the above framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Since these shocks will also generate magnetic (micro)turbulence, the effective magnetization (𝜎 := 𝐵2/(4𝜋⟨𝛾⟩𝑛𝑚𝑐2) could in principle be higher than the pre-shock one, yet decaying downstream with distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Accordingly, the magnetization in the radiation zone, 𝜎rad, may be in the range 𝜎 <∼ 𝜎rad <∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' For high (Alfvenic) Mach numbers, significant thermal coupling can occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' In large 2d PIC simulations of quasi-parallel, mildly relativistic (𝛽𝑠 ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='8), weakly magnetized (𝜎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='007, defined downstream) electron-ion shocks (with reduced mass ration 𝑚 𝑝/𝑚𝑒 = 64), for example, 𝑇𝑒 ∼ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='2−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='3) 𝑇𝑝 and 𝑘𝐵𝑇𝑝 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='2𝑚 𝑝𝑐2 has been seen, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' 7 [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' For smaller magnetizations and larger shock speeds, electron preheating can be further increased [69, 70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Figure 8 shows an exemplary reproduction of the SED of the prototypical EHBL 1ES 0229+200 motivated by such a framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' In this case, re-acceleration at multiple (𝑛 ≥ 2) shocks is needed to account for the hard particle spectra required for spectral fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' This leads to a modification (hardening and power-law deviation) of the input, single shock electron spectrum (of power law momentum index 𝑠 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='2), and also increases the effective injection Lorentz factor to 𝛾min,eff ∼ Γ2𝑛/3 𝑠 𝛾min,e [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' 9 t/(r t/(r =191.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='55 t/(rg /c) =191.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='97 t/(ra /C) =192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='29 101 100 △n/△n 10 :01X ( 0 5 10 15 2 3 4 2 3 4 2 3 4 2 3 4 YTheory of Gamma-Ray Loud AGNs Figure 7: 2d PIC simulations of a weakly magnetized, mildly relativistic, high Mach number (𝑀𝐴 = 15) shock with Γ𝑠 ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Right: Shock evolution and magnetic field generation (initially Weibel-, later Bell- mediated) with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Right: Downstream particle distributions 𝑝4 𝑓 (𝑝) at different times, with the formation of a non-thermal power-law tail on top of a thermalized particle distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' At late times, the downstream electron temperature is 20 − 30% of the ion temperature, indicating significant coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Based on ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Figure 8: Reproduction of the SED of the EHBL source 1ES 0229+200 in a multiple shock scenario with input parameters for the initial electron distribution as shown in the table to the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' From ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [67], reproduced with permission © ESO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' The noted framework represents an exemplary approach to utilise microphysical insights for blazar SED modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' One central assumption is that the emission arises in a jet region that is kinetically dominated by an electron-proton composition (with protonic emission expected to be suppressed) rather than an 𝑒+𝑒− one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Similar indications have been found in the case of another prominent 𝛾-ray blazar, PKS 2155-304, based on a statistical analysis of its X-ray emission [71], though there is also other evidence (based on optical circular polarization) suggesting that the (non-thermal) positron fraction cannot be too low [72], cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' also ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' A possible caveat for the noted approach relates to the model requirement of a region significantly out of equipartition and characterized by a very low magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' In principle, variability studies could allow to probe into these assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' A related framework considers the electrons in the jet to be (initially) diffusively accelerated at a recollimation-type shock, but then further energized through stochastic (2nd order Fermi) 10 downstream spectrum 25 y [clopi] Ions n/no 10-2 6 io1 3375 y [clOpi], 100 10-4 0 101 50 75 100 125 150 175 200 225 250 x [clopi] 10-6 50 12 Electrons 10-2 Wpit = 4148 n/no 6 0 Te~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='23T, 50 iol f(p) αp-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' 2 10- 4 100 25 n(E) α E-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='2 0 101 150 200 250 300 350 400 450 500 x [clopi] 10-6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='0 10-1 100 101 10-2 p/(yoβomic)-9 1 ES 0229+200 Modelling parameters 10 (multiple shocks) 50 Rsrc [1016cm] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='3 B[mG] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='4 12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='0 × 10-4 O rad 270 rad/o .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='8 × 103 Ye,min Ye,max 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='9 × 106 14 ne [cm3] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='13 12 14 16 18 20 22 24 26 28 log (v [Hz])Theory of Gamma-Ray Loud AGNs acceleration in the downstream flow [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Hence, instead of acceleration at multiple shocks, recollimation in a weakly magnetized flow is presumed to result in strong turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' While the synchrotron spectrum can be satisfactorily reproduced in a SSC model where synchrotron cooling is comparable to particle escape (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' also [41] for another variant), the Compton SED component appears somewhat too hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Since EHBL SED models are in general far from equipartition (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', 𝜎 ≪ 1), characteristic Alfven speeds are sub-relativistic (𝑣 𝐴 ∝ √𝜎 ≪ 𝑐) making stochastic acceleration less efficient (𝑡 ∝ 1/𝑣2 𝐴), so that it remains to be explored to what extent a self- consistent SED fit can be achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Nevertheless, these approaches represent instructive examples of how to fruitfully combine insights from jet simulation, particle acceleration physics and SED modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='3 Understanding the large-scale jet emission of AGNs The origin of extended X-ray emission along the large-scale jets in AGNs has for long been a matter of debate, with electron-synchrotron and IC-CMB scenarios as the leading contenders, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [75, 76] for review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Within more recent years, however, IC-CMB models have been disfavoured in an increasing number of sources as they tend to, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', over-predict Fermi-LAT gamma-ray flux limits [77, 78] or are not supported by detailed X-ray spectral information [79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' An electron synchrotron origin, however, requires to maintain electrons with Lorentz factors 𝛾𝑒 ∼ 108 all along the jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' The short cooling timescales 𝑡cool ∝ 1/𝛾 of these electrons, and correspondingly short cooling lengths 𝑐𝑡cool ≪ 1 kpc, calls for the operation of a "distributed" acceleration mechanism along the jet, beyond "localised" shock acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' The recent detection of extended VHE emission along the kpc-scale jet of Cen A [80] has provided an important validation to this picture, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' 9, by directly tracing the electrons and removing the degeneracy (𝐵, 𝛾) inherent in an X-ray synchrotron model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' IC up-scattering of dust by ultra-relativistic (synchrotron X-ray emitting) electrons with 𝛾𝑒 = 108 accounts for the observed VHE emission and confirms the X-ray synchrotron interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' The results imply a jet that is only weakly magnetized (𝜎 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='5) on kiloparsec scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Stochastic (2nd order Fermi) and shear particle acceleration processes are among the most promising mechanisms facilitating continuous re-acceleration in kinetically dominated jets [82–85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Particle acceleration in the latter framework essentially draws on the jet flow velocity difference that a charged particles experiences as it moves across the jet shear (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='2), see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [86] for an accessible introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' It can be viewed as a second-order (Δ𝐸 ∝ 𝑢2 sc𝐸) Fermi-type acceleration process, in which the common scattering center speed, 𝑢sc, is replaced by an effective velocity, ¯𝑢, determined by the shear flow profile, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', by the characteristic flow velocity change sampled over the mean free path 𝜆 ≃ 𝜏𝑐 of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Hence, for instance ¯𝑢 = (𝜕𝑢𝑧/𝜕𝑟) 𝜆 for a continuous shear profile �𝑢 = 𝑢𝑧(𝑟) �𝑒𝑧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' This results in a characteristic acceleration timescale 𝑡acc ≃ 𝐸 𝑑𝐸/𝑑𝑡 ≃ 𝜏 ⟨Δ𝐸/𝐸⟩ ∝ 𝜏 ¯𝑢2 ∝ 1 𝜆 , (3) scaling inversely with the particle mean free path 𝜆 or scattering time 𝜏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' While this makes shear flows also interesting for UHE cosmic-ray acceleration in the jets of AGNs [87], it typically requires some energetic seed injection for electron acceleration to proceed efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' In principle, this could be achieved by shock or classical 2nd order Fermi acceleration (for which 𝑡acc ∝ 𝜆) [82], or possibly, by kinetic-scale reconnection processes [88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Phenomenologically, this could result 11 Theory of Gamma-Ray Loud AGNs Figure 9: Extended VHE emission along the kpc-scale jet of Cen A as seen by H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' The SED of the kpc-scale jet can be reproduced by means of an (exponential-cutoff) broken-power law electron distribution with 𝛾𝑏 ≃ 106 and cut-off energy 𝛾𝑐 ≃ 108, for a characteristic jet magnetic field strength of 𝐵 ≃ 20𝜇G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Interestingly, IC emission by the kpc-scale jet may be behind the unusual spectral hardening [81] seen at GeV energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Based on ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' in multi-component particle distributions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', broken power-laws with changes in spectral indices different from a classical cooling break.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' A recent application shows that in the presence of a Kolmogorov-type 𝜆 ∝ 𝛾1/3, the SED of the kpc-scale jet in Cen A can be successfully reproduced within a linear shear model for spine velocities of 𝑣 𝑗 ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='7c and shear layer widths of Δ𝑟 ≃ 100 pc [84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' In general, efficient shear acceleration requires relativistic flow speeds [89, 90] and this confines its applicability to astrophysical objects with fast outflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' There is currently a promising trend to incorporate particle acceleration and non-thermal emission recipes in large-scale MHD jet simulations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' While such a hybrid approach is vital, it should be kept in mind that it still involves significant extrapolations (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=') that may need to be further explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' The prospects are good, though, for achieving further progress soon as several new observational facilities are coming online [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Conclusions By providing new and often unexpected insights, multi-wavelengh and multi-messenger obser- vations have become an indispensable tool for progress in many areas of astronomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' The field of AGNs, and in particular jetted AGNs, to which most extragalactic gamma-ray sources belong, is no exception in this regard [92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' In the present review, I have highlighted some of the developments that are of particular relevance for our understanding of the nature of these sources, and/or directly driven and inspired by gamma-ray observations, ranging from new observational findings to conceptual and methodological advances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' I believe that we have seen true progress in our understanding of the physics of gamma-ray loud AGNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' As for jetted AGNs in general and for gamma-ray loud AGNs in particular: "The observational prospects for securing our understanding of AGN jets are bright.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='" (Blandford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', [15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' In particular, given available (from e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Fermi-LAT and the current IACTs, HESS, MAGIC and VERITAS to HAWC, LHAASO) and upcoming capabilities (CTA) at highest photon energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' 12 10-10 radio convolved with H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=" PSF core 42°56'00." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='0" 10-11 synchrotron IC 58\'00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='0" E2 dN/dE [erg s-1 cm-2] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='2 kpc 10-12 optical H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=" (2000) kpc jet radio X-rays 43°00'00." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='0" 10-13 Dec 02\'00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='0" 10-14 IC (CMB) IC (starlight) IC (dust) 10-15, IC (SSC) 04\'00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='0" IC (total) Sync 10-16 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=" 10-8 10-6 10-4 10-2 100 102 104 106 108 1010 10121014 06'00." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='0" Photon energy [eV] 26m00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='00s 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='00s 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='00s 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='00s 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='00s 13h25m0iTheory of Gamma-Ray Loud AGNs Acknowledgments I would like to thank Josep Maria Parades, Valenti Bosch-Ramon, Pol Bordas, Marc Ribo and the whole Barcelona LOC team for a truly pleasant and inspiring conference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' I am grateful to Felix Aharonian for fruitful collaboration over the years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Funding by a DFG Fellowship under RI 1187/8-1 is gratefully acknowledged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' References [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Padovani, On the two main classes of active galactic nuclei, Nature Astronomy 1 (2017) 0194 [1707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='08069].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Biteau, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Prandini, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Costamante, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Lemoine, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Padovani, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Pueschel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', Progress in unveiling extreme particle acceleration in persistent astrophysical jets, Nature Astronomy 4 (2020) 124 [2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='09222].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [3] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Urry and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Padovani, Unified Schemes for Radio-Loud Active Galactic Nuclei, PASP 107 (1995) 803 [astro-ph/9506063].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [4] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Costamante, Blazars - an updated review, in Multifrequency Behaviour of High Energy Cosmic Sources - XIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' 3-8 June 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Palermo, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' 35, Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', 2020, DOI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [5] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Rulten, Radio Galaxies at TeV Energies, Galaxies 10 (2022) 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [6] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Abdollahi, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Acero, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Ackermann, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Ajello, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Atwood, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Axelsson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', Fermi Large Area Telescope Fourth Source Catalog, ApJS 247 (2020) 33 [1902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='10045].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [7] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Abdollahi, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Acero, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Baldini, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Ballet, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Bastieri, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Bellazzini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', Incremental Fermi Large Area Telescope Fourth Source Catalog, ApJS 260 (2022) 53 [2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='11184].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [8] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Rieger and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Levinson, Radio Galaxies at VHE Energies, Galaxies 6 (2018) 116 [1810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='05409].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [9] MAGIC Collaboration, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Acciari, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Ansoldi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Antonelli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Babić, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Banerjee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', Study of the variable broadband emission of Markarian 501 during the most extreme Swift X-ray activity, A&A 637 (2020) A86 [2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='07729].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [10] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Aharonian, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Akhperjanian, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Bazer-Bachi, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Behera, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Beilicke, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Benbow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', An Exceptional Very High Energy Gamma-Ray Flare of PKS 2155-304, ApJL 664 (2007) L71 [0706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='0797].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [11] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Komossa, Narrow-line Seyfert 1 Galaxies, in Revista Mexicana de Astronomia y Astrofisica Conference Series, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' 32 of Revista Mexicana de Astronomia y Astrofisica Conference Series, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' 86–92, Apr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', 2008, DOI [0710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='3326].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [12] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Varglund, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Järvelä, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Lähteenmäki, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Berton, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Ciroi and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Congiu, Jetted narrow-line Seyfert 1 galaxies breaking the jet paradigm – a comprehensive study of host galaxy morphologies, arXiv e-prints (A&A accepted) (2022) arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='09270 [2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='09270].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' 13 Theory of Gamma-Ray Loud AGNs [13] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' D’Ammando, Relativistic Jets in Gamma-Ray-Emitting Narrow-Line Seyfert 1 Galaxies, Galaxies 7 (2019) 87 [1911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='03500].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [14] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Foschini, Jetted Narrow-Line Seyfert 1 Galaxies & Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=': Where Do We Stand?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', Universe 6 (2020) 136 [2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='13383].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [15] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Blandford, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Meier and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Readhead, Relativistic Jets from Active Galactic Nuclei, ARA&A 57 (2019) 467 [1812.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='06025].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [16] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Porth, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Chatterjee, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Narayan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Gammie, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Mizuno, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Anninos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', The Event Horizon General Relativistic Magnetohydrodynamic Code Comparison Project, ApJS 243 (2019) 26 [1904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='04923].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [17] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Nishikawa, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Duţan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Köhn and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Mizuno, PIC methods in astrophysics: simulations of relativistic jets and kinetic physics in astrophysical systems, Living Reviews in Computational Astrophysics 7 (2021) 1 [2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='02105].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [18] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Ji and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Daughton, Phase diagram for magnetic reconnection in heliophysical, astrophysical, and laboratory plasmas, Physics of Plasmas 18 (2011) 111207 [1109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='0756].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [19] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Kormendy and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Richstone, Inward Bound—The Search For Supermassive Black Holes In Galactic Nuclei, ARA&A 33 (1995) 581.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [20] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Katsoulakos, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Rieger and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Reville, Constraining Cosmic-Ray Acceleration in the Magnetospheric Gaps of Sgr A*, ApJL 899 (2020) L7 [2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='12189].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [21] Event Horizon Telescope Collaboration, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Akiyama, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Alberdi, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Alef, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Asada, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Azulay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', First M87 Event Horizon Telescope Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' The Shadow of the Supermassive Black Hole, ApJL 875 (2019) L1 [1906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='11238].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [22] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Blandford and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Znajek, Electromagnetic extraction of energy from Kerr black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', MNRAS 179 (1977) 433.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [23] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Mizuno, GRMHD Simulations and Modeling for Jet Formation and Acceleration Region in AGNs, Universe 8 (2022) 85 [2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='12608].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [24] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Boccardi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Krichbaum, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Ros and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Zensus, Radio observations of active galactic nuclei with mm-VLBI, A&ARv 25 (2017) 4 [1711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='07548].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [25] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Kim, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Krichbaum, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Lu, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Ros, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Bach, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Bremer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', The limb-brightened jet of M87 down to the 7 Schwarzschild radii scale, A&A 616 (2018) A188 [1805.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='02478].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [26] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Sol, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Pelletier and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Asseo, Two-flow model for extragalactic radio jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', MNRAS 237 (1989) 411.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [27] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Ghisellini, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Tavecchio and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Chiaberge, Structured jets in TeV BL Lac objects and radiogalaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Implications for the observed properties, A&A 432 (2005) 401 [astro-ph/0406093].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' 14 Theory of Gamma-Ray Loud AGNs [28] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Mertens, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Lobanov, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Walker and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Hardee, Kinematics of the jet in M 87 on scales of 100-1000 Schwarzschild radii, A&A 595 (2016) A54 [1608.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='05063].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [29] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Janssen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Falcke, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Kadler, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Ros, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Wielgus, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Akiyama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', Event Horizon Telescope observations of the jet launching and collimation in Centaurus A, Nature Astronomy 5 (2021) 1017 [2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='03356].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [30] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Henri and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Saugé, The Bulk Lorentz Factor Crisis of TeV Blazars: Evidence for an Inhomogeneous Pileup Energy Distribution?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', ApJ 640 (2006) 185 [astro-ph/0511610].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [31] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Hada, Relativistic Jets from AGN Viewed at Highest Angular Resolution, Galaxies 8 (2019) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [32] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Kovalev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Pushkarev, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Nokhrina, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Plavin, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Beskin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Chernoglazov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', A transition from parabolic to conical shape as a common effect in nearby AGN jets, MNRAS 495 (2020) 3576 [1907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='01485].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [33] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Nakamura, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Asada, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Hada, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Pu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Noble, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Tseng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', Parabolic Jets from the Spinning Black Hole in M87, ApJ 868 (2018) 146 [1810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='09963].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [34] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Park, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Hada, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Kino, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Nakamura, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Hodgson, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Ro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', Kinematics of the M87 Jet in the Collimation Zone: Gradual Acceleration and Velocity Stratification, ApJ 887 (2019) 147 [1911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='02279].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [35] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Beskin and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Nokhrina, The effective acceleration of plasma outflow in the paraboloidal magnetic field, MNRAS 367 (2006) 375.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [36] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Komissarov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Barkov, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Vlahakis and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Königl, Magnetic acceleration of relativistic active galactic nucleus jets, MNRAS 380 (2007) 51 [astro-ph/0703146].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [37] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Georganopoulos, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Perlman and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Kazanas, Is the Core of M87 the Source of Its TeV Emission?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Implications for Unified Schemes, ApJL 634 (2005) L33 [astro-ph/0510783].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [38] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Meyer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Fossati, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Georganopoulos and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Lister, From the Blazar Sequence to the Blazar Envelope: Revisiting the Relativistic Jet Dichotomy in Radio-loud Active Galactic Nuclei, ApJ 740 (2011) 98 [1107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='5105].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [39] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Meyer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Georganopoulos, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Sparks, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Perlman, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' van der Marel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Anderson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', A kiloparsec-scale internal shock collision in the jet of a nearby radio galaxy, Nature 521 (2015) 495.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [40] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Archer, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Benbow, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Bird, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Brill, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Buchovecky, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Buckley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', VERITAS Discovery of VHE Emission from the Radio Galaxy 3C 264: A Multiwavelength Study, ApJ 896 (2020) 41 [2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='03110].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [41] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Lefa, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Rieger and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Aharonian, Formation of Very Hard Gamma-Ray Spectra of Blazars in Leptonic Models, ApJ 740 (2011) 64 [1106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='4201].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' 15 Theory of Gamma-Ray Loud AGNs [42] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Lefa, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Aharonian and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Rieger, “Leading Blob” Model in a Stochastic Acceleration Scenario: The Case of the 2009 Flare of Mkn 501, ApJL 743 (2011) L19 [1108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='4568].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [43] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Biteau and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Meyer, Gamma-Ray Cosmology and Tests of Fundamental Physics, Galaxies 10 (2022) 39 [2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='00523].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [44] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Rieger, Gamma-Ray Astrophysics in the Time Domain, Galaxies 7 (2019) 28 [1901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='10216].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [45] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Aharonian, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Barkov and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Khangulyan, Scenarios for Ultrafast Gamma-Ray Variability in AGN, ApJ 841 (2017) 61 [1704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='08148].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [46] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Ait Benkhali, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Hofmann, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Rieger and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Chakraborty, Evaluating quasi-periodic variations in the 𝛾-ray light curves of Fermi-LAT blazars, A&A 634 (2020) A120 [1901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='10246].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [47] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Peñil, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Ajello, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Buson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Domínguez, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Westernacher-Schneider and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Zrake, Evidence of Periodic Variability in Gamma-ray Emitting Blazars with Fermi-LAT, arXiv e-prints (2022) arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='01894 [2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='01894].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [48] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Sobacchi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Sormani and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Stamerra, A model for periodic blazars, MNRAS 465 (2017) 161 [1610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='04709].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [49] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Tavani, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Cavaliere, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Munar-Adrover and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Argan, The Blazar PG 1553+113 as a Binary System of Supermassive Black Holes, ApJ 854 (2018) 11 [1801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='03335].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [50] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Collaboration, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Abdalla, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Abramowski, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Aharonian, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Ait Benkhali, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Akhperjanian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', Characterizing the 𝛾-ray long-term variability of PKS 2155-304 with H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' and Fermi-LAT, A&A 598 (2017) A39 [1610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='03311].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [51] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Polkas, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Petropoulou, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Vasilopoulos, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Mastichiadis, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Urry, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Coppi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', A numerical study of long-term multiwavelength blazar variability, MNRAS 505 (2021) 6103 [2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='07030].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [52] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Dmytriiev, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Sol and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Zech, Connecting steady emission and very high energy flaring states in blazars: the case of Mrk 421, MNRAS 505 (2021) 2712 [2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='12480].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [53] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Goldreich and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Julian, Pulsar Electrodynamics, ApJ 157 (1969) 869.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [54] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Katsoulakos and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Rieger, Gap-type Particle Acceleration in the Magnetospheres of Rotating Supermassive Black Holes, ApJ 895 (2020) 99 [2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='05076].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [55] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Katsoulakos and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Rieger, Magnetospheric Gamma-Ray Emission in Active Galactic Nuclei, ApJ 852 (2018) 112 [1712.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='04203].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [56] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Levinson and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Rieger, Variable TeV Emission as a Manifestation of Jet Formation in M87?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', ApJ 730 (2011) 123 [1011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='5319].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' 16 Theory of Gamma-Ray Loud AGNs [57] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Contopoulos, The immediate environment of an astrophysical black hole, MNRAS 473 (2018) L146 [1711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='00302].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [58] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Hirotani, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Pu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Lin, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Chang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Inoue, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Kong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', Lepton Acceleration in the Vicinity of the Event Horizon: High-energy and Very-high-energy Emissions from Rotating Black Holes with Various Masses, ApJ 833 (2016) 142 [1610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='07819].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [59] Event Horizon Telescope Collaboration, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Akiyama, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Alberdi, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Alef, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Asada, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Azulay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', First M87 Event Horizon Telescope Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Physical Origin of the Asymmetric Ring, ApJL 875 (2019) L5 [1906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='11242].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [60] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Ait Benkhali, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Chakraborty and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Rieger, Complex gamma-ray behavior of the radio galaxy M 87, A&A 623 (2019) A2 [1802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='03103].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [61] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Chen and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Yuan, Physics of Pair Producing Gaps in Black Hole Magnetospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' General Relativity, ApJ 895 (2020) 121 [1908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='06919].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [62] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Crinquand, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Cerutti, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Philippov, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Parfrey and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Dubus, Multidimensional Simulations of Ergospheric Pair Discharges around Black Holes, Phys Rev Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' 124 (2020) 145101 [2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='03548].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [63] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Kisaka, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Levinson and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Toma, Comprehensive Analysis of Magnetospheric Gaps around Kerr Black Holes Using 1D GRPIC Simulations, ApJ 902 (2020) 80 [2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='02838].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [64] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Kisaka, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Levinson, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Toma and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Niv, The Response of Black Hole Spark Gaps to External Changes: A Production Mechanism of Rapid TeV Flares?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', ApJ 924 (2022) 28 [2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='02971].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [65] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Cerruti, Leptonic and Hadronic Radiative Processes in Supermassive-Black-Hole Jets, Galaxies 8 (2020) 72 [2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='13302].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [66] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Sol and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Zech, Blazars at Very High Energies: Emission Modelling, Galaxies 10 (2022) 105 [2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='03580].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [67] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Zech and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Lemoine, Electron-proton co-acceleration on relativistic shocks in extreme-TeV blazars, A&A 654 (2021) A96 [2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='12271].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [68] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Crumley, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Caprioli, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Markoff and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Spitkovsky, Kinetic simulations of mildly relativistic shocks - I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Particle acceleration in high Mach number shocks, MNRAS 485 (2019) 5105 [1809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='10809].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [69] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Sironi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Spitkovsky and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Arons, The Maximum Energy of Accelerated Particles in Relativistic Collisionless Shocks, ApJ 771 (2013) 54 [1301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='5333].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [70] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Vanthieghem, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Lemoine, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Plotnikov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Grassi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Grech, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Gremillet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', Physics and Phenomenology of Weakly Magnetized, Relativistic Astrophysical Shock Waves, Galaxies 8 (2020) 33 [2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='01141].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' 17 Theory of Gamma-Ray Loud AGNs [71] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Jagan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Sahayanathan, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Rieger and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Ravikumar, Convex X-ray spectra of PKS 2155-304 and constraints on the minimum electron energy, MNRAS 506 (2021) 3996 [2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='04534].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [72] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Liodakis, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Blinov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Potter and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Rieger, Constraints on magnetic field and particle content in blazar jets through optical circular polarization, MNRAS 509 (2022) L21 [2110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='11434].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [73] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Romero, The content of astrophysical jets, Astronomische Nachrichten 342 (2021) 727 [2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='14346].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [74] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Tavecchio, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Costa and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Sciaccaluga, Extreme blazars: the result of unstable recollimated jets?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', MNRAS 517 (2022) L16 [2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='12766].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [75] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Harris and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Krawczynski, X-Ray Emission from Extragalactic Jets, ARA&A 44 (2006) 463 [astro-ph/0607228].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [76] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Georganopoulos, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Meyer and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Perlman, Recent Progress in Understanding the Large Scale Jets of Powerful Quasars, Galaxies 4 (2016) 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [77] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Breiding, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Meyer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Georganopoulos, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Keenan, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' DeNigris and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Hewitt, Fermi Non-detections of Four X-Ray Jet Sources and Implications for the IC/CMB Mechanism, ApJ 849 (2017) 95 [1710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='04250].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [78] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Breiding, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Meyer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Georganopoulos, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Reddy, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Kollmann and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Roychowdhury, A multiwavelength study of multiple spectral component jets in AGN: testing the IC/CMB model for the large-scale-jet X-ray emission, MNRAS 518 (2023) 3222 [2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='13104].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [79] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='-N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Sun, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='-Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Yang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Rieger, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Liu and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Aharonian, Energy distribution of relativistic electrons in the kiloparsec scale jet of M 87 with Chandra, A&A 612 (2018) A106 [1712.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='06390].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [80] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Collaboration, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Abdalla, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Adam, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Aharonian, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Ait Benkhali, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Angüner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', Resolving acceleration to very high energies along the jet of Centaurus A, Nature 582 (2020) 356 [2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='04823].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [81] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Collaboration, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Abdalla, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Abramowski, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Aharonian, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Ait Benkhali, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Angüner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=', The 𝛾-ray spectrum of the core of Centaurus A as observed with H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' and Fermi-LAT, A&A 619 (2018) A71 [1807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='07375].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [82] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Liu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Rieger and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Aharonian, Particle Acceleration in Mildly Relativistic Shearing Flows: The Interplay of Systematic and Stochastic Effects, and the Origin of the Extended High-energy Emission in AGN Jets, ApJ 842 (2017) 39 [1706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='01054].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [83] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Tavecchio, Constraining the shear acceleration model for the X-ray emission of large-scale extragalactic jets, MNRAS 501 (2021) 6199 [2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='03264].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' 18 Theory of Gamma-Ray Loud AGNs [84] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Wang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Reville, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Liu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Rieger and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Aharonian, Particle acceleration in shearing flows: the case for large-scale jets, MNRAS 505 (2021) 1334 [2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='08600].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [85] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Wang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Reville, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Mizuno, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Rieger and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Aharonian, Particle acceleration in shearing flows: the self-generation of turbulent spine-sheath structures in relativistic MHD jet simulations, arXiv e-prints (2022) arXiv:2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='03226 [2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='03226].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [86] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Rieger, An Introduction to Particle Acceleration in Shearing Flows, Galaxies 7 (2019) 78 [1909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='07237].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [87] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Rieger, Active Galactic Nuclei as Potential Sources of Ultra-High Energy Cosmic Rays, Universe 8 (2022) 607 [2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='12202].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [88] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Sironi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Rowan and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Narayan, Reconnection-driven Particle Acceleration in Relativistic Shear Flows, ApJL 907 (2021) L44 [2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='11877].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [89] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Webb, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Barghouty, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Hu and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' le Roux, Particle Acceleration Due to Cosmic-ray Viscosity and Fluid Shear in Astrophysical Jets, ApJ 855 (2018) 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [90] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Rieger and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Duffy, Particle Acceleration in Relativistic Shearing Flows: Energy Spectrum, ApJ 933 (2022) 149 [2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='13098].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [91] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Mukherjee, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Bodo, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Rossi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Mignone and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Vaidya, Simulating the dynamics and synchrotron emission from relativistic jets - II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Evolution of non-thermal electrons, MNRAS 505 (2021) 2267 [2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='02836].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' [92] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' Böttcher, Progress in Multi-wavelength and Multi-Messenger Observations of Blazars and Theoretical Challenges, Galaxies 7 (2019) 20 [1901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content='04178].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} +page_content=' 19' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFRT4oBgHgl3EQfJDc4/content/2301.13494v1.pdf'} diff --git a/QNE4T4oBgHgl3EQfkw0x/vector_store/index.faiss b/QNE4T4oBgHgl3EQfkw0x/vector_store/index.faiss new file mode 100644 index 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sha256:0b0dc44590bd93eb5d179990cdbebd887c951e14d437768db0a0c93cfdf59eac +size 1572909 diff --git a/X9E2T4oBgHgl3EQfuwii/content/tmp_files/2301.04084v1.pdf.txt b/X9E2T4oBgHgl3EQfuwii/content/tmp_files/2301.04084v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ff8706407d79bc155060c697595468581952bf90 --- /dev/null +++ b/X9E2T4oBgHgl3EQfuwii/content/tmp_files/2301.04084v1.pdf.txt @@ -0,0 +1,1441 @@ +arXiv:2301.04084v1 [math.CA] 10 Jan 2023 +THE DIMENSION OF HARMONIC MEASURE ON SOME AD-REGULAR +FLAT SETS OF FRACTIONAL DIMENSION +XAVIER TOLSA +Abstract. In this paper it is shown that if E ⊂ Rn+1 is an s-AD regular compact set, with +s ∈ [n − 1 +2, n), and E is contained in a hyperplane or, more generally, in an n-dimensional C1 +manifold, then the Hausdorff dimension of the harmonic measure for the domain Rn+1 \ E is +strictly smaller than s, i.e., than the Hausdorff dimension of E. +1. Introduction +The study of the metric and geometric properties of harmonic measure is a classical topic +in analysis. At least, this goes back to the work of the Riesz brothers [RR] about the mutual +absolute continuity between harmonic measure and arc-length measure on Jordan domains with +rectifiable boundaries. In the last years, the application of new ideas and techniques originating +from harmonic analysis, PDE’s, and geometric measure theory has allowed to obtain remarkable +advances, especially when the boundary of the domain has codimension 1. +See for example, +[AHM3TV], [AHMMT], [Az1], [GMT], [HMM]. The case of codimension different from one is +less studied and presents more difficulties, due to the fact that notions such as rectifiability or L2 +boundedness of Riesz transforms seem to play no role. In this paper we will focus on the behavior +of harmonic measure on AD-regular boundaries of codimension larger than one. +Recall that, given Ω ⊂ Rn+1 and p ∈ Ω, the harmonic measure ωp for Ω with pole in p is the +Borel measure supported in ∂Ω such that, for any f ∈ Cc(∂Ω) which extends continuously to the +whole Ω and is harmonic in Ω (and vanishes at ∞ in case that n > 1 and Ω is unbounded), we +have +f(p) = +ˆ +∂Ω +f dωp. +One of the most important problems about harmonic measure consists in estimating its dimension. +Recall that, for any Borel measure ν in Rn+1, its (Hausdorff) dimension, denoted by dim ν, is +defined by +dim ν = inf {dim F : F ⊂ Rn+1 Borel, ν(F c) = 0}, +where dim F stands for the Hausdorff dimension of F. Notice that dim ωp does not depend on the +precise pole p, assuming Ω to be connected. So quite often we will write dim ω instead of dim ωp. +One of the first relevant results about the dimension of harmonic measure was obtained by +Makarov [Mak] in 1985, when he showed that for any simply connected domain in the plane, +dim ω = 1 (in spite of the fact that the dimension of the boundary of the domain may have +dimension larger than 1). Later on, Jones and Wolff [JW] showed that, for any arbitrary domain +The author is supported by the European Research Council (ERC) under the European Union’s Horizon 2020 +research and innovation programme (grant agreement 101018680). Also partially supported by MICINN (Spain) +under the grant PID2020-114167GB-I00 and the Mar´ıa de Maeztu Program for units of excellence (Spain) (CEX2020- +001084-M). . +1 + +2 +XAVIER TOLSA +in the plane, dim ω ≤ 1. Subsequently, Wolff [Wo1] sharpened this result by proving that ω must +be concentrated on a set of σ-finite length. In the higher dimensional case n > 1, the situation +is more complicated. +On the one hand, Bourgain [Bo] proved in 1987 that there exists some +constant εn > 0 just depending on n such that dim ω ≤ n − εn for any Ω ⊂ Rn+1. A natural guess +would be that one could take εn = 1, so that dim ω ≤ n for any domain of Rn+1, analogously to +what happens in the plane. However, this was disproved by Wolff in his celebrated work [Wo2], +where he managed to construct a snowflake type domain Ω ⊂ Rn+1 satisfying dim ω > n. A +difficult open question in the area consists in finding the optimal value of the constant εn such +that dim ω ≤ n − εn for any Ω ⊂ Rn+1. See for example [Jo]. +When the (Hausdorff) codimension of ∂Ω is not 1 or ∂Ω is of fractal type, many examples +show that we may have dim ω < dim ∂Ω. This is the so-called “dimension drop” for harmonic +measure, which seems to be a frequent phenomenon. This was first observed by Carleson [Ca] +for some domains defined as complements of suitable Cantor type sets in the plane. Later on, +Jones and Wolff showed a similar result for some planar boundaries ∂Ω satisfying some uniform +disconnectedness property (see [GM, Section X.I.2]. In [Vo1] and [Vo2] (see also [MV]) Volberg +studied the dimension drop for a large class of Cantor repellers in the plane (see [Vo2] for the +notion of Cantor repeller and the precise statement of the result). Later on, Batakis [Ba] proved +analogous results for the harmonic measure for a large class of (complements of) self-similar sets +in the plane. In [UZ] Urba´nski and Zdunik showed that the dimension drop also occurs for the +attractors of conformal iterated function systems (IFS) when either the limit set is contained in +a real-analytic curve, if the IFS consists of similarities only, or if the IFS is irregular. Another +related result was obtained more recently in by Batakis and Zdunik in [BZ] for another class of +IFS. +In view of the results described above, it is natural to wonder if the dimension drop for harmonic +measure occurs for more general, “not dynamically generated”, subsets of Rn+1 with boundaries +with fractional dimension, like domains with AD-regular boundaries of fractional dimension. Re- +call that, given s > 0 and C0 > 1, a set E ⊂ Rn+1 is called AD-regular (or s-AD regular, or +(s, C0)-AD regular, if we want to be more precise) if +C−1 +0 rs ≤ Hs(E ∩ B(x, r)) ≤ C0 rs +for all x ∈ E and 0 < r ≤ diam(E). +An answer in the affirmative to the above question was given by Azzam [Az2] in the case s ∈ +(n, n + 1) (i.e., for codimension smaller than 1). The case of codimension larger than 1 (with1 +s ∈ (n − 1, n)) is more challenging. In fact, very recently Guy David, Cole Jeznach, and Antoine +Julia [DJJ] have informed me that they have managed to construct some s-AD-regular sets E ⊂ R2, +with s ∈ (0, 1), such that the harmonic measure for Ω = R2 \ E is also s-AD-regular, and thus +mutually absolutely continuous with Hs|E. A previous (unpublished) example of Chris Bishop +also showed that harmonic measure can be mutually absolutely continuous with Hs for s < 1 for +some domains in the plane. However in Bishops’s example ∂Ω is not s-AD regular. +The main result of the present paper goes in the converse direction, and it shows that the +dimension drop occurs for s-AD regular subsets of hyperplanes in a suitable range of values of s of +codimension larger than 1. In the plane, for example, this occurs for s-AD regular sets contained +in a line, for s ∈ [1 +2, 1). The precise result is the following. +1When s ≤ n − 1 and ∂Ω is s-AD regular, this is a polar set and harmonic measure on ∂Ω is not defined. + +DIMENSION OF HARMONIC MEASURE ON SOME FLAT SETS OF FRACTIONAL DIMENSION +3 +Theorem 1.1. For n ≥ 1 and s ∈ [n − 1 +2, n), let E ⊂ Rn+1 be an s-AD regular compact set +contained in a hyperplane. Let Ω = Rn+1 \ E and denote by ω the harmonic measure for Ω. Then +dim ω < s. +One can quantify the result above and show that dim ω ≤ s − κ, with κ = κ(n, s, C0) > 0, for +E ⊂ Rn+1 being (s, C0)-AD regular. Further, the assumption that E is contained in a hyperplane +can be replaced by E being contained in a C1 n-dimensional manifold in Rn+1 (see Theorem 5.1 +below). +I don’t know if the threshold n − 1 +2 is sharp in Theorem 1.1. However, we can go a bit below +the threshold n − 1 +2 in the following sense: +Theorem 1.2. For s > 0, C0 > 1, let E ⊂ Rn+1 be an (s, C0)-AD regular compact set contained +in a hyperplane, and let Ω = Rn+1 \ E. Then there exists ε > 0 small enough depending on n and +C0 such that if s ∈ [n − 1 +2 − ε, n), we have dim ω < s. +On the other hand, it seems that Theorem 1.1 does not hold for s close enough to n − 1. +Indeed, it seems that some of the aforementioned examples of the authors in [DJJ] are s-AD +regular subsets of the real line in the plane. So an interesting open problem consists in finding +the sharp threshold s0 such that for all s-AD regular sets with s ∈ (s0, 1) contained in a line the +dimension drop for harmonic measure occurs. Also, for s-AD regular sets not contained in a line, +it is an open question if a similar threshold exists. +The proofs of Theorem 1.1 and Theorem 1.2 rely on an idea originating from Bourgain [Bo] +(see Lemma 2.8) and they follow an approach similar to the one of [Ba] and [Az2]. Using touching +point arguments, the maximum principle, and suitable modifications of the domain, we are led +to some estimates involving the harmonic measure for the complement of a segment in the plane +and for the complement of a suitable flat annulus for n > 1. In the plane, such harmonic measure +can be computed explicitly by means of a conformal transformation while in higher dimensions +we need more elaborated estimates which also rely on the conformal transformation from the +planar case. Our arguments make an essential us of the fact the boundary ∂Ω is contained in a +hyperplane and so they cannot be extended to arbitrary s-AD regular sets. +2. Preliminaries +In the paper, constants denoted by C or c depend just on the dimension and perhaps other +fixed parameters, such as the parameter s in Theorem 1.1, for example. We will write a ≲ b if +there is C > 0 such that a ≤ Cb . We write a ≈ b if a ≲ b ≲ a. +2.1. Capacities and the capacity density condition. The set of (positive) Radon measure in +Rn+1 is denoted by M+(Rn+1). The Hausdorff s-dimensional measure and Hausdorff s-dimensional +content are denoted by Hs and Hs +∞, respectively. +The fundamental solution of the negative Laplacian in R2 is +E2(x) = 1 +2π log 1 +|x|, +while in higher dimensions, i.e., in Rn+1, n ≥ 2, it equals +En+1(x) = +cn +|x|n−1 , + +4 +XAVIER TOLSA +where cn = (n−1)Hn(Sn), with Sn being the unit hypersphere in Rn+1. In any case, for a measure +µ in Rn+1, we consider the energy +I(µ) = +¨ +En+1(x − y) dµ(x) dµ(y) +and, for F ⊂ Rn+1 we define the capacity +Cap(F) = +1 +infµ∈M1(F ) I(µ), +where the infimum is taken over all probability measures µ supported on F. In the planar case, +Cap(F) is the Wiener capacity of F, and in higher dimensions this is the Newtonian capacity. +From now on, in the plane we will write CapW (F) instead of Cap(F). In fact, in the plane it is +more convenient to work with the logarithmic capacity, defined by +CapL(F) = e− +2π +CapW (F ) . +Recall that we denote by ω (and sometimes ωΩ) the harmonic measure on an open set Ω. The +following result is well known. See for example Lemma 2.1 from [To]. +Lemma 2.1. For n ≥ 2, let Ω ⊂ Rn+1 be open and let B be a closed ball centered in ∂Ω. Then +ωx(B) ≥ c(n)Cap(1 +4B \ Ω) +rad(B)n−1 +for all x ∈ 1 +4B ∩ Ω, +with c(n) > 0. +An analogous result holds in the plane. The precise statement is the following. +Lemma 2.2. Let Ω ⊂ R2 be open and let B be a closed ball centered in ∂Ω. Then +ωx(B) ≳ +1 +log +rad(B) +CapL(1 +4B \ Ω) +for all x ∈ 1 +4B ∩ Ω. +I provide the detailed proof below because it is not easy to find in the literature. +Proof. Denote r = rad(B). Replacing Ω by +1 +4r Ω if necessary, we can assume that diam(B) < 1. +Then, denoting F = B \ Ω, the following identity holds: +CapW (F) = sup +� +µ(F) : µ ∈ M+(Rn+1), supp µ ⊂ F, ∥E ∗ µ∥∞ ≤ 1 +� +. +Let µ be the optimal measure for this supremum, so that suppµ ⊂ F, µ(F) = CapW (F), and +the function u := E ∗ µ is harmonic out of F and it satisfies ∥u∥∞ ≤ 1. For all z ∈ 1 +4B and all +y ∈ F we have |z − y| ≤ 1 +2 r. Therefore, +u(z) = 1 +2π +ˆ +log +1 +|z − y| dµ(y) ≥ 1 +2π +ˆ +log 2 +r dµ(y) = µ(F) +2π +log 2 +r +for all z ∈ 1 +4B. +Also, for z ∈ Bc, we have dist(z, supp F) ≥ 3 +4r(B), and thus +u(z) ≤ 1 +2π +ˆ +log 4 +3r dµ(y) = µ(F) +2π +log 4 +3r +for all z ∈ Bc. +Consider now the function +v = u − µ(F) +2π +log 4 +3r. + +DIMENSION OF HARMONIC MEASURE ON SOME FLAT SETS OF FRACTIONAL DIMENSION +5 +Observe that +v(z) ≥ µ(F) +2π +log 2 +r − µ(F) +2π +log 4 +3r = µ(F) +2π +log 3 +2 +for all z ∈ 1 +4B +and +v(z) ≤ 0 +for all z ∈ Bc. +By the maximum principle and the fact that x ∈ 1 +4B we deduce that +ωx(B) ≥ v(x) +sup v ≥ µ(F) +2π +log 3 +2 = c CapW(F) +sup v +. +Regarding sup v, taking into account that ∥u∥∞ ≤ 1, it is clear that +sup v ≤ 1 − 1 +2π log 4 +3r µ(F) = 1 − 1 +2π log 4 +3r CapW(F) ≤ 1 − 1 +2π log 1 +r CapW(F). +Therefore, +ωx(B) ≥ c +CapW(F) +1 − 1 +2π log 1 +r CapW (F) = c′ +1 +log +1 +CapL(F) − log 1 +r += c′ +1 +log +r +CapL(F) +. +□ +Lemma 2.3. Let E ⊂ Rn+1 be compact and n − 1 < s ≤ n + 1. In the case n > 1, we have +Cap(E) ≳s,n Hs +∞(E) +n−1 +s . +In the case n = 1, we have +CapL(E) ≳s Hs +∞(E) +1 +s . +The proof of this result is an immediate consequence of Frostman’s Lemma. See [Mat, Chapter +8] for the case n > 1, and [CTV, Lemma 4] for the case n = 1, for example. +Let Ω ⊊ Rn+1 be open, and let ξ ∈ ∂Ω and r0 > 0. We say that Ω satisfies the (ξ, r0)-local +capacity density condition (CDC) if there exists some constant c > 0 such that, for any r ∈ (0, r0), +Cap(B(ξ, r) \ Ω) ≥ c rn−1 +in the case n > 1, +and +CapL(B(ξ, r) \ Ω) ≥ c r +in the case n = 1. +We say that Ω satisfies the capacity density condition (CDC) if it satisfies the (ξ, r0)-local capacity +density condition for all ξ ∈ ∂Ω and all r0 ∈ (0, diam∂Ω). +Remark that, by the previous lemmas, if Ω satisfies the CDC, then it holds +ωx(B) ≳ 1 +for all x ∈ 1 +4B ∩ Ω +In particular, if ∂Ω is s-AD regular for some s > n − 1, then Ω satisfies the CDC. The following +result is also standard and well known. See [An], for example. + +6 +XAVIER TOLSA +Lemma 2.4. Let Ω ⊂ Rn+1, let ξ ∈ ∂Ω, and let r > 0. Suppose that Ω satisfies the (ξ, r0)-local +capacity density condition. Let u be a nonnegative function which is continuous in B(ξ, r) ∩ Ω +and harmonic in B(ξ, r) ∩ Ω, and vanishes continuously on B(ξ, r) ∩ ∂Ω. Then there is α > 0 +such that for all r ∈ (0, r0), +(2.1) +u(x) ≲ +�|x − ξ| +r +�α +sup +B(ξ,r)∩Ω +u +for all x ∈ Ω ∩ B(ξ, r). +2.2. Corkscrews, connectivity conditions, and uniform domains. A domain Ω ⊂ Rn+1 is +called uniform (or C-uniform) if for every x, y ∈ Ω there is a curve γ ⊂ Rn+1 connecting x and y +such that +(a) H1(γ) ≤ C |x − y|, and +(b) for all z ∈ γ, dist(z, ∂Ω) ≥ C−1 dist(z, {x, y}). +We say that Ω satisfies the corkscrew condition (or c-corkscrew condition) if there exists some +constant c > 0 such that for all ξ ∈ Ω and all 0 < r ≤ diam(∂Ω) there is a ball B(x, cr) ⊂ +B(ξ, r) ∩ Ω. The point x is called a “corkscrew point” relative to the ball B(ξ, r). Further, we +say that x is a John point relative to B(ξ, r) if for every y ∈ B(ξ, r) ∩ Ω there exists a curve γ +satisfying the properties (a), (b) above. It is immediate to check that if Ω is a uniform domain, +then it satisfies the corkscrew condition. Also, for these domains, any corkscrew point is a John +point relative to the ball associated to the corkscrew. +We will need to use the following result below. This is proven exactly in the same way as the +analogous result for NTA domains in [JK, Lemma 4.4]. +Lemma 2.5. For n ≥ 1, let Ω ⊂ Rn+1 be a domain and ξ ∈ ∂Ω, 0 < r ≤ diam(∂Ω) such that +the (ξ, r)-local CDC holds. Let x ∈ Ω ∩ B(ξ, r) be a corkscrew John point relative to B(ξ, r). Let +u be a nonnegative function which is continuous in B(ξ, r) ∩ Ω and harmonic in B(ξ, r) ∩ Ω, and +vanishes continuously on B(ξ, r) ∩ ∂Ω. Then we have +sup +B(ξ,r/2)∩Ω +u ≲ u(x), +with the implicit constant depending on the local CDC, and the corkscrew and John properties of +x. +In [DFM, Lemma 2.2] the following result has been proved. +Lemma 2.6. For 0 < s < n and C0 > 1, let E be an (s, C0)-AD regular set. Then the domain +Ω = Rn+1 \ E is C-uniform, with C depending just on n, s, C0. +So the the assumptions in Theorems 1.1 or 1.2 ensure that the domain Ω in those theorems +is uniform and satisfies the CDC. This is very useful because it implies some nice properties for +the associated harmonic measure. For example, it implies that given p ∈ Rn+1 with dist(p, E) ≳ +diam(E), the doubling property ωp(2B) ≲ ω(B) holds for any ball B centered in ∂Ω. +The following theorem contains the so called “change of pole formula” for uniform domains. +This is proven in [JK] for NTA domains (i.e., for uniform domains Ω such that Rn+1 \ Ω is a +corkscrew domain), while the more general statement below is from [MT]. + +DIMENSION OF HARMONIC MEASURE ON SOME FLAT SETS OF FRACTIONAL DIMENSION +7 +Theorem 2.7. For n ≥ 1, let Ω ⊂ Rn+1 be a Wiener regular uniform domain and let B be a ball +centered at ∂Ω. Let p1, p2 ∈ Ω such that dist(pi, B ∩ ∂Ω) ≥ c−1 +0 +r(B) for i = 1, 2. Then, for any +Borel set E ⊂ B ∩ ∂Ω, +ωp1(E) +ωp1(B) ≈ ωp2(E) +ωp2(B), +with the implicit constant depending only on c0 and the uniform behavior of Ω. +2.3. The Kelvin transform. Given x ∈ Rn+1 \ {0}, we let x∗ = +1 +|x|2 x. Notice that the map +defined by T(x) = x∗ is an involution of Rn+1 ∪ {∞}, understanding that T(0) = ∞. For r > 0, +we denote r∗ = 1 +r. Given Ω ⊂ Rn+1 such that ∂Ω is compact and 0 ̸∈ ∂Ω, we also set +Ω∗ = +� +x∗ : x ∈ Ω +� +∪ {0} +(so identifying Ω with Ω ∪ {∞}, we have Ω∗ = T(Ω)). Given a function f : Rn+1 ⊃ E → R, its +Kelvin transform is defined by +f ∗(x∗) = +1 +|x∗|n−1 f(x), +understanding that f(∞) = 0. It is well known that, if u : Ω → R vanishes at ∞, then ∆u = 0 in +Ω if and only if ∆(u∗) = 0 in Ω∗. Further, (u∗)∗ = u. See [ABR, Chapter 4], for example. Hence, +if we take u(x) = ωx +Ω(F) for F ⊂ ∂Ω, it follows that u∗ is a harmonic function in Ω, comparable +to 1 in F, and vanishing away from F, with the implicit constant depending on dist(0, ∂Ω) and +diam(∂Ω). Hence, for x ∈ Ω, we have +(2.2) +ωx∗ +Ω∗(F ∗) ≈ u∗(x∗) = +1 +|x∗|n−1 u(x) ≈ ωx +Ω(F), +with the first implicit constant depending on dist(0, ∂Ω), diam(∂Ω), and the second one on |x∗|. +2.4. The Main Lemma. To prove Theorems 1.1 and 1.2 we will use the following result, first +implicitly used by Bourgain in [Bo], later by Batakis [Ba], and more recently by Azzam [Az2]. +Lemma 2.8. For n ≥ 1, s > 0, C0 > 1, there exists an M = M(n, s, C0) > 1 (sufficiently big) +such that the following holds. Let E ⊂ Rn+1 be an (s, C0)-AD regular set. Let ν be a measure +supported on E and c1 ∈ (0, 1) such that, for all x ∈ E, 0 < r ≤ diam(E), there exists a ball +B(y, ρ) with y ∈ B(x, r), c1 r ≤ ρ ≤ r, satisfying either +(2.3) +ν(B(y, ρ)) +ρs +≥ M ν(B(x, r)) +rs +or +ν(B(y, ρ)) +ρs +≤ M−1 ν(B(x, r)) +rs +. +Then dim ν < s. +Although this result is not stated as above in [Az2], the detailed arguments of the proof are +contained in Section 5 of that paper. +Given Ω and E as in Theorems 1.1 and 1.2, we denote by p0 the pole for harmonic measure +and we assume that this is far away from E (in the plane we could take p0 = ∞), and we denote +ω = ωp0. Recall that Ω = Ec is a uniform domain satisfying the CDC. Thanks to Lemma 2.8, the +fact that dim ω < s is an immediate consequence of the following result. + +8 +XAVIER TOLSA +Main Lemma 2.9. For s > 0, C0 > 1, let E ⊂ Rn+1 be an (s, C0)-AD regular closed set contained +in a hyperplane. Let M ≥ 1 and suppose either that s ∈ [n − 1 +2, n) or s ∈ (n − 1 +2 − ε, n − 1 +2) for +some ε = ε(n, s, C0, M) > 0 small enough. +For any x ∈ E, 0 < r ≤ diam(E), and p0 ∈ +Rn+1 \E ∪B(x, 2r)), there exists a ball B(y, ρ) with y ∈ B(x, r), c r ≤ ρ ≤ r, with c > 0 depending +just on n, s, C0, M, satisfying either +(2.4) +ωp0(B(y, ρ)) +ρs +≥ M ωp0(B(x, r)) +rs +or +ωp0(B(y, ρ)) +ρs +≤ M−1 ωp0(B(x, r)) +rs +. +To prove the Main Lemma, first we will show that, given E, B(x, r), and p0 as above, there +exists y ∈ B(x, r) such that +(2.5) +ωp0(B(y, ρ)) +ωp0(B(x, r)) ≥ c(s, C0) +�ρ +r +�n− 1 +2 +for all ρ ∈ (0, c′ r), for some c′ ∈ (0, 1) depending just on s and C0. Clearly, this yields (2.4) for +s ∈ (n − 1 +2, n) and ρ small enough. For the cases s = n − 1 +2 and s ∈ (n − 1 +2 − ε, n − 1 +2) we will need +more careful estimates. Notice also that the estimate (2.5) is independent of the pole p0, modulo +a constant factor (as soon as p0 ∈ Rn+1 \ (E ∪ B(x, 2r))), because Rn+1 \ E is a uniform domain. +3. The arguments for the planar case n = 1 with s ∈ [1 +2, 1) +This section and the next one are devoted to the proof of Main Lemma 2.9. +3.1. Proof of (2.5). Without loss of generality, we assume that E ⊂ R ≡ R × {0}. Let x ∈ E +and 0 < r ≤ diamE. Taking into account that s < 1, by a pigeon-hole argument, there is an +open interval I = (a, b) ⊂ [x − r, x] which does not intersect E and satisfies ℓ := H1(I) ≈s r. By +enlarging I if necessary, we can assume that b ∈ E. Notice that b is contained in [x − (1 − c)r, x] +because x ∈ E, for some c > 0 depending on s. +We choose y = b. Again by the s-AD regularity of E and the pigeon hole principle, there exist +radii r1, r2 with ℓ/2 ≤ r1 < r2 ≤ ℓ, r2 − r1 ≈s ℓ ≈ r such that +A(y, r1, r2) ∩ E = ∅. +Observe that the left component of A(y, r1, r2) ∩ R is contained in I. +Next we apply a “localization argument”. +We denote E1 = E ∩ ¯B(y, r1), Ω1 = Ec +1, r′ = +(r1 +r2)/2. It is immediate to check that E1 is still s-AD regular and thus Ω1 is a uniform domain +too. We claim that for any subset F ⊂ E1 and any p ∈ ∂B(y, r′), +(3.1) +ωp +1(F) ≈s ωp(F), +where ω1 stands for the harmonic measure for Ω1. To prove the claim, consider first p ∈ ∂B(y, r′) +such that +ωp +1(F) = +max +q∈∂B(y,r′) ωq +1(F). +Using that ωz +1(F) is harmonic in Ω and vanishes in E1 \ F and the maximum principle, we get +ωp +1(F) = +ˆ +E +ωz +1(F) dωp(z) = ωp(F) + +ˆ +E\E1 +ωz +1(F) dωp(z) ≤ ωp(F) + +sup +z∈E\E1 +ωz +1(F) ωp(E \ E1). + +DIMENSION OF HARMONIC MEASURE ON SOME FLAT SETS OF FRACTIONAL DIMENSION +9 +Observe that, by the CDC and a Harnack chain argument, ωp(E1) ≥ δ0, for some δ0 > 0 depending +just on s. +Hence, ωp(E \ E1) ≤ 1 − δ0. +Also, since ωz +1(F) is harmonic in C∞ \ B(y, r′) and +E \ E1 ⊂ C∞ \ B(y, r′), by the maximum principle we have +sup +z∈E\E1 +ωz +1(F) ≤ +max +q∈∂B(y,r′) ωq +1(F) = ωp +1(F). +Therefore, +ωp +1(F) ≤ ωp(F) + ωp +1(F) (1 − δ0), +or equivalently, ωp +1(F) ≤ δ−1 +0 +ωp(F). By the definition of r′ and Harnack’s inequality, we infer +ωp +1(F) ≲ ωp(F) +for all p ∈ ∂B(y, r′). +On the other hand, by the maximum principle, we have trivially that +ωp +1(F) ≥ ωp(F), which concludes the proof of the claimed estimate (3.1). +Next we will perform another modification of the domain Ω1. For a fixed ρ ∈ (0, r1/4), consider +the intervals J = [y, y + ρ/2], J′ = [y, y + ρ] and define E2 = E1 ∪ J and Ω2 = Ec +2 = Ω1 \ J. By +the CDC and the uniformity of Ω1, we infer that, for all q ∈ ∂B(y, ρ/2), +ωq +1(J′ ∩ E1) ≳ 1 ≥ ωq +2(J). +We also have ωq +1(J′ ∩ E1) ≥ ωq +2(J) = 0 for all q ∈ Jc ∩ E1. Then, by the maximum principle, since +both ωz +1(J′ ∩ E1) and ωz +2(J) are harmonic in Ω1 \ ¯B(y, ρ/2) = Ω2 \ ¯B(y, ρ/2) we deduce that +ωq +1(J′ ∩ E1) ≳ ωq +2(J) +for all q ∈ Ω2 \ ¯B(y, ρ/2), and in particular for all p ∈ ∂B(y, r′). +Finally we let E3 = [y, y + r1] and Ω3 = Ec +3, so that E2 ⊂ E3. By the maximum principle, we +have +ωp +2(J) ≥ ωp +3(J) +for all p ∈ ∂B(y, r′). Hence, gathering the above estimates, we infer that, for all p ∈ ∂B(y, r′), +(3.2) +ωp(J′ ∩ E) ≈s ωp +1(J′ ∩ E) ≳ ωp +2(J) ≥ ωp +3(J). +Now it just remains to estimate ωp +3(J). +We can do this by means of a conformal transfor- +mation. Indeed, observe first that, by a Harnack chain argument and the maximum principle, +ωp +3(J) ≈ ω∞ +3 (J) for all p ∈ ∂B(y, r′). +Next, suppose for simplicity that y = −r1/2, so that +E3 = [−r1/2, r1/2]. The map f : ¯B(0, 1) → Ω3 defined by +(3.3) +f(z) = +� +z + 1 +z +�r1 +4 +is a conformal transformation from B(0, 1) to Ω3 such that f(0) = ∞, with f(∂B(0, 1)) = +f(∂Ω3) = f(E3). Thus, +ω∞ +3 (J) = 1 +2π H1(f −1(J)). +An easy computation shows that +f −1(J) = {eiα : π − θ ≤ α ≤ π + θ}, +with +(3.4) +θ = arccos +� +1 − H1(J) +2r1 +� += arccos +� +1 − ρ +4r1 +� +≈ +� ρ +r1 +�1/2 +. + +10 +XAVIER TOLSA +Thus, +(3.5) +ω∞ +3 (J) = θ +π ≈ +� ρ +r1 +�1/2 +≈ +�ρ +r +�1/2 +. +Consequently, by the change of pole formula for uniform CDC domains and (3.2), we deduce that, +for p ∈ ∂B(y, r′), +(3.6) +ωp0(B(y, ρ)) +ωp0(B(x, r)) ≈ ωp( ¯B(y, ρ)) = ωp(J′ ∩ E) ≳ ωp +3(J) ≈ +�ρ +r +�1/2 +, +which completes the proof of (2.5). +3.2. The case s = 1/2. In this case the inequality (2.5) does not suffice to prove (2.4) and we need +a better estimate. We consider the preceding domains Ω1, Ω2, Ω3, so that, for all p ∈ ∂B(y, r′), +(3.2) holds. However, the estimate ωp +2(J) ≥ ωp +3(J) is too coarse for our purposes. Instead, we +write +ωp +2(J) ≈ ω∞ +2 (J) = +ˆ +E3 +ωz +2(J) dω∞ +3 (z). +The density +dω∞ +3 +dH1|E3 can be computed explicitly by means of the conformal transformation in (3.3). +Using the identity ω∞ +3 (J) = π−1 arccos +� +1 − H1(J) +2r1 +� +and differentiating, it follows that +dω∞ +3 +dH1|E3 +(t) = +1 +π +� +(r1 +2 − t)(t + r1 +2 ). +Thus, +(3.7) +dω∞ +3 +dH1|E3 +(t) ≈ +1 +� +r1(t + r1 +2 ) +for t ∈ [−r1/2, 0], +and so +(3.8) +ωp +2(J) ≳ +ˆ 0 +−r1 +2 +ωt +2(J) +dt +� +r1(t + r1 +2 ) +(recall that we are identifying R ≡ R × {0}). +To estimate the integral in (3.8) from below, consider the annuli Ak = A(y, 2kρ, 2k+1ρ), for +k ≥ 1. Let N = [log2 +r1 +ρ ]. By the s-AD regularity of E and pigeonholing, for every k ∈ [1, N] +there exists an interval Ik ⊂ Ak ∩ E3 (recall E3 is an interval) such that H1(Ik) ≈ 2kρ and +Ik ∩ E = Ik ∩ E2 = ∅. Let ˆIk be another interval concentric with Ik and half length. Then we +write +(3.9) +ˆ 0 +−r1 +2 +ωt +2(J) +dt +� +r1(t + r1 +2 ) ≥ +N +� +k=1 +ˆ +ˆIk +ωt +2(J) +dt +� +r1(t + r1 +2 ). +We claim that +(3.10) +ωt +2(J) ≳ +� +ρ +|t − y| +�1/2 +for all t ∈ �N +k=1 ˆIk. + +DIMENSION OF HARMONIC MEASURE ON SOME FLAT SETS OF FRACTIONAL DIMENSION +11 +Assuming this for the moment, we obtain +ωp +2(J) ≳ +N +� +k=1 +ˆ +ˆIk +� +ρ +|t − y| +�1/2 +dt +� +r1(t + r1 +2 ) +≈ +N +� +k=1 +ˆ +ˆIk +� +ρ +H1(ˆIk) +�1/2 +dt +r1/2 +1 +H1(ˆIk)1/2 = N +� ρ +r1 +�1/2 +≈ log r +ρ +�ρ +r +�1/2 +. +By (3.2) and the change of pole formula, arguing as in the preceding subsection, we obtain +(3.11) +ωp0(B(y, ρ)) +ωp0(B(x, r)) ≈ ωp( ¯B(y, ρ)) = ωp(J′ ∩ E) ≳ ωp +2(J) ≈ log r +ρ +�ρ +r +�1/2 +, +which implies (2.4) for ρ small enough. +It remains to prove (3.10). To this end, for each t ∈ ˆIk, let t′ ∈ R be the point symmetric to +t with respect to y. That is, t′ = −r1 − t. Notice that t′ is on the left of the interval E3 (recall +that the leftmost point of E3 is y = −r1/2). By a Harnack chain argument and the maximum +principle, we have +ωt +2(J) ≈ ωt′ +2 (J) ≥ ωt′ +3 (J). +Now we can compute explicitly ωt′ +3 (J) by means of the conformal transformation in (3.3). Indeed, +consider the change of variable t′ = y − r1 +2 h. Then, it follows easily that +f −1(t′) = +−1 +1 + h + +� +h(2 + h) +. +So f −1(t′) is a point in the unit disk belonging to the segment (−1, 0) such that +| − 1 − f −1(t′)| = 1 − +1 +1 + h + +� +h(2 + h) +≈ h1/2 = +�2|t′ − y| +r1 +�1/2 +. +Recall that f −1(J) = [π−θ, π+θ], with θ ≈ +� ρ +r1 +�1/2, by (3.4). Hence, |−1−f −1(t′)| ≳ H1(f −1(J)). +Taking into account that, for any point q ∈ B(0, 1) and η := 10 dist(q, ∂B(0, 1)), ωq +B(0,1)|B(q,η) is +comparable to η−1H1|∂B(0,1)∩B(q,η), we deduce that +ωt′ +3 (J) = ωf−1(t′) +B(0,1) (f −1(J)) ≈ +θ +| − 1 − f −1(t′)| ≈ +� ρ +r1 +�1/2 +� +2|t′−y| +r1 +�1/2 ≈ +ρ1/2 +|t′ − y|1/2 , +which yields (3.10). +□ +4. The higher dimensional case n > 1 +In this section we complete the proof of Main Lemma 2.9. Although we will focus mainly in +the case n > 1, the arguments in this section are also valid for the case n = 1. In the preceding +section we have studied separately the case n = 1, s ∈ [1/2, 1) because the arguments are more +elementary, specially for s = 1/2. + +12 +XAVIER TOLSA +4.1. An auxiliary lemma. In the case n > 1, we cannot use conformal transformations as in +the planar case. Instead, we will use the following auxiliary result. +Lemma 4.1. For n > 1, r > 0, let E = (B(0, 2r)\B(0, r))∩(Rn ×{0}) and Ω = Rn+1 \E. Let B +a ball centered in E such that ρ := rad(B) ≈ dist(B, ∂B(0, r)), with 0 < ρ ≤ r/2. Then we have +ω0(B) ≳ +�ρ +r +�n− 1 +2 , +where ω0 stands for the harmonic measure for Ω with pole in 0. +Quite likely this result can be proven by an explicit computation of the density +dω0 +dHn|E . However, +I have not found in the literature such computation, which seems to be non-trivial. For this reason, +below I show a different argument relying on the planar case. +Proof. Observe first that, by rescaling, we can assume that r = 1, and we can estimate ωp0(B) for +p0 = (0, . . . , 0, 1/2), say, instead of ω0(B), since ω0(B) ≈ ωp0(B). Next, by applying the Kelvin +transform Tx = +x +|x|2, the statement in the lemma is equivalent to saying that, for Ω1 = Rn+1 \ E1 +with E1 = T(E) = (B(0, 1) \ B(0, 1/2)) ∩ (Rn × {0}), and any ball B centered in E1 such that +ρ := rad(B) ≈ dist(B, ∂B(0, 1)), with 0 < ρ ≤ r/2, we have +ωp1 +1 (B) ≳ ρn− 1 +2, +where ωp1 +1 +stands for the harmonic measure for Ω1 with pole in p1 := T(p0) = (0, . . . , 0, 2). +Consider now Ω2 = Rn+1 \ E2 with E2 = B(0, 1) ∩ (Rn × {0}). By the maximum principle, it is +clear that ωp1 +1 (B) ≥ ωp1 +2 (B). Hence it suffices to prove that +(4.1) +ωp1 +2 (B) ≳ ρn− 1 +2. +To prove (4.1), we may assume that the ball B is centered in the segment L connecting the +origin to the point (1, 0, . . . , 0). Then we let Q be an n-dimensional cube concentric with B (with +sides parallel to the axes), contained in E2 ∩ B, with side length comparable to rad(B), and we +set R = [−1, 1]n × {0}. Since Q ⊂ B ∩ E2 and E2 ⊂ R, by the maximum principle we have +(4.2) +ωp1 +2 (B) ≥ ωp1 +R (Q), +where ωR stands for the harmonic measure for the domain Rn+1 \ R. To estimate ωp1 +R (Q) from +below, we can assume that this is an n-dimensional dyadic cube descendant of R, by translating +and reducing Q slightly if necessary. We let I0 = [−1, 1] and J = L ∩ Q, so that J ⊂ I0 and +dist(J, ∂I0) ≈ ℓ(J). We consider I0, J as subsets of R ≡ R × {0} ⊂ C and we let +u(z) = ωz +I0(J), +where ωz +I0 is the harmonic measure for the domain C \ I0 with pole in z. Next we denote �I0 = +I0 × Rn−1 × {0}, �Ω0 = Rn+1 \ �I0, �J = J × Rn−1 × {0} and we take the function f : �Ω0 → R +defined by f(x) = u(x1, xn+1), for x = (x1, . . . , xn+1). Notice that f is non-negative, bounded +by 1, and harmonic in Rn+1 \ �I0. Moreover, it extends continuously to 0 in �I0 \ �J, and to 1 in +J◦ × Rn−1 × {0}. Then it follows that f(x) = �ωx( �J), where �ωx is the harmonic measure for �Ω0 +with pole in x. +We claim that +(4.3) +�ωp1( �J) ≈ �ωp1( �J ∩ R). + +DIMENSION OF HARMONIC MEASURE ON SOME FLAT SETS OF FRACTIONAL DIMENSION +13 +To prove this, denote g(x) = �ωx( �J ∩ R), so that +f(x) − g(x) = �ωx( �J \ R). +For a given δ ∈ (0, 1/2), let zδ ∈ C \ I0 be such that dist(zδ, I0) = δ and +u(zδ) = +max +z∈C:dist(z,I0)=δ u(z) = +max +z∈C:dist(z,I0)=δ ωz +I0(J), +so that by the maximum principle u(z) ≤ u(z0) for any z ∈ C such that dist(z, I0) ≥ δ. Let +qδ = (ℜzδ, 0, . . . , 0, ℑzδ). By the definition of f, it turns out that f(x) ≤ f(qδ) for all x ∈ Rn+1 +such that dist(x, �I0) ≥ δ. Then, by the (local) CDC of �I0, since f − g vanishes identically in +R × {0}, we deduce that +|f(x) − g(x)| ≲ +sup +y∈∈3I0×( 3 +4 I0)n +|f(y) − g(y)| dist(x, R)α +for all x ∈ 2I0 × (1 +2I0)n. +Since p1 is a local John point for �Ω0 in 4I0×In +0 , the supremum above is bounded by C|f(p1)−g(p1)|, +by Lemma 2.5. Then, choosing x = qδ we obtain +|f(qδ) − g(qδ)| ≲ |f(p1) − g(p1)| dist(qδ, R)α ≤ f(p1) δα ≤ f(qδ) δα. +So, choosing δ small enough (independent of f and g), we derive f(qδ) ≈ g(qδ). Since f(p1) ≈ f(qδ) +and g(p1) ≈ g(qδ) (with implicit constants depending on δ), we deduce (4.3), as claimed. +Next we split R = �N +i=1 Pi, where Pi’s are dyadic n-dimensional cubes, descendants of R, with +the same side length as Q, and N = ℓ(Q)−n. Observe that J ∩ R coincides with the union of +NJ := ℓ(Q)1−n of the cubes Pi. So reordering the family of such cubes if necessary, we have +�ωp1( �J ∩ R) = +NJ +� +i=1 +�ωp1(Pi). +It is easy to check that �ωp1(Pi) ≈ �ωp1(Pj) for all 1 ≤ i, j ≤ NJ. Indeed, if ci stands for the center of +Pi and we take the corkscrew point zi = ci + (0, . . . , 0, 1), by symmetry we have �ωzi(Pi) = �ωzj(Pj) +for all i, j, and by Harnack, �ωzi(Pi) ≈ �ωp1(Pi) for all i. Recalling that Q coincides with one of the +cubes Pi, using also (4.3), we get +�ωp1(Q) ≳ +1 +NJ +NJ +� +i=1 +�ωp1(Pi) = ℓ(Q)n−1 �ωp1( �J ∩ R) ≈ ℓ(Q)n−1 �ωp1( �J). +By construction, we have �ωp1( �J) = ω(0,1) +I0 +(J), and using the conformal transformation in (3.3) and +the identity (3.7), it is immediate to check that ω(0,1) +I0 +(J) ≈ ℓ(J)1/2 = ℓ(Q)1/2. Therefore, +�ωp1(Q) ≳ ℓ(Q)n− 1 +2. +By the maximum principle, we have ωp1 +R (Q) ≥ �ωp1(Q), and then by (4.2), we deduce that +ωp1 +2 (B) ≳ ℓ(Q)n− 1 +2 ≈ ρn− 1 +2, +which proves (4.1) and the lemma. +□ + +14 +XAVIER TOLSA +4.2. Proof of (2.5). We assume that E ⊂ Rn ≡ Rn × {0}. By a pigeon-hole argument, there is +an open ball B1 := B(x1, r1) centered in Rn, which satisfies: +2 ¯B1 ⊂ B(x, r), +B1 ∩ E = ∅, +∂B1 ∩ E ̸= ∅, +r1 ≈ r. +We choose y ∈ ∂B1 ∩ E. +As in the case n = 1, we intend to apply a localization argument. We denote E1 = E∩ ¯B(x1, 2r1), +Ω1 = Ec +1. We claim that for any subset F ⊂ E1 ∩ B(x1, 1.5r1), +(4.4) +ωx1 +1 (F) ≈s ωp(F), +where ω1 stands for the harmonic measure for Ω1. +To prove the claim, consider first p ∈ +∂B(x1, 1.8r1) such that +ωp +1(F) = +max +q∈∂B(x1,1.8r1) ωq +1(F). +Using that ωz +1(F) is harmonic in Ω and vanishes in E1 \ F, we get +ωp +1(F) = +ˆ +E +ωz +1(F) dωp(z) = ωp(F) + +ˆ +E\E1 +ωz +1(F) dωp(z) ≤ ωp(F) + +sup +z∈E\E1 +ωz +1(F) ωp(E \ E1). +Since ωz +1(F) is harmonic in Rn+1 \ E1, vanishes in E1 \ ¯B(x1, 1.5r1) and at ∞, and E \ E1 ⊂ +Rn+1 \ ¯B(x1, 1.5r1), by the maximum principle we have +sup +z∈E\E1 +ωz +1(F) ≤ +max +q∈∂B(x1,1.5r1) ωq +1(F) = ωp +1(F). +By the H¨older continuity of ωz +1(F) in ∂B(x1, 1.8r1), which is far away from F and E \ E1, and by +the uniformity of Ω and Lemma 2.5, we derive +sup +q∈∂B(x1,1.8r1):dist(q,∂Ω)≤δ +ωq +1(F) ≲ +� δ +r1 +�α +sup +q∈A(x1,1.6r1,2r1) +ωq +1(F) ≈ +� δ +r1 +�α +sup +q∈∂B(x1,1.8r1): +dist(q,∂Ω)≥r1 +ωq +1(F), +for some α > 0 depending on s. So we derive that dist(p, ∂Ω) ≈ r1. +Since p is a corkscrew point for Ω relative to B(y, r1), by the CDC and a Harnack chain +argument, ωp(E1) ≥ δ0, for some δ0 > 0 depending just on s. +Hence, ωp(E \ E1) ≤ 1 − δ0. +Therefore, +ωp +1(F) ≤ ωp(F) + ωp +1(F) (1 − δ0), +or equivalently, ωp +1(F) ≤ δ−1 +0 +ωp(F). Since both x1 and p are corkscrew points for Ω relative to +B(y, 2r1), we deduce that +ωx1 +1 (F) ≲ ωx1(F). +On the other hand, by the maximum principle, we have trivially that ωx1 +1 (F) ≥ ωx1(F), which +concludes the proof of the claimed estimate (4.4). +Next we will perform another modification of the domain Ω1. For 0 < ρ ≤ r1/4, we denote +E2 = E1 ∪ +� ¯B(y, ρ/2) ∩ Rn \ B1 +� +and we let Ω2 = Rn+1 \ E2. By arguments analogous to the ones +for the case n = 1, we infer that +(4.5) +ωx1 +1 (B(y, ρ)) ≳ ωx1 +2 (B(y, ρ/2)) ≥ ωx1 +1 (B(y, ρ/2)). +Finally, we let E3 = (2 ¯B1 \B1) ∩ Rn and Ω3 = Rn+1 \E3. By the maximum principle and Lemma +4.1, we have +ωx1 +2 (B(y, ρ/2)) ≥ ωx1 +3 (B(y, ρ/2)) ≳ +�ρ +r +�n− 1 +2. +Together with (4.4) and (4.5), and the change of pole formula, as in (3.6), this yields (2.5). + +DIMENSION OF HARMONIC MEASURE ON SOME FLAT SETS OF FRACTIONAL DIMENSION +15 +4.3. The case s ∈ (n − 1 +2 − ε, n − 1 +2]. In this case, instead of obtaining an explicit estimate for +ω(B(y,ρ)) +ω(B(x,r)), we will argue by contradiction. +We assume that ε ≤ 1/4. In this situation it is easy to check that the estimates obtained in +Subsection 4.2 involving the domains Ω1, Ω2, Ω3 are uniform in s, assuming the AD-regularity +constant to be also uniform on s. In the rest of this subsection we assume also that all the implicit +constants involved in the relations ≲, ≳, ≈ are uniform on s, for s ∈ [1/2 − ε, 1/2]. +For a given M ≫ 1, suppose that +(4.6) +M−1�ρ +r +�s +≤ ωp0(B(y, η)) +ωp0(B(x, r)) ≤ M +�ρ +r +�s +for ρ ≤ η ≤ r. +For x1 as above, we write +ωx1 +2 (B(y, ρ/2)) = +ˆ +E3 +ωz +2(B(y, ρ/2)) dωx1 +3 (z). +Consider the annuli Ak = A(y, 2kρ, 2k+1ρ), for k ≥ 1. Let N = [log2 +r1 +ρ ]. By the s-AD regularity +of E and pigeon-holing, for every k ∈ [1, N] there exists a ball Bk centered in E3 such that +2Bk ∩ Rn ⊂ Ak ∩ E3 \ E with rad(Bk) ≈ 2kρ. So we have +ωx1 +2 (B(y, ρ/2)) ≥ +N +� +k=1 +ˆ +E3∩Bk +ωz +2(B(y, ρ/2)) dωx1 +3 (z) ≳ +N +� +k=1 +ωx1 +3 (Bk) inf +z∈Bk +ωz +2(B(y, ρ/2)). +By Lemma 4.1, we have +ωx1 +3 (Bk) ≳ +�2kρ +r +�n− 1 +2 . +By the maximum principle and the change of pole formula, for any z ∈ Bk, +ωz +2(B(y, ρ/2)) ≥ ωz +1(B(y, ρ/2)) ≥ ωz(B(y, ρ/2)) ≈ ωp0(B(y, ρ/2)) +ωp0(B(y, 2kρ)) ≥ M−2 +ρs +(2kρ)s = M−2 2−ks. +In the case s = n − 1 +2, we obtain +ωx1 +2 (B(y, ρ/2)) ≳ M−2 +N +� +k=1 +�2kρ +r +�n− 1 +2 2−k(n− 1 +2) = M−2N +�ρ +r +�n− 1 +2. +By the change of pole formula and the relationship between ω1, ω2, ω3, this yields +(4.7) +ωp0(B(y, ρ)) +ωp0(B(x, r)) ≈ ωx1(B(y, ρ)) ≈ ωx1 +1 (B(y, ρ)) ≳ ωx1 +2 (B(y, ρ/2)) ≳ M−2N +�ρ +r +�n− 1 +2. +For N big enough, this contradicts the assumption (4.6). +Hence (2.4) holds, with ρ in (2.4) +replaced by some η such that ρ ≤ η ≤ r. +For s ∈ (n − 1 +2 − ε, n − 1 +2), we have +ωx1 +2 (B(y, ρ/2)) ≳ M−2 +N +� +k=1 +�2kρ +r +�n− 1 +2 2−ks = M−2�ρ +r +�n− 1 +2 +N +� +k=1 +2k(n− 1 +2−s). +Assuming N big enough, or equivalently ρ small enough (depending on |n − 1 +2 − s|), we write +N +� +k=1 +2k(n− 1 +2−s) ≥ +N−1 +� +k=0 +2k(n− 1 +2−s) = 2N(n− 1 +2 −s) − 1 +2n− 1 +2−s − 1 +≈ 2N(n− 1 +2 −s) +2n− 1 +2−s − 1 +≈ +rn− 1 +2−s +(n − 1 +2 − s) ρn− 1 +2 −s . + +16 +XAVIER TOLSA +Hence, +ωx1 +2 (B(y, ρ/2)) ≳ M−2 +ρs +(n − 1 +2 − s) rs. +As in (4.7), this implies +ωp0(B(y, ρ)) +ωp0(B(x, r)) ≳ M−2 +ρs +(n − 1 +2 − s) rs. +For N big enough, this contradicts (4.6) and so (2.4) holds, with ρ in (2.4) replaced by some η +such that ρ ≤ η ≤ r. +□ +5. Extension to subsets of C1 manifolds +In this section we extend the main results of the paper to s-AD regular subsets of n-dimensional +C1 manifolds. We prove the following. +Theorem 5.1. For n ≥ 1 and s ∈ [n − 1 +2, n), let E ⊂ Rn+1 be an (s, C0)-AD regular compact set +contained in an n-dimensional C1 manifold. Let Ω = Rn+1 \ E and denote by ω the harmonic +measure for Ω. Then dim ω < s. +Given E ⊂ Rn+1, x ∈ Rn+1 and r > 0, we denote +β∞,E(x, r) = inf +L +sup +y∈B(x,r) +dist(y, L) +r +, +where the infimum is taken over all the hyperplanes L ⊂ Rn+1. +To prove Theorem 5.1, we will use the following variant of the Main Lemma 2.9. +Lemma 5.2. For n ≥ 1, let s ∈ [n − 1 +2, n) and C0 > 0. For all M ≥ 1 there are constants +γ > 0, c > 0 both depending on n, s, C0, M such that the following holds. Let E ⊂ Rn+1 be an +(s, C0)-AD regular compact set, and x ∈ E, 0 < r ≤ diam(E) such that β∞,E(x, γ−1r) ≤ γ2, and +p ∈ Rn+1 \(E ∪ B(x, 2r)). Then there exists a ball B(y, ρ) with y ∈ B(x, r), c r ≤ ρ ≤ r, such that +either +(5.1) +ωp(B(y, ρ)) +ρs +≥ M ωp(B(x, r)) +rs +or +ωp(B(y, ρ)) +ρs +≤ M−1 ωp(B(x, r)) +rs +. +Proof. Suppose the lemma fails. That is, there exist n > 1, s ∈ [n − 1 +2, n), C0 > 0, and M > 1 +such that for all γ = 1/k and c = 1/k there exists an (s, C0)-AD regular set Ek ⊂ Rn+1 and +xk ∈ Ek, 0 < rk ≤ diam(Ek), such that β∞,Ek(x, kr) ≤ k−2, and pk ∈ Rn+1 \ (Ek ∪ B(x, 2rk)), +and moreover for any ball B(y, ρ) with y ∈ B(xk, rk), 1 +k rk ≤ ρ ≤ rk, it holds +(5.2) +M−1 ωpk +k (B(x, r)) +rs +< ωpk +k (B(y, ρ)) +ρs +< M ωpk +k (B(x, r)) +rs +, +where ωk is the harmonic measure for Rn+1 \ Ek. By dilating and translating suitably Ek, we can +assume that xk = 0 and rk = 1. Also, by the uniformity and the s-AD-regularity of the domain +Rn+1 \ Ek, we can also assume that pk ∈ A(0, 2, 3) and dist(pk, Ek) ≥ c′ > 0. By considering a +subsequence, we can assume that the sets Ek converge in the Attouch-Wets topology (this is a +local variant of the Hausdorff metric topology) to some (s, C0)-AD regular set �E ⊂ Rn+1, and +also that pk converges to some point �p ∈ A(0, 2, 3) and dist(�p, �E) ≥ c′ > 0. It easily follows then +that ωpk +k +converges weakly * to �ω�p, the harmonic measure for �Ω = Rn+1 \ �E with pole in �p. + +DIMENSION OF HARMONIC MEASURE ON SOME FLAT SETS OF FRACTIONAL DIMENSION +17 +Observe that, for 0 < R < k, we have +β∞,Ek(0, R) ≤ k +R β∞,Ek(0, k) ≤ 1 +kR → 0 +as k → ∞. +Hence we infer thatβ∞, � +E(0, R) = 0 for every R > 0. That is to say, �E is contained in some line, +which passes through the origin because 0 ∈ �E. On the other hand, from (5.2) and the weak * +convergence of ωpk +k +to �ω�p we deduce that, for x = 0, r = 1, +M−1 �ω�p(B(x, r)) +rs +≤ �ω�p(B(y, ρ)) +ρs +≤ M �ω�p(B(x, r)) +rs +for all y ∈ B(0, r), 0 < ρ ≤ r. +This contradicts Main Lemma 2.9. +□ +Proof of Theorem 5.1. Just notice that, if E is compact, s-AD regular with s ∈ [n− 1 +2, n), and it is +contained in an n-dimensional C1 manifold, then the assumptions of Lemma 5.2 hold for any ball +B(x, r) centered in E with small enough radius r. It is easy to check that then the assumptions of +Lemma 2.8 hold (by adjusting suitably the constant c1 in the lemma if necessary), which ensures +that dim ω < s. +□ +Finally remark that, by similar compactness arguments, one can also extend Theorem 1.2 to +subsets of n-dimensional C1 manifolds: +Theorem 5.3. Let E ⊂ Rn+1 be a compact set contained in an n-dimensional C1 manifold which +is (s, C0)-AD regular for some s > 0. That is, +C−1 +0 rs ≤ Hs(E ∩ B(x, r)) ≤ C0 rs +for all x ∈ E and 0 < r ≤ diam(E). +Then there exists ε > 0 small enough depending on C0 such that if s ∈ (n − 1 +2 − ε, n), we have +dim ω < s. +To prove this, it suffices to consider the case s ∈ (n − 1 +2 − ε, n − 1) and then one can use +arguments similar to ones in Lemma 5.2. The details are left for the reader. +References +[ABR] +S. Axler, P. Bourdon, and W. Ramey. Harmonic function theory. Second edition. Graduate Texts in +Mathematics, 137. Springer-Verlag, New York, 2001. xii+259 pp. 7 +[AHM3TV] J. Azzam, S. Hofmann, J.M. Martell, S. Mayboroda, M. Mourgoglou, X. Tolsa, and A. Volberg. Recti- +fiability of harmonic measure. Geom. Funct. Anal. (GAFA), 26 (2016), no. 3, 703–728. 1 +[AHMMT] +J. Azzam, S. Hofmann, J.M. Martell, M. Mourgoglou, and X. Tolsa. Harmonic measure and quantitative +connectivity: geometric characterization of the Lp-solvability of the Dirichlet problem. Invent. Math. +222 (2020), no. 3, 881–993. 1 +[An] +A. Ancona, On strong barriers and an inequality of Hardy for domains in Rn, J. London Math. Soc. +(2) 34 (1986), no. 2, 274–290. 5 +[Az1] +J. Azzam. Semi-uniform domains and the A∞ property for harmonic measure. Int. Math. Res. Not. +IMRN 2021, no. 9, 6717–6771. 1 +[Az2] +J. Azzam. Dimension drop for harmonic measure on Ahlfors regular boundaries. Potential Anal. +53:1025–104 (2020). 2, 3, 7 +[Ba] +A. Batakis. Harmonic measure of some Cantor type sets. Ann. Acad. Sci. Fenn. Math. 21(2) (1996), +255–270. 2, 3, 7 +[Bo] +J. Bourgain. On the Hausdorff dimension of harmonic measure in higher dimension. Invent. Math. 87 +(1987), no. 3, 477–483. 2, 3, 7 + +18 +XAVIER TOLSA +[BZ] +A. Batakis and A. Zdunik. Hausdorff and harmonic measures on non-homogeneous Cantor sets. Ann. +Acad. Sci. Fenn. Math. 40 (2015), no. 1, 279–303. 2 +[Ca] +L. Carleson. On the support of harmonic measure for sets of Cantor type. Ann. Acad. Sci. Fenn. Ser. +A I Math., 10 (1985), 113–123. 2 +[CTV] +J. Cuf´ı, J. Verdera, X. Tolsa. About the Jones-Wolff Theorem on the Hausdorff dimension of harmonic +measure. Preprint arXiv:1809.08026 (2018). 5 +[DFM] +G. David, J. Feneuil, and S. Mayboroda. Elliptic theory for sets with higher co-dimensional boundaries. +Mem. Amer. Math. Soc. 274 (2021), no. 1346, vi+123 pp. 6 +[DJJ] +G. David, C. Jeznach, and A. Julia. Private communication. 2022. 2, 3 +[GM] +J. B. Garnett and D. E. Marshall. Harmonic Measure. New Mathematical Monographs 2. Cambridge +University Press, Cambridge, 2005. 2 +[GMT] +J. Garnett, M. Mourgoglou, and X. Tolsa. Uniform rectifiability in terms of Carleson measure estimates +and ε-approximability of bounded harmonic functions. Duke Math. J. 167 (2018), no. 8, 1473–1524. 1 +[HMM] +S. Hofmann, J.M. Martell, and S. Mayboroda. Uniform rectifiability, Carleson measure estimates, and +approximation of harmonic functions. Duke Math. J. 165 (2016), no. 12, 2331–2389. 1 +[JK] +D. S. Jerison and C. E. Kenig. Boundary behavior of harmonic functions in nontangentially accessible +domains, Adv. in Math. 46 (1982), no. 1, 80–147. 6 +[Jo] +P.W. Jones. On scaling properties of harmonic measure. Perspectives in analysis, 73?81, Math. Phys. +Stud., 27, Springer, Berlin, 2005. 2 +[JW] +P. W. Jones and T.H. Wolff. Hausdorff dimension of harmonic measures in the plane. Acta Math., +161(1-2) (1988), 131–144. 1 +[Mak] +N. G. Makarov. On the distortion of boundary sets under conformal mappings. Proc. London Math. +Soc. (3), 51(2) (1985), 369–384. 1 +[Mat] +P. Mattila. Geometry of sets and measures in Euclidean spaces. Cambridge Studies in Advanced Math- +ematics, vol. 44, Cambridge University Press, Cambridge, 1995. 5 +[MT] +M. Mourgoglou and X. Tolsa. Harmonic measure and Riesz transform in uniform and general domains. +J. Reine Angew. Math. 758 (2020), 183–221. 6 +[MV] +N. Makarov and A. Volberg. On the harmonic measure of discontinuous fractals. Preprint LOMI E-6-86, +Leningrad, 1986. 2 +[RR] +F. Riesz and M. Riesz. ¨Uber die randwerte einer analtischen funktion. Compte Rendues du Quatri`eme +Congr`es des Math´ematiciens Scandinaves, Stockholm 1916, Almqvists and Wilksels, Upsala, 1920. 1 +[To] +Tolsa, Xavier The mutual singularity of harmonic measure and Hausdorff measure of codimension +smaller than one. Int. Math. Res. Not. IMRN 2021, no. 18, 13783—13811. 4 +[UZ] +M. Urba´nski and A. Zdunik. Hausdorff dimension of harmonic measure for self-conformal sets. Adv. +Math., 171(1) (2002), 1–58. 2 +[Vo1] +A.L. Volberg. On the harmonic measure of self-similar sets on the plane. In Harmonic Analysis and +Discrete Potential theory, p. 267–280. Springer, 1992. 2 +[Vo2] +A. Volberg. On the dimension of harmonic measure of Cantor repellers. Michigan Math. J., 40(2) +(1993), 239–258. 2 +[Wo1] +T.H. Wolff. Plane harmonic measures live on sets of σ-finite length. Ark. Mat. 31 (1993), no. 1, 137–172. +2 +[Wo2] +T.H. Wolff. Counterexamples with harmonic gradients in R3. In: Essays on Fourier Analysis in Honor +of Elias M. Stein, Princeton Math. Ser. 42, Princeton Univ. Press, 321–384 (1995). 2 +ICREA, Barcelona, Dept. de Matem`atiques, Universitat Aut`onoma de Barcelona, and Centre de +Recerca Matem`atica, Barcelona, Catalonia. +Email address: xtolsa@mat.uab.cat + diff --git a/X9E2T4oBgHgl3EQfuwii/content/tmp_files/load_file.txt b/X9E2T4oBgHgl3EQfuwii/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b5a43710fbc80d8afd2665737b5bd6400e1e7baf --- /dev/null +++ b/X9E2T4oBgHgl3EQfuwii/content/tmp_files/load_file.txt @@ -0,0 +1,790 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf,len=789 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='04084v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='CA] 10 Jan 2023 THE DIMENSION OF HARMONIC MEASURE ON SOME AD-REGULAR FLAT SETS OF FRACTIONAL DIMENSION XAVIER TOLSA Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' In this paper it is shown that if E ⊂ Rn+1 is an s-AD regular compact set, with s ∈ [n − 1 2, n), and E is contained in a hyperplane or, more generally, in an n-dimensional C1 manifold, then the Hausdorff dimension of the harmonic measure for the domain Rn+1 \\ E is strictly smaller than s, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=', than the Hausdorff dimension of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Introduction The study of the metric and geometric properties of harmonic measure is a classical topic in analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' At least, this goes back to the work of the Riesz brothers [RR] about the mutual absolute continuity between harmonic measure and arc-length measure on Jordan domains with rectifiable boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' In the last years, the application of new ideas and techniques originating from harmonic analysis, PDE’s, and geometric measure theory has allowed to obtain remarkable advances, especially when the boundary of the domain has codimension 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' See for example, [AHM3TV], [AHMMT], [Az1], [GMT], [HMM].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' The case of codimension different from one is less studied and presents more difficulties, due to the fact that notions such as rectifiability or L2 boundedness of Riesz transforms seem to play no role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' In this paper we will focus on the behavior of harmonic measure on AD-regular boundaries of codimension larger than one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Recall that, given Ω ⊂ Rn+1 and p ∈ Ω, the harmonic measure ωp for Ω with pole in p is the Borel measure supported in ∂Ω such that, for any f ∈ Cc(∂Ω) which extends continuously to the whole Ω and is harmonic in Ω (and vanishes at ∞ in case that n > 1 and Ω is unbounded), we have f(p) = ˆ ∂Ω f dωp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' One of the most important problems about harmonic measure consists in estimating its dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Recall that, for any Borel measure ν in Rn+1, its (Hausdorff) dimension, denoted by dim ν, is defined by dim ν = inf {dim F : F ⊂ Rn+1 Borel, ν(F c) = 0}, where dim F stands for the Hausdorff dimension of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Notice that dim ωp does not depend on the precise pole p, assuming Ω to be connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' So quite often we will write dim ω instead of dim ωp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' One of the first relevant results about the dimension of harmonic measure was obtained by Makarov [Mak] in 1985, when he showed that for any simply connected domain in the plane, dim ω = 1 (in spite of the fact that the dimension of the boundary of the domain may have dimension larger than 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Later on, Jones and Wolff [JW] showed that, for any arbitrary domain The author is supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement 101018680).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Also partially supported by MICINN (Spain) under the grant PID2020-114167GB-I00 and the Mar´ıa de Maeztu Program for units of excellence (Spain) (CEX2020- 001084-M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 1 2 XAVIER TOLSA in the plane, dim ω ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Subsequently, Wolff [Wo1] sharpened this result by proving that ω must be concentrated on a set of σ-finite length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' In the higher dimensional case n > 1, the situation is more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' On the one hand, Bourgain [Bo] proved in 1987 that there exists some constant εn > 0 just depending on n such that dim ω ≤ n − εn for any Ω ⊂ Rn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' A natural guess would be that one could take εn = 1, so that dim ω ≤ n for any domain of Rn+1, analogously to what happens in the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' However, this was disproved by Wolff in his celebrated work [Wo2], where he managed to construct a snowflake type domain Ω ⊂ Rn+1 satisfying dim ω > n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' A difficult open question in the area consists in finding the optimal value of the constant εn such that dim ω ≤ n − εn for any Ω ⊂ Rn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' See for example [Jo].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' When the (Hausdorff) codimension of ∂Ω is not 1 or ∂Ω is of fractal type, many examples show that we may have dim ω < dim ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' This is the so-called “dimension drop” for harmonic measure, which seems to be a frequent phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' This was first observed by Carleson [Ca] for some domains defined as complements of suitable Cantor type sets in the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Later on, Jones and Wolff showed a similar result for some planar boundaries ∂Ω satisfying some uniform disconnectedness property (see [GM, Section X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' In [Vo1] and [Vo2] (see also [MV]) Volberg studied the dimension drop for a large class of Cantor repellers in the plane (see [Vo2] for the notion of Cantor repeller and the precise statement of the result).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Later on, Batakis [Ba] proved analogous results for the harmonic measure for a large class of (complements of) self-similar sets in the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' In [UZ] Urba´nski and Zdunik showed that the dimension drop also occurs for the attractors of conformal iterated function systems (IFS) when either the limit set is contained in a real-analytic curve, if the IFS consists of similarities only, or if the IFS is irregular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Another related result was obtained more recently in by Batakis and Zdunik in [BZ] for another class of IFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' In view of the results described above, it is natural to wonder if the dimension drop for harmonic measure occurs for more general, “not dynamically generated”, subsets of Rn+1 with boundaries with fractional dimension, like domains with AD-regular boundaries of fractional dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Re- call that, given s > 0 and C0 > 1, a set E ⊂ Rn+1 is called AD-regular (or s-AD regular, or (s, C0)-AD regular, if we want to be more precise) if C−1 0 rs ≤ Hs(E ∩ B(x, r)) ≤ C0 rs for all x ∈ E and 0 < r ≤ diam(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' An answer in the affirmative to the above question was given by Azzam [Az2] in the case s ∈ (n, n + 1) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=', for codimension smaller than 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' The case of codimension larger than 1 (with1 s ∈ (n − 1, n)) is more challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' In fact, very recently Guy David, Cole Jeznach, and Antoine Julia [DJJ] have informed me that they have managed to construct some s-AD-regular sets E ⊂ R2, with s ∈ (0, 1), such that the harmonic measure for Ω = R2 \\ E is also s-AD-regular, and thus mutually absolutely continuous with Hs|E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' A previous (unpublished) example of Chris Bishop also showed that harmonic measure can be mutually absolutely continuous with Hs for s < 1 for some domains in the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' However in Bishops’s example ∂Ω is not s-AD regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' The main result of the present paper goes in the converse direction, and it shows that the dimension drop occurs for s-AD regular subsets of hyperplanes in a suitable range of values of s of codimension larger than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' In the plane, for example, this occurs for s-AD regular sets contained in a line, for s ∈ [1 2, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' The precise result is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 1When s ≤ n − 1 and ∂Ω is s-AD regular, this is a polar set and harmonic measure on ∂Ω is not defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' DIMENSION OF HARMONIC MEASURE ON SOME FLAT SETS OF FRACTIONAL DIMENSION 3 Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' For n ≥ 1 and s ∈ [n − 1 2, n), let E ⊂ Rn+1 be an s-AD regular compact set contained in a hyperplane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Let Ω = Rn+1 \\ E and denote by ω the harmonic measure for Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Then dim ω < s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' One can quantify the result above and show that dim ω ≤ s − κ, with κ = κ(n, s, C0) > 0, for E ⊂ Rn+1 being (s, C0)-AD regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Further, the assumption that E is contained in a hyperplane can be replaced by E being contained in a C1 n-dimensional manifold in Rn+1 (see Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='1 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' I don’t know if the threshold n − 1 2 is sharp in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' However, we can go a bit below the threshold n − 1 2 in the following sense: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' For s > 0, C0 > 1, let E ⊂ Rn+1 be an (s, C0)-AD regular compact set contained in a hyperplane, and let Ω = Rn+1 \\ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Then there exists ε > 0 small enough depending on n and C0 such that if s ∈ [n − 1 2 − ε, n), we have dim ω < s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' On the other hand, it seems that Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='1 does not hold for s close enough to n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Indeed, it seems that some of the aforementioned examples of the authors in [DJJ] are s-AD regular subsets of the real line in the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' So an interesting open problem consists in finding the sharp threshold s0 such that for all s-AD regular sets with s ∈ (s0, 1) contained in a line the dimension drop for harmonic measure occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Also, for s-AD regular sets not contained in a line, it is an open question if a similar threshold exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' The proofs of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='1 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='2 rely on an idea originating from Bourgain [Bo] (see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='8) and they follow an approach similar to the one of [Ba] and [Az2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Using touching point arguments, the maximum principle, and suitable modifications of the domain, we are led to some estimates involving the harmonic measure for the complement of a segment in the plane and for the complement of a suitable flat annulus for n > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' In the plane, such harmonic measure can be computed explicitly by means of a conformal transformation while in higher dimensions we need more elaborated estimates which also rely on the conformal transformation from the planar case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Our arguments make an essential us of the fact the boundary ∂Ω is contained in a hyperplane and so they cannot be extended to arbitrary s-AD regular sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Preliminaries In the paper, constants denoted by C or c depend just on the dimension and perhaps other fixed parameters, such as the parameter s in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='1, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' We will write a ≲ b if there is C > 0 such that a ≤ Cb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' We write a ≈ b if a ≲ b ≲ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Capacities and the capacity density condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' The set of (positive) Radon measure in Rn+1 is denoted by M+(Rn+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' The Hausdorff s-dimensional measure and Hausdorff s-dimensional content are denoted by Hs and Hs ∞, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' The fundamental solution of the negative Laplacian in R2 is E2(x) = 1 2π log 1 |x|, while in higher dimensions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=', in Rn+1, n ≥ 2, it equals En+1(x) = cn |x|n−1 , 4 XAVIER TOLSA where cn = (n−1)Hn(Sn), with Sn being the unit hypersphere in Rn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' In any case, for a measure µ in Rn+1, we consider the energy I(µ) = ¨ En+1(x − y) dµ(x) dµ(y) and, for F ⊂ Rn+1 we define the capacity Cap(F) = 1 infµ∈M1(F ) I(µ), where the infimum is taken over all probability measures µ supported on F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' In the planar case, Cap(F) is the Wiener capacity of F, and in higher dimensions this is the Newtonian capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' From now on, in the plane we will write CapW (F) instead of Cap(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' In fact, in the plane it is more convenient to work with the logarithmic capacity, defined by CapL(F) = e− 2π CapW (F ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Recall that we denote by ω (and sometimes ωΩ) the harmonic measure on an open set Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' The following result is well known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' See for example Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='1 from [To].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' For n ≥ 2, let Ω ⊂ Rn+1 be open and let B be a closed ball centered in ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Then ωx(B) ≥ c(n)Cap(1 4B \\ Ω) rad(B)n−1 for all x ∈ 1 4B ∩ Ω, with c(n) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' An analogous result holds in the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' The precise statement is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Let Ω ⊂ R2 be open and let B be a closed ball centered in ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Then ωx(B) ≳ 1 log rad(B) CapL(1 4B \\ Ω) for all x ∈ 1 4B ∩ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' I provide the detailed proof below because it is not easy to find in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Denote r = rad(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Replacing Ω by 1 4r Ω if necessary, we can assume that diam(B) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Then, denoting F = B \\ Ω, the following identity holds: CapW (F) = sup � µ(F) : µ ∈ M+(Rn+1), supp µ ⊂ F, ∥E ∗ µ∥∞ ≤ 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Let µ be the optimal measure for this supremum, so that suppµ ⊂ F, µ(F) = CapW (F), and the function u := E ∗ µ is harmonic out of F and it satisfies ∥u∥∞ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' For all z ∈ 1 4B and all y ∈ F we have |z − y| ≤ 1 2 r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Therefore, u(z) = 1 2π ˆ log 1 |z − y| dµ(y) ≥ 1 2π ˆ log 2 r dµ(y) = µ(F) 2π log 2 r for all z ∈ 1 4B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Also, for z ∈ Bc, we have dist(z, supp F) ≥ 3 4r(B), and thus u(z) ≤ 1 2π ˆ log 4 3r dµ(y) = µ(F) 2π log 4 3r for all z ∈ Bc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Consider now the function v = u − µ(F) 2π log 4 3r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' DIMENSION OF HARMONIC MEASURE ON SOME FLAT SETS OF FRACTIONAL DIMENSION 5 Observe that v(z) ≥ µ(F) 2π log 2 r − µ(F) 2π log 4 3r = µ(F) 2π log 3 2 for all z ∈ 1 4B and v(z) ≤ 0 for all z ∈ Bc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' By the maximum principle and the fact that x ∈ 1 4B we deduce that ωx(B) ≥ v(x) sup v ≥ µ(F) 2π log 3 2 = c CapW(F) sup v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Regarding sup v, taking into account that ∥u∥∞ ≤ 1, it is clear that sup v ≤ 1 − 1 2π log 4 3r µ(F) = 1 − 1 2π log 4 3r CapW(F) ≤ 1 − 1 2π log 1 r CapW(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Therefore, ωx(B) ≥ c CapW(F) 1 − 1 2π log 1 r CapW (F) = c′ 1 log 1 CapL(F) − log 1 r = c′ 1 log r CapL(F) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Let E ⊂ Rn+1 be compact and n − 1 < s ≤ n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' In the case n > 1, we have Cap(E) ≳s,n Hs ∞(E) n−1 s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' In the case n = 1, we have CapL(E) ≳s Hs ∞(E) 1 s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' The proof of this result is an immediate consequence of Frostman’s Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' See [Mat, Chapter 8] for the case n > 1, and [CTV, Lemma 4] for the case n = 1, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Let Ω ⊊ Rn+1 be open, and let ξ ∈ ∂Ω and r0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' We say that Ω satisfies the (ξ, r0)-local capacity density condition (CDC) if there exists some constant c > 0 such that, for any r ∈ (0, r0), Cap(B(ξ, r) \\ Ω) ≥ c rn−1 in the case n > 1, and CapL(B(ξ, r) \\ Ω) ≥ c r in the case n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' We say that Ω satisfies the capacity density condition (CDC) if it satisfies the (ξ, r0)-local capacity density condition for all ξ ∈ ∂Ω and all r0 ∈ (0, diam∂Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Remark that, by the previous lemmas, if Ω satisfies the CDC, then it holds ωx(B) ≳ 1 for all x ∈ 1 4B ∩ Ω In particular, if ∂Ω is s-AD regular for some s > n − 1, then Ω satisfies the CDC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' The following result is also standard and well known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' See [An], for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 6 XAVIER TOLSA Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Let Ω ⊂ Rn+1, let ξ ∈ ∂Ω, and let r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Suppose that Ω satisfies the (ξ, r0)-local capacity density condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Let u be a nonnegative function which is continuous in B(ξ, r) ∩ Ω and harmonic in B(ξ, r) ∩ Ω, and vanishes continuously on B(ξ, r) ∩ ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Then there is α > 0 such that for all r ∈ (0, r0), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='1) u(x) ≲ �|x − ξ| r �α sup B(ξ,r)∩Ω u for all x ∈ Ω ∩ B(ξ, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Corkscrews, connectivity conditions, and uniform domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' A domain Ω ⊂ Rn+1 is called uniform (or C-uniform) if for every x, y ∈ Ω there is a curve γ ⊂ Rn+1 connecting x and y such that (a) H1(γ) ≤ C |x − y|, and (b) for all z ∈ γ, dist(z, ∂Ω) ≥ C−1 dist(z, {x, y}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' We say that Ω satisfies the corkscrew condition (or c-corkscrew condition) if there exists some constant c > 0 such that for all ξ ∈ Ω and all 0 < r ≤ diam(∂Ω) there is a ball B(x, cr) ⊂ B(ξ, r) ∩ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' The point x is called a “corkscrew point” relative to the ball B(ξ, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Further, we say that x is a John point relative to B(ξ, r) if for every y ∈ B(ξ, r) ∩ Ω there exists a curve γ satisfying the properties (a), (b) above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' It is immediate to check that if Ω is a uniform domain, then it satisfies the corkscrew condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Also, for these domains, any corkscrew point is a John point relative to the ball associated to the corkscrew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' We will need to use the following result below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' This is proven exactly in the same way as the analogous result for NTA domains in [JK, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' For n ≥ 1, let Ω ⊂ Rn+1 be a domain and ξ ∈ ∂Ω, 0 < r ≤ diam(∂Ω) such that the (ξ, r)-local CDC holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Let x ∈ Ω ∩ B(ξ, r) be a corkscrew John point relative to B(ξ, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Let u be a nonnegative function which is continuous in B(ξ, r) ∩ Ω and harmonic in B(ξ, r) ∩ Ω, and vanishes continuously on B(ξ, r) ∩ ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Then we have sup B(ξ,r/2)∩Ω u ≲ u(x), with the implicit constant depending on the local CDC, and the corkscrew and John properties of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' In [DFM, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='2] the following result has been proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' For 0 < s < n and C0 > 1, let E be an (s, C0)-AD regular set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Then the domain Ω = Rn+1 \\ E is C-uniform, with C depending just on n, s, C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' So the the assumptions in Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='1 or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='2 ensure that the domain Ω in those theorems is uniform and satisfies the CDC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' This is very useful because it implies some nice properties for the associated harmonic measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' For example, it implies that given p ∈ Rn+1 with dist(p, E) ≳ diam(E), the doubling property ωp(2B) ≲ ω(B) holds for any ball B centered in ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' The following theorem contains the so called “change of pole formula” for uniform domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' This is proven in [JK] for NTA domains (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=', for uniform domains Ω such that Rn+1 \\ Ω is a corkscrew domain), while the more general statement below is from [MT].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' DIMENSION OF HARMONIC MEASURE ON SOME FLAT SETS OF FRACTIONAL DIMENSION 7 Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' For n ≥ 1, let Ω ⊂ Rn+1 be a Wiener regular uniform domain and let B be a ball centered at ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Let p1, p2 ∈ Ω such that dist(pi, B ∩ ∂Ω) ≥ c−1 0 r(B) for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Then, for any Borel set E ⊂ B ∩ ∂Ω, ωp1(E) ωp1(B) ≈ ωp2(E) ωp2(B), with the implicit constant depending only on c0 and the uniform behavior of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' The Kelvin transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Given x ∈ Rn+1 \\ {0}, we let x∗ = 1 |x|2 x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Notice that the map defined by T(x) = x∗ is an involution of Rn+1 ∪ {∞}, understanding that T(0) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' For r > 0, we denote r∗ = 1 r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Given Ω ⊂ Rn+1 such that ∂Ω is compact and 0 ̸∈ ∂Ω, we also set Ω∗ = � x∗ : x ∈ Ω � ∪ {0} (so identifying Ω with Ω ∪ {∞}, we have Ω∗ = T(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Given a function f : Rn+1 ⊃ E → R, its Kelvin transform is defined by f ∗(x∗) = 1 |x∗|n−1 f(x), understanding that f(∞) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' It is well known that, if u : Ω → R vanishes at ∞, then ∆u = 0 in Ω if and only if ∆(u∗) = 0 in Ω∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Further, (u∗)∗ = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' See [ABR, Chapter 4], for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Hence, if we take u(x) = ωx Ω(F) for F ⊂ ∂Ω, it follows that u∗ is a harmonic function in Ω, comparable to 1 in F, and vanishing away from F, with the implicit constant depending on dist(0, ∂Ω) and diam(∂Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Hence, for x ∈ Ω, we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='2) ωx∗ Ω∗(F ∗) ≈ u∗(x∗) = 1 |x∗|n−1 u(x) ≈ ωx Ω(F), with the first implicit constant depending on dist(0, ∂Ω), diam(∂Ω), and the second one on |x∗|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' The Main Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' To prove Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='2 we will use the following result, first implicitly used by Bourgain in [Bo], later by Batakis [Ba], and more recently by Azzam [Az2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' For n ≥ 1, s > 0, C0 > 1, there exists an M = M(n, s, C0) > 1 (sufficiently big) such that the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Let E ⊂ Rn+1 be an (s, C0)-AD regular set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Let ν be a measure supported on E and c1 ∈ (0, 1) such that, for all x ∈ E, 0 < r ≤ diam(E), there exists a ball B(y, ρ) with y ∈ B(x, r), c1 r ≤ ρ ≤ r, satisfying either (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='3) ν(B(y, ρ)) ρs ≥ M ν(B(x, r)) rs or ν(B(y, ρ)) ρs ≤ M−1 ν(B(x, r)) rs .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Then dim ν < s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Although this result is not stated as above in [Az2], the detailed arguments of the proof are contained in Section 5 of that paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Given Ω and E as in Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='2, we denote by p0 the pole for harmonic measure and we assume that this is far away from E (in the plane we could take p0 = ∞), and we denote ω = ωp0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Recall that Ω = Ec is a uniform domain satisfying the CDC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Thanks to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='8, the fact that dim ω < s is an immediate consequence of the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 8 XAVIER TOLSA Main Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' For s > 0, C0 > 1, let E ⊂ Rn+1 be an (s, C0)-AD regular closed set contained in a hyperplane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Let M ≥ 1 and suppose either that s ∈ [n − 1 2, n) or s ∈ (n − 1 2 − ε, n − 1 2) for some ε = ε(n, s, C0, M) > 0 small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' For any x ∈ E, 0 < r ≤ diam(E), and p0 ∈ Rn+1 \\E ∪B(x, 2r)), there exists a ball B(y, ρ) with y ∈ B(x, r), c r ≤ ρ ≤ r, with c > 0 depending just on n, s, C0, M, satisfying either (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='4) ωp0(B(y, ρ)) ρs ≥ M ωp0(B(x, r)) rs or ωp0(B(y, ρ)) ρs ≤ M−1 ωp0(B(x, r)) rs .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' To prove the Main Lemma, first we will show that, given E, B(x, r), and p0 as above, there exists y ∈ B(x, r) such that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='5) ωp0(B(y, ρ)) ωp0(B(x, r)) ≥ c(s, C0) �ρ r �n− 1 2 for all ρ ∈ (0, c′ r), for some c′ ∈ (0, 1) depending just on s and C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Clearly, this yields (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='4) for s ∈ (n − 1 2, n) and ρ small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' For the cases s = n − 1 2 and s ∈ (n − 1 2 − ε, n − 1 2) we will need more careful estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Notice also that the estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='5) is independent of the pole p0, modulo a constant factor (as soon as p0 ∈ Rn+1 \\ (E ∪ B(x, 2r))), because Rn+1 \\ E is a uniform domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' The arguments for the planar case n = 1 with s ∈ [1 2, 1) This section and the next one are devoted to the proof of Main Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Proof of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Without loss of generality, we assume that E ⊂ R ≡ R × {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Let x ∈ E and 0 < r ≤ diamE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Taking into account that s < 1, by a pigeon-hole argument, there is an open interval I = (a, b) ⊂ [x − r, x] which does not intersect E and satisfies ℓ := H1(I) ≈s r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' By enlarging I if necessary, we can assume that b ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Notice that b is contained in [x − (1 − c)r, x] because x ∈ E, for some c > 0 depending on s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' We choose y = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Again by the s-AD regularity of E and the pigeon hole principle, there exist radii r1, r2 with ℓ/2 ≤ r1 < r2 ≤ ℓ, r2 − r1 ≈s ℓ ≈ r such that A(y, r1, r2) ∩ E = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Observe that the left component of A(y, r1, r2) ∩ R is contained in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Next we apply a “localization argument”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' We denote E1 = E ∩ ¯B(y, r1), Ω1 = Ec 1, r′ = (r1 +r2)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' It is immediate to check that E1 is still s-AD regular and thus Ω1 is a uniform domain too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' We claim that for any subset F ⊂ E1 and any p ∈ ∂B(y, r′), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='1) ωp 1(F) ≈s ωp(F), where ω1 stands for the harmonic measure for Ω1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' To prove the claim, consider first p ∈ ∂B(y, r′) such that ωp 1(F) = max q∈∂B(y,r′) ωq 1(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Using that ωz 1(F) is harmonic in Ω and vanishes in E1 \\ F and the maximum principle, we get ωp 1(F) = ˆ E ωz 1(F) dωp(z) = ωp(F) + ˆ E\\E1 ωz 1(F) dωp(z) ≤ ωp(F) + sup z∈E\\E1 ωz 1(F) ωp(E \\ E1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' DIMENSION OF HARMONIC MEASURE ON SOME FLAT SETS OF FRACTIONAL DIMENSION 9 Observe that, by the CDC and a Harnack chain argument, ωp(E1) ≥ δ0, for some δ0 > 0 depending just on s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Hence, ωp(E \\ E1) ≤ 1 − δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Also, since ωz 1(F) is harmonic in C∞ \\ B(y, r′) and E \\ E1 ⊂ C∞ \\ B(y, r′), by the maximum principle we have sup z∈E\\E1 ωz 1(F) ≤ max q∈∂B(y,r′) ωq 1(F) = ωp 1(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Therefore, ωp 1(F) ≤ ωp(F) + ωp 1(F) (1 − δ0), or equivalently, ωp 1(F) ≤ δ−1 0 ωp(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' By the definition of r′ and Harnack’s inequality, we infer ωp 1(F) ≲ ωp(F) for all p ∈ ∂B(y, r′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' On the other hand, by the maximum principle, we have trivially that ωp 1(F) ≥ ωp(F), which concludes the proof of the claimed estimate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Next we will perform another modification of the domain Ω1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' For a fixed ρ ∈ (0, r1/4), consider the intervals J = [y, y + ρ/2], J′ = [y, y + ρ] and define E2 = E1 ∪ J and Ω2 = Ec 2 = Ω1 \\ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' By the CDC and the uniformity of Ω1, we infer that, for all q ∈ ∂B(y, ρ/2), ωq 1(J′ ∩ E1) ≳ 1 ≥ ωq 2(J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' We also have ωq 1(J′ ∩ E1) ≥ ωq 2(J) = 0 for all q ∈ Jc ∩ E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Then, by the maximum principle, since both ωz 1(J′ ∩ E1) and ωz 2(J) are harmonic in Ω1 \\ ¯B(y, ρ/2) = Ω2 \\ ¯B(y, ρ/2) we deduce that ωq 1(J′ ∩ E1) ≳ ωq 2(J) for all q ∈ Ω2 \\ ¯B(y, ρ/2), and in particular for all p ∈ ∂B(y, r′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Finally we let E3 = [y, y + r1] and Ω3 = Ec 3, so that E2 ⊂ E3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' By the maximum principle, we have ωp 2(J) ≥ ωp 3(J) for all p ∈ ∂B(y, r′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Hence, gathering the above estimates, we infer that, for all p ∈ ∂B(y, r′), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='2) ωp(J′ ∩ E) ≈s ωp 1(J′ ∩ E) ≳ ωp 2(J) ≥ ωp 3(J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Now it just remains to estimate ωp 3(J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' We can do this by means of a conformal transfor- mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Indeed, observe first that, by a Harnack chain argument and the maximum principle, ωp 3(J) ≈ ω∞ 3 (J) for all p ∈ ∂B(y, r′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Next, suppose for simplicity that y = −r1/2, so that E3 = [−r1/2, r1/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' The map f : ¯B(0, 1) → Ω3 defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='3) f(z) = � z + 1 z �r1 4 is a conformal transformation from B(0, 1) to Ω3 such that f(0) = ∞, with f(∂B(0, 1)) = f(∂Ω3) = f(E3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Thus, ω∞ 3 (J) = 1 2π H1(f −1(J)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' An easy computation shows that f −1(J) = {eiα : π − θ ≤ α ≤ π + θ}, with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='4) θ = arccos � 1 − H1(J) 2r1 � = arccos � 1 − ρ 4r1 � ≈ � ρ r1 �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 10 XAVIER TOLSA Thus, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='5) ω∞ 3 (J) = θ π ≈ � ρ r1 �1/2 ≈ �ρ r �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Consequently, by the change of pole formula for uniform CDC domains and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='2), we deduce that, for p ∈ ∂B(y, r′), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='6) ωp0(B(y, ρ)) ωp0(B(x, r)) ≈ ωp( ¯B(y, ρ)) = ωp(J′ ∩ E) ≳ ωp 3(J) ≈ �ρ r �1/2 , which completes the proof of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' The case s = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' In this case the inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='5) does not suffice to prove (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='4) and we need a better estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' We consider the preceding domains Ω1, Ω2, Ω3, so that, for all p ∈ ∂B(y, r′), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='2) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' However, the estimate ωp 2(J) ≥ ωp 3(J) is too coarse for our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Instead, we write ωp 2(J) ≈ ω∞ 2 (J) = ˆ E3 ωz 2(J) dω∞ 3 (z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' The density dω∞ 3 dH1|E3 can be computed explicitly by means of the conformal transformation in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Using the identity ω∞ 3 (J) = π−1 arccos � 1 − H1(J) 2r1 � and differentiating, it follows that dω∞ 3 dH1|E3 (t) = 1 π � (r1 2 − t)(t + r1 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Thus, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='7) dω∞ 3 dH1|E3 (t) ≈ 1 � r1(t + r1 2 ) for t ∈ [−r1/2, 0], and so (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='8) ωp 2(J) ≳ ˆ 0 −r1 2 ωt 2(J) dt � r1(t + r1 2 ) (recall that we are identifying R ≡ R × {0}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' To estimate the integral in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='8) from below, consider the annuli Ak = A(y, 2kρ, 2k+1ρ), for k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Let N = [log2 r1 ρ ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' By the s-AD regularity of E and pigeonholing, for every k ∈ [1, N] there exists an interval Ik ⊂ Ak ∩ E3 (recall E3 is an interval) such that H1(Ik) ≈ 2kρ and Ik ∩ E = Ik ∩ E2 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Let ˆIk be another interval concentric with Ik and half length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Then we write (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='9) ˆ 0 −r1 2 ωt 2(J) dt � r1(t + r1 2 ) ≥ N � k=1 ˆ ˆIk ωt 2(J) dt � r1(t + r1 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' We claim that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='10) ωt 2(J) ≳ � ρ |t − y| �1/2 for all t ∈ �N k=1 ˆIk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' DIMENSION OF HARMONIC MEASURE ON SOME FLAT SETS OF FRACTIONAL DIMENSION 11 Assuming this for the moment, we obtain ωp 2(J) ≳ N � k=1 ˆ ˆIk � ρ |t − y| �1/2 dt � r1(t + r1 2 ) ≈ N � k=1 ˆ ˆIk � ρ H1(ˆIk) �1/2 dt r1/2 1 H1(ˆIk)1/2 = N � ρ r1 �1/2 ≈ log r ρ �ρ r �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='2) and the change of pole formula, arguing as in the preceding subsection, we obtain (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='11) ωp0(B(y, ρ)) ωp0(B(x, r)) ≈ ωp( ¯B(y, ρ)) = ωp(J′ ∩ E) ≳ ωp 2(J) ≈ log r ρ �ρ r �1/2 , which implies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='4) for ρ small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' It remains to prove (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' To this end, for each t ∈ ˆIk, let t′ ∈ R be the point symmetric to t with respect to y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' That is, t′ = −r1 − t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Notice that t′ is on the left of the interval E3 (recall that the leftmost point of E3 is y = −r1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' By a Harnack chain argument and the maximum principle, we have ωt 2(J) ≈ ωt′ 2 (J) ≥ ωt′ 3 (J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Now we can compute explicitly ωt′ 3 (J) by means of the conformal transformation in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Indeed, consider the change of variable t′ = y − r1 2 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Then, it follows easily that f −1(t′) = −1 1 + h + � h(2 + h) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' So f −1(t′) is a point in the unit disk belonging to the segment (−1, 0) such that | − 1 − f −1(t′)| = 1 − 1 1 + h + � h(2 + h) ≈ h1/2 = �2|t′ − y| r1 �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Recall that f −1(J) = [π−θ, π+θ], with θ ≈ � ρ r1 �1/2, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Hence, |−1−f −1(t′)| ≳ H1(f −1(J)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Taking into account that, for any point q ∈ B(0, 1) and η := 10 dist(q, ∂B(0, 1)), ωq B(0,1)|B(q,η) is comparable to η−1H1|∂B(0,1)∩B(q,η), we deduce that ωt′ 3 (J) = ωf−1(t′) B(0,1) (f −1(J)) ≈ θ | − 1 − f −1(t′)| ≈ � ρ r1 �1/2 � 2|t′−y| r1 �1/2 ≈ ρ1/2 |t′ − y|1/2 , which yields (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' The higher dimensional case n > 1 In this section we complete the proof of Main Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Although we will focus mainly in the case n > 1, the arguments in this section are also valid for the case n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' In the preceding section we have studied separately the case n = 1, s ∈ [1/2, 1) because the arguments are more elementary, specially for s = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 12 XAVIER TOLSA 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' An auxiliary lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' In the case n > 1, we cannot use conformal transformations as in the planar case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Instead, we will use the following auxiliary result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' For n > 1, r > 0, let E = (B(0, 2r)\\B(0, r))∩(Rn ×{0}) and Ω = Rn+1 \\E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Let B a ball centered in E such that ρ := rad(B) ≈ dist(B, ∂B(0, r)), with 0 < ρ ≤ r/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Then we have ω0(B) ≳ �ρ r �n− 1 2 , where ω0 stands for the harmonic measure for Ω with pole in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Quite likely this result can be proven by an explicit computation of the density dω0 dHn|E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' However, I have not found in the literature such computation, which seems to be non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' For this reason, below I show a different argument relying on the planar case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Observe first that, by rescaling, we can assume that r = 1, and we can estimate ωp0(B) for p0 = (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' , 0, 1/2), say, instead of ω0(B), since ω0(B) ≈ ωp0(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Next, by applying the Kelvin transform Tx = x |x|2, the statement in the lemma is equivalent to saying that, for Ω1 = Rn+1 \\ E1 with E1 = T(E) = (B(0, 1) \\ B(0, 1/2)) ∩ (Rn × {0}), and any ball B centered in E1 such that ρ := rad(B) ≈ dist(B, ∂B(0, 1)), with 0 < ρ ≤ r/2, we have ωp1 1 (B) ≳ ρn− 1 2, where ωp1 1 stands for the harmonic measure for Ω1 with pole in p1 := T(p0) = (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' , 0, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Consider now Ω2 = Rn+1 \\ E2 with E2 = B(0, 1) ∩ (Rn × {0}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' By the maximum principle, it is clear that ωp1 1 (B) ≥ ωp1 2 (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Hence it suffices to prove that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='1) ωp1 2 (B) ≳ ρn− 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' To prove (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='1), we may assume that the ball B is centered in the segment L connecting the origin to the point (1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' , 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Then we let Q be an n-dimensional cube concentric with B (with sides parallel to the axes), contained in E2 ∩ B, with side length comparable to rad(B), and we set R = [−1, 1]n × {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Since Q ⊂ B ∩ E2 and E2 ⊂ R, by the maximum principle we have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='2) ωp1 2 (B) ≥ ωp1 R (Q), where ωR stands for the harmonic measure for the domain Rn+1 \\ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' To estimate ωp1 R (Q) from below, we can assume that this is an n-dimensional dyadic cube descendant of R, by translating and reducing Q slightly if necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' We let I0 = [−1, 1] and J = L ∩ Q, so that J ⊂ I0 and dist(J, ∂I0) ≈ ℓ(J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' We consider I0, J as subsets of R ≡ R × {0} ⊂ C and we let u(z) = ωz I0(J), where ωz I0 is the harmonic measure for the domain C \\ I0 with pole in z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Next we denote �I0 = I0 × Rn−1 × {0}, �Ω0 = Rn+1 \\ �I0, �J = J × Rn−1 × {0} and we take the function f : �Ω0 → R defined by f(x) = u(x1, xn+1), for x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' , xn+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Notice that f is non-negative, bounded by 1, and harmonic in Rn+1 \\ �I0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Moreover, it extends continuously to 0 in �I0 \\ �J, and to 1 in J◦ × Rn−1 × {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Then it follows that f(x) = �ωx( �J), where �ωx is the harmonic measure for �Ω0 with pole in x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' We claim that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='3) �ωp1( �J) ≈ �ωp1( �J ∩ R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' DIMENSION OF HARMONIC MEASURE ON SOME FLAT SETS OF FRACTIONAL DIMENSION 13 To prove this, denote g(x) = �ωx( �J ∩ R), so that f(x) − g(x) = �ωx( �J \\ R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' For a given δ ∈ (0, 1/2), let zδ ∈ C \\ I0 be such that dist(zδ, I0) = δ and u(zδ) = max z∈C:dist(z,I0)=δ u(z) = max z∈C:dist(z,I0)=δ ωz I0(J), so that by the maximum principle u(z) ≤ u(z0) for any z ∈ C such that dist(z, I0) ≥ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Let qδ = (ℜzδ, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' , 0, ℑzδ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' By the definition of f, it turns out that f(x) ≤ f(qδ) for all x ∈ Rn+1 such that dist(x, �I0) ≥ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Then, by the (local) CDC of �I0, since f − g vanishes identically in R × {0}, we deduce that |f(x) − g(x)| ≲ sup y∈∈3I0×( 3 4 I0)n |f(y) − g(y)| dist(x, R)α for all x ∈ 2I0 × (1 2I0)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Since p1 is a local John point for �Ω0 in 4I0×In 0 , the supremum above is bounded by C|f(p1)−g(p1)|, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Then, choosing x = qδ we obtain |f(qδ) − g(qδ)| ≲ |f(p1) − g(p1)| dist(qδ, R)α ≤ f(p1) δα ≤ f(qδ) δα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' So, choosing δ small enough (independent of f and g), we derive f(qδ) ≈ g(qδ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Since f(p1) ≈ f(qδ) and g(p1) ≈ g(qδ) (with implicit constants depending on δ), we deduce (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='3), as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Next we split R = �N i=1 Pi, where Pi’s are dyadic n-dimensional cubes, descendants of R, with the same side length as Q, and N = ℓ(Q)−n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Observe that J ∩ R coincides with the union of NJ := ℓ(Q)1−n of the cubes Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' So reordering the family of such cubes if necessary, we have �ωp1( �J ∩ R) = NJ � i=1 �ωp1(Pi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' It is easy to check that �ωp1(Pi) ≈ �ωp1(Pj) for all 1 ≤ i, j ≤ NJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Indeed, if ci stands for the center of Pi and we take the corkscrew point zi = ci + (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' , 0, 1), by symmetry we have �ωzi(Pi) = �ωzj(Pj) for all i, j, and by Harnack, �ωzi(Pi) ≈ �ωp1(Pi) for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Recalling that Q coincides with one of the cubes Pi, using also (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='3), we get �ωp1(Q) ≳ 1 NJ NJ � i=1 �ωp1(Pi) = ℓ(Q)n−1 �ωp1( �J ∩ R) ≈ ℓ(Q)n−1 �ωp1( �J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' By construction, we have �ωp1( �J) = ω(0,1) I0 (J), and using the conformal transformation in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='3) and the identity (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='7), it is immediate to check that ω(0,1) I0 (J) ≈ ℓ(J)1/2 = ℓ(Q)1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Therefore, �ωp1(Q) ≳ ℓ(Q)n− 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' By the maximum principle, we have ωp1 R (Q) ≥ �ωp1(Q), and then by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='2), we deduce that ωp1 2 (B) ≳ ℓ(Q)n− 1 2 ≈ ρn− 1 2, which proves (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='1) and the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' □ 14 XAVIER TOLSA 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Proof of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' We assume that E ⊂ Rn ≡ Rn × {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' By a pigeon-hole argument, there is an open ball B1 := B(x1, r1) centered in Rn, which satisfies: 2 ¯B1 ⊂ B(x, r), B1 ∩ E = ∅, ∂B1 ∩ E ̸= ∅, r1 ≈ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' We choose y ∈ ∂B1 ∩ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' As in the case n = 1, we intend to apply a localization argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' We denote E1 = E∩ ¯B(x1, 2r1), Ω1 = Ec 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' We claim that for any subset F ⊂ E1 ∩ B(x1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='5r1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='4) ωx1 1 (F) ≈s ωp(F), where ω1 stands for the harmonic measure for Ω1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' To prove the claim, consider first p ∈ ∂B(x1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='8r1) such that ωp 1(F) = max q∈∂B(x1,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='8r1) ωq 1(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Using that ωz 1(F) is harmonic in Ω and vanishes in E1 \\ F, we get ωp 1(F) = ˆ E ωz 1(F) dωp(z) = ωp(F) + ˆ E\\E1 ωz 1(F) dωp(z) ≤ ωp(F) + sup z∈E\\E1 ωz 1(F) ωp(E \\ E1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Since ωz 1(F) is harmonic in Rn+1 \\ E1, vanishes in E1 \\ ¯B(x1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='5r1) and at ∞, and E \\ E1 ⊂ Rn+1 \\ ¯B(x1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='5r1), by the maximum principle we have sup z∈E\\E1 ωz 1(F) ≤ max q∈∂B(x1,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='5r1) ωq 1(F) = ωp 1(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' By the H¨older continuity of ωz 1(F) in ∂B(x1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='8r1), which is far away from F and E \\ E1, and by the uniformity of Ω and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='5, we derive sup q∈∂B(x1,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='8r1):dist(q,∂Ω)≤δ ωq 1(F) ≲ � δ r1 �α sup q∈A(x1,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='6r1,2r1) ωq 1(F) ≈ � δ r1 �α sup q∈∂B(x1,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='8r1): dist(q,∂Ω)≥r1 ωq 1(F), for some α > 0 depending on s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' So we derive that dist(p, ∂Ω) ≈ r1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Since p is a corkscrew point for Ω relative to B(y, r1), by the CDC and a Harnack chain argument, ωp(E1) ≥ δ0, for some δ0 > 0 depending just on s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Hence, ωp(E \\ E1) ≤ 1 − δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Therefore, ωp 1(F) ≤ ωp(F) + ωp 1(F) (1 − δ0), or equivalently, ωp 1(F) ≤ δ−1 0 ωp(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Since both x1 and p are corkscrew points for Ω relative to B(y, 2r1), we deduce that ωx1 1 (F) ≲ ωx1(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' On the other hand, by the maximum principle, we have trivially that ωx1 1 (F) ≥ ωx1(F), which concludes the proof of the claimed estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Next we will perform another modification of the domain Ω1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' For 0 < ρ ≤ r1/4, we denote E2 = E1 ∪ � ¯B(y, ρ/2) ∩ Rn \\ B1 � and we let Ω2 = Rn+1 \\ E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' By arguments analogous to the ones for the case n = 1, we infer that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='5) ωx1 1 (B(y, ρ)) ≳ ωx1 2 (B(y, ρ/2)) ≥ ωx1 1 (B(y, ρ/2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Finally, we let E3 = (2 ¯B1 \\B1) ∩ Rn and Ω3 = Rn+1 \\E3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' By the maximum principle and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='1, we have ωx1 2 (B(y, ρ/2)) ≥ ωx1 3 (B(y, ρ/2)) ≳ �ρ r �n− 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Together with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='4) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='5), and the change of pole formula, as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='6), this yields (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' DIMENSION OF HARMONIC MEASURE ON SOME FLAT SETS OF FRACTIONAL DIMENSION 15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' The case s ∈ (n − 1 2 − ε, n − 1 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' In this case, instead of obtaining an explicit estimate for ω(B(y,ρ)) ω(B(x,r)), we will argue by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' We assume that ε ≤ 1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' In this situation it is easy to check that the estimates obtained in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='2 involving the domains Ω1, Ω2, Ω3 are uniform in s, assuming the AD-regularity constant to be also uniform on s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' In the rest of this subsection we assume also that all the implicit constants involved in the relations ≲, ≳, ≈ are uniform on s, for s ∈ [1/2 − ε, 1/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' For a given M ≫ 1, suppose that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='6) M−1�ρ r �s ≤ ωp0(B(y, η)) ωp0(B(x, r)) ≤ M �ρ r �s for ρ ≤ η ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' For x1 as above, we write ωx1 2 (B(y, ρ/2)) = ˆ E3 ωz 2(B(y, ρ/2)) dωx1 3 (z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Consider the annuli Ak = A(y, 2kρ, 2k+1ρ), for k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Let N = [log2 r1 ρ ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' By the s-AD regularity of E and pigeon-holing, for every k ∈ [1, N] there exists a ball Bk centered in E3 such that 2Bk ∩ Rn ⊂ Ak ∩ E3 \\ E with rad(Bk) ≈ 2kρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' So we have ωx1 2 (B(y, ρ/2)) ≥ N � k=1 ˆ E3∩Bk ωz 2(B(y, ρ/2)) dωx1 3 (z) ≳ N � k=1 ωx1 3 (Bk) inf z∈Bk ωz 2(B(y, ρ/2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='1, we have ωx1 3 (Bk) ≳ �2kρ r �n− 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' By the maximum principle and the change of pole formula, for any z ∈ Bk, ωz 2(B(y, ρ/2)) ≥ ωz 1(B(y, ρ/2)) ≥ ωz(B(y, ρ/2)) ≈ ωp0(B(y, ρ/2)) ωp0(B(y, 2kρ)) ≥ M−2 ρs (2kρ)s = M−2 2−ks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' In the case s = n − 1 2, we obtain ωx1 2 (B(y, ρ/2)) ≳ M−2 N � k=1 �2kρ r �n− 1 2 2−k(n− 1 2) = M−2N �ρ r �n− 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' By the change of pole formula and the relationship between ω1, ω2, ω3, this yields (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='7) ωp0(B(y, ρ)) ωp0(B(x, r)) ≈ ωx1(B(y, ρ)) ≈ ωx1 1 (B(y, ρ)) ≳ ωx1 2 (B(y, ρ/2)) ≳ M−2N �ρ r �n− 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' For N big enough, this contradicts the assumption (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Hence (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='4) holds, with ρ in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='4) replaced by some η such that ρ ≤ η ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' For s ∈ (n − 1 2 − ε, n − 1 2), we have ωx1 2 (B(y, ρ/2)) ≳ M−2 N � k=1 �2kρ r �n− 1 2 2−ks = M−2�ρ r �n− 1 2 N � k=1 2k(n− 1 2−s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Assuming N big enough, or equivalently ρ small enough (depending on |n − 1 2 − s|), we write N � k=1 2k(n− 1 2−s) ≥ N−1 � k=0 2k(n− 1 2−s) = 2N(n− 1 2 −s) − 1 2n− 1 2−s − 1 ≈ 2N(n− 1 2 −s) 2n− 1 2−s − 1 ≈ rn− 1 2−s (n − 1 2 − s) ρn− 1 2 −s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 16 XAVIER TOLSA Hence, ωx1 2 (B(y, ρ/2)) ≳ M−2 ρs (n − 1 2 − s) rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' As in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='7), this implies ωp0(B(y, ρ)) ωp0(B(x, r)) ≳ M−2 ρs (n − 1 2 − s) rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' For N big enough, this contradicts (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='6) and so (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='4) holds, with ρ in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='4) replaced by some η such that ρ ≤ η ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Extension to subsets of C1 manifolds In this section we extend the main results of the paper to s-AD regular subsets of n-dimensional C1 manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' We prove the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' For n ≥ 1 and s ∈ [n − 1 2, n), let E ⊂ Rn+1 be an (s, C0)-AD regular compact set contained in an n-dimensional C1 manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Let Ω = Rn+1 \\ E and denote by ω the harmonic measure for Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Then dim ω < s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Given E ⊂ Rn+1, x ∈ Rn+1 and r > 0, we denote β∞,E(x, r) = inf L sup y∈B(x,r) dist(y, L) r , where the infimum is taken over all the hyperplanes L ⊂ Rn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' To prove Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='1, we will use the following variant of the Main Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' For n ≥ 1, let s ∈ [n − 1 2, n) and C0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' For all M ≥ 1 there are constants γ > 0, c > 0 both depending on n, s, C0, M such that the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Let E ⊂ Rn+1 be an (s, C0)-AD regular compact set, and x ∈ E, 0 < r ≤ diam(E) such that β∞,E(x, γ−1r) ≤ γ2, and p ∈ Rn+1 \\(E ∪ B(x, 2r)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Then there exists a ball B(y, ρ) with y ∈ B(x, r), c r ≤ ρ ≤ r, such that either (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='1) ωp(B(y, ρ)) ρs ≥ M ωp(B(x, r)) rs or ωp(B(y, ρ)) ρs ≤ M−1 ωp(B(x, r)) rs .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Suppose the lemma fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' That is, there exist n > 1, s ∈ [n − 1 2, n), C0 > 0, and M > 1 such that for all γ = 1/k and c = 1/k there exists an (s, C0)-AD regular set Ek ⊂ Rn+1 and xk ∈ Ek, 0 < rk ≤ diam(Ek), such that β∞,Ek(x, kr) ≤ k−2, and pk ∈ Rn+1 \\ (Ek ∪ B(x, 2rk)), and moreover for any ball B(y, ρ) with y ∈ B(xk, rk), 1 k rk ≤ ρ ≤ rk, it holds (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='2) M−1 ωpk k (B(x, r)) rs < ωpk k (B(y, ρ)) ρs < M ωpk k (B(x, r)) rs , where ωk is the harmonic measure for Rn+1 \\ Ek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' By dilating and translating suitably Ek, we can assume that xk = 0 and rk = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Also, by the uniformity and the s-AD-regularity of the domain Rn+1 \\ Ek, we can also assume that pk ∈ A(0, 2, 3) and dist(pk, Ek) ≥ c′ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' By considering a subsequence, we can assume that the sets Ek converge in the Attouch-Wets topology (this is a local variant of the Hausdorff metric topology) to some (s, C0)-AD regular set �E ⊂ Rn+1, and also that pk converges to some point �p ∈ A(0, 2, 3) and dist(�p, �E) ≥ c′ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' It easily follows then that ωpk k converges weakly * to �ω�p, the harmonic measure for �Ω = Rn+1 \\ �E with pole in �p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' DIMENSION OF HARMONIC MEASURE ON SOME FLAT SETS OF FRACTIONAL DIMENSION 17 Observe that, for 0 < R < k, we have β∞,Ek(0, R) ≤ k R β∞,Ek(0, k) ≤ 1 kR → 0 as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Hence we infer thatβ∞, � E(0, R) = 0 for every R > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' That is to say, �E is contained in some line, which passes through the origin because 0 ∈ �E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' On the other hand, from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='2) and the weak * convergence of ωpk k to �ω�p we deduce that, for x = 0, r = 1, M−1 �ω�p(B(x, r)) rs ≤ �ω�p(B(y, ρ)) ρs ≤ M �ω�p(B(x, r)) rs for all y ∈ B(0, r), 0 < ρ ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' This contradicts Main Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' □ Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Just notice that, if E is compact, s-AD regular with s ∈ [n− 1 2, n), and it is contained in an n-dimensional C1 manifold, then the assumptions of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='2 hold for any ball B(x, r) centered in E with small enough radius r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' It is easy to check that then the assumptions of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='8 hold (by adjusting suitably the constant c1 in the lemma if necessary), which ensures that dim ω < s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' □ Finally remark that, by similar compactness arguments, one can also extend Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='2 to subsets of n-dimensional C1 manifolds: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Let E ⊂ Rn+1 be a compact set contained in an n-dimensional C1 manifold which is (s, C0)-AD regular for some s > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' That is, C−1 0 rs ≤ Hs(E ∩ B(x, r)) ≤ C0 rs for all x ∈ E and 0 < r ≤ diam(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Then there exists ε > 0 small enough depending on C0 such that if s ∈ (n − 1 2 − ε, n), we have dim ω < s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' To prove this, it suffices to consider the case s ∈ (n − 1 2 − ε, n − 1) and then one can use arguments similar to ones in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' The details are left for the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' References [ABR] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Axler, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Bourdon, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Ramey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Harmonic function theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Second edition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Graduate Texts in Mathematics, 137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Springer-Verlag, New York, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' xii+259 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 7 [AHM3TV] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Azzam, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Hofmann, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Martell, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Mayboroda, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Mourgoglou, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Tolsa, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Volberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Recti- fiability of harmonic measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' (GAFA), 26 (2016), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 3, 703–728.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 1 [AHMMT] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Azzam, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Hofmann, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Martell, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Mourgoglou, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Tolsa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Harmonic measure and quantitative connectivity: geometric characterization of the Lp-solvability of the Dirichlet problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Invent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 222 (2020), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 3, 881–993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 1 [An] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Ancona, On strong barriers and an inequality of Hardy for domains in Rn, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' London Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' (2) 34 (1986), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 2, 274–290.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 5 [Az1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Azzam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Semi-uniform domains and the A∞ property for harmonic measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' IMRN 2021, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 9, 6717–6771.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 1 [Az2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Azzam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Dimension drop for harmonic measure on Ahlfors regular boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Potential Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 53:1025–104 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 2, 3, 7 [Ba] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Batakis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Harmonic measure of some Cantor type sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Fenn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 21(2) (1996), 255–270.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 2, 3, 7 [Bo] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Bourgain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' On the Hausdorff dimension of harmonic measure in higher dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Invent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 87 (1987), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 3, 477–483.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 2, 3, 7 18 XAVIER TOLSA [BZ] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Batakis and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Zdunik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Hausdorff and harmonic measures on non-homogeneous Cantor sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Fenn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 40 (2015), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 1, 279–303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 2 [Ca] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Carleson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' On the support of harmonic measure for sets of Cantor type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Fenn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' A I Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=', 10 (1985), 113–123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 2 [CTV] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Cuf´ı, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Verdera, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Tolsa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' About the Jones-Wolff Theorem on the Hausdorff dimension of harmonic measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Preprint arXiv:1809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='08026 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 5 [DFM] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' David, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Feneuil, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Mayboroda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Elliptic theory for sets with higher co-dimensional boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 274 (2021), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 1346, vi+123 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 6 [DJJ] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' David, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Jeznach, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Julia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Private communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 2, 3 [GM] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Garnett and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Marshall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Harmonic Measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' New Mathematical Monographs 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Cambridge University Press, Cambridge, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 2 [GMT] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Garnett, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Mourgoglou, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Tolsa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Uniform rectifiability in terms of Carleson measure estimates and ε-approximability of bounded harmonic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Duke Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 167 (2018), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 8, 1473–1524.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 1 [HMM] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Hofmann, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Martell, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Mayboroda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Uniform rectifiability, Carleson measure estimates, and approximation of harmonic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Duke Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 165 (2016), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 12, 2331–2389.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 1 [JK] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Jerison and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Kenig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Boundary behavior of harmonic functions in nontangentially accessible domains, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' in Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 46 (1982), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 1, 80–147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 6 [Jo] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Jones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' On scaling properties of harmonic measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Perspectives in analysis, 73?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='81, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Stud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=', 27, Springer, Berlin, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 2 [JW] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Jones and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Wolff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Hausdorff dimension of harmonic measures in the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Acta Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=', 161(1-2) (1988), 131–144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 1 [Mak] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Makarov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' On the distortion of boundary sets under conformal mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' London Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' (3), 51(2) (1985), 369–384.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 1 [Mat] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Mattila.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Geometry of sets and measures in Euclidean spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Cambridge Studies in Advanced Math- ematics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 44, Cambridge University Press, Cambridge, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 5 [MT] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Mourgoglou and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Tolsa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Harmonic measure and Riesz transform in uniform and general domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Reine Angew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 758 (2020), 183–221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 6 [MV] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Makarov and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Volberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' On the harmonic measure of discontinuous fractals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Preprint LOMI E-6-86, Leningrad, 1986.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 2 [RR] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Riesz and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Riesz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' ¨Uber die randwerte einer analtischen funktion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Compte Rendues du Quatri`eme Congr`es des Math´ematiciens Scandinaves, Stockholm 1916, Almqvists and Wilksels, Upsala, 1920.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 1 [To] Tolsa, Xavier The mutual singularity of harmonic measure and Hausdorff measure of codimension smaller than one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' IMRN 2021, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 18, 13783—13811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 4 [UZ] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Urba´nski and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Zdunik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Hausdorff dimension of harmonic measure for self-conformal sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=', 171(1) (2002), 1–58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 2 [Vo1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Volberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' On the harmonic measure of self-similar sets on the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' In Harmonic Analysis and Discrete Potential theory, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 267–280.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Springer, 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 2 [Vo2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Volberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' On the dimension of harmonic measure of Cantor repellers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Michigan Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=', 40(2) (1993), 239–258.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 2 [Wo1] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Wolff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Plane harmonic measures live on sets of σ-finite length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Ark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 31 (1993), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 1, 137–172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 2 [Wo2] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Wolff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Counterexamples with harmonic gradients in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' In: Essays on Fourier Analysis in Honor of Elias M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Stein, Princeton Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 42, Princeton Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Press, 321–384 (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' 2 ICREA, Barcelona, Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' de Matem`atiques, Universitat Aut`onoma de Barcelona, and Centre de Recerca Matem`atica, Barcelona, Catalonia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content=' Email address: xtolsa@mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='uab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} +page_content='cat' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfuwii/content/2301.04084v1.pdf'} diff --git a/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf b/YtA0T4oBgHgl3EQfFv9O/content/2301.02036v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..d67f08f8565f55a02bb72e4b7dad747f62a4acb7 --- 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Hydro-Bio-Mechanical Systems Research Center, Budapest, Hungary +Abstract +The analytic implicit function theorem is extended. The function f of the theorem is integrated +with respect to the dependent variable of the implicit function. A geometrical interpretation is +given for the sub-geometry of the integral function F by using the Morse lemma. The result is +used in the analysis of the hierarchical technique related to the minimization inon-linear +parameter identification. +Keywords: Implicit, Morse, non-linear minimization, hierarchical parameter identification + +Introduction +The hierarchical solution of non-linear inverse problems +The solution of the non-linear inverse problems can be determined by non-linear minimization +which is hindered by the fact that the LS merit function has an unimaginable complexity of the M- +dimensional topography due to the noise, where M is the parameter number. +The minima can be global or local. Two heuristics are used: (i) to find local extrema starting +from varying initial values of the independent variables, and then pick the most extreme [1]; or (ii) +to perturb a local extremum by taking a step away from it, and then see if the iteration returns to a +better point, or to the same one [1]. Another approach is to fit locally a nicer hyper-surface and +descending along it [2]. Some additional geometrical difficulties may arise in case of a quasi- +degenerated global minimum, which can theoretically be treated by regularization ([3]) needing +modified algorithms. +No hierarchical solution method has been reported for the non-linear case despite of the fact that +the number of the parameters in the non-linear algorithm can be decreased resulting in less critical +points due to the noise and less numerical work besides other advantages (the one-dimensional +sections of the merit functions so determined can be used for parameter error estimation and implicit +regularization). +The aim of the paper +The concept of the hierarchical method [4] in well-known for linear system of equation eg., int +he context of linear inverse problems and is started to be introduced into the solution of non-linear +inverse problems here. The use of the hierarchical solution is visualized as follows (Fig.1). + +We may assume that F : RM = Rn+m → {0U R+} is an analytic Least Squares merit function +which has a single minimum pmin within the parameter domain. Let us assume that the parameter +space is split into direct sum p=(x,y) and the solution of the inverse problem is split into two, +smaller dimensional parts (in Fig. 1: x1, x2), accordingly. + + + + +2 + + +Figure 1. The implicit function x2(x1) is defined by f(x , y)= f (x1, x2) =F’y (x1, x2)= 0. The +implicit function x2(x1) is the inverse image of the set of the conditional minimum points at +fixed x=x1 values. The minimum of F (x1, x2) with respect to the two parameters is the same +as the minimum of the ‘minimal section’ F [x1, x2(x1)] if the sub-minima along each vertical +(y=x2 directional) hyper-plane is unique. + +One kind of minimization happens along the y directional parallel plane sections of the +parameter space: the y2 solution part is searched in every fixed value of x by sub-minimization (using +minimization method 1). The set of the so determined sub-minima are found along the graph of an +implicit function y(x) related to the condition F’y = f = 0. The so called minimal section F[x,y(x)] +is minimized (with method 2). + +The paper treats the condition of the equivalence of the original and the hierarchical inverse +problem solutions, by the analysis of the geometry of the merit function along the y planes. It is +found that for the equivalence, F is needed to be convex in the y direction. The condition is met in +case of strictly convex merit functions, or if the model depends linearly on y when the merit +function is partly strictly convex (ie., in the y direction). + +Moreover, some consequences and inferences are mentioned. In the strictly convex case, the +set of the graphs of the implicit functions related to the various subspaces is isomorphic with the +algebraic lattice of the subspaces of the parameter space RM. The geometry and the analytic nature +of the one-dimensional minimal sections can be used in reliability testing and in regularization. +The extension of the implicit function theorem +Assumptions, concepts, theorems +The basic concepts used are as follows [5 to 8]. For the smooth function F : RM → R the +points where the first derivative vanishes are called critical points and their images are called +critical values. If at a critical point the matrix of second derivative (the Hessian matrix) is non- +singular, then it is called a non-degenerate critical point; if the Hessian is singular then it is a +degenerate critical point. Some basic theorems are presented which are coupled in the extended +implicit function theorem. + +Morse lemma + ++2 +hyper-planes +leval lines +nyper-planes +Parameter 1 + + +3 + +Let p be a non-degenerate critical point of F. Then there exists a chart in a neighborhood U of p +such that for all xi(p) = 0 where xi constitute a local map around p and + + + + + (1) +throughout U. Here + is equal to the index of F at p (the number of the negative eigenvalues of +the Hessian matrix). As a corollary of the Morse lemma, one sees that non-degenerate critical +points are isolated. +The last Morse inequality on Sn, relating the number of critical points with index k and the Euler +characteristic of Sn – being an equality – is as follows: + + + + (2) +From this, the equation for Dn (n-dimensional ball) can be derived. A 1 must be subtracted from +both sides if a minimum (critical point with index 0) and (-1)n if a maximum (critical point with +index n) is left out. If the function "curls up" at the edge of the ball (the gradient is outward at the +edge): + + + + + (3) +In the strict convex case, the edge Dn (n-dimensional ball, which is a half Sn) has a single critical +point with index of 0. +The implicit function theorem +Let f : Rn+m → Rm be an analytic function. Rn+m is the direct sum of Rn and Rm, a point of this is +(x, y) = (x1, ..., xn, y1, ..., ym). Starting from the given function f, the goal is to construct a function +g: Rn → Rm whose graph (x, g(x)) is precisely the set of all (x, y) such that f(x, y) =0. +Fix a point (a,b) = (a1, ..., an, b1, ..., bm) with f(a, b) = 0, where 0 is the element of Rm. If the matrix +[(∂fi/∂yj)(a, b)] is invertible, then there exists an open set U containing a, an open set V containing +b, and a unique function g: U → V such that + +  + +0 +y +x, +f +x +y +x +x +x +g +x += + += + +) +( +, +) +, +( +) +( +, +U +U + + + + + (4) +Whenever we have the hypothesis that f is analytic inside U × V, then the same holds true for the +implicit function g inside U. +Extension +Statement 1 (The extension of the implicit function theorem with geometrical interpretation) +Let us assume that the parameter space is split into the direct sum p=(x,y) and the conditions of the +analytic implicit function theorem are met in (a,b) = (a1, ..., an, b1, ..., bm), being f(a, b) = 0, where +0 is the element of Rm and being f’’y regular. As a result, there exists an open set U containing a, +and an open set V containing b, and a unique, analytic function g: U → V , y=g(x), g: Rn → Rm , +such that f[x,g(x)] = 0. +(i) Let us assume that the function f : Rn+m → Rm in the analytic implicit function theorem has an +analytic partial integral function with respect to y, denoted by F : Rn+m → R such that f= F’y . The +F[x,g(x)] is called minimal section of F with respect to x. The points (a,b) or [x,g(x)] are non- + + + + +4 + +degenerate, “partial” critical points of F in the y direction, with local sub-geometry determined by +the index of F’’yy. (The points can be the degenerate critical points of F). +Explanation: +Let us consider the point (a,b) and it environment U, V of the analytic implicit function theorem. +Let us introduce here the partial function Fx(y) : Rm → R1, describing the variation of y at fixed x. +Its first derivative is equal to F’y and its second derivative is equal to F’’yy. The slice function +Fa(y) : Rm → R is a Morse function, being its first derivative zero, its Hessian non-degenerate in b +having an isolated, non-degenerate critical point there. The same index can be expected along the +implicit function due to the nice features of the function F. +Statement 2 (The application for the hierarchical minimization) +If F has a non-degenerate minimum in the point (a,b), then for any direct sum decomposition +p=(x,y), every possible implicit function locally exists with a local sub-minimum in terms of y. +However, the implicit function for a special direct sum decomposition p=(x,y) exists globally if +F’’yy is positive definite globally on the whole parameter domain. +Explanation: +Since F’’yy is globally positive definite, the implicit function can locally be extended repetitively +(for any x). The index of Fx(y) is 0, having a non-degenerate local sub-minimum where F’y = +f[x,g(x)] = 0. Moreover, F is strictly convex in the y direction since F’’yy is positive definite in the +y direction globally. Therefore, Fx(y) has only one critical point, the minimum (the statement also +follows from the last Morse inequality). +Statement 3 (The case of hierarchical minimization of convex functions on RM) +The conditions of the implicit function theorem are met in the non-degenerate minimum of F for +any direct sum decomposition p=(x,y) and, therefore, the related implicit functions locally exist. +(i) In case of a strictly convex function F, the partial derivative F’’yy is globally positive definite for +any direct sum decomposition p=(x,y). Therefore, every possible minimal section globally and +uniquely exists. The set of these is isomorphic with the algebraic lattice of the sub-spaces of RM +with partial ordering of the containing. If x’ is contained by x then the smaller dimensional +minimal section of x’ is contained by a larger dimensional minimal section of x. Therefore, the +hierarchical technique can be applied in a repetitive way. +(ii) Let us consider the Euclidean space generated by the graph of F. For every z level line there are +two coordinate hyper-planes being normal to the coordinate direction of parameter xi and tangent +to the given z level line (Fig. 1). Such hyper-plane pair series bracket decreasing z level lines. The +tangent point pairs are the points of the 1-dimensional minimal section of xi, the interval defined +by the point pairs [xi z-, xi z+] is the orthogonal projection of the sub-level set z of F onto the axis of +the parameter xi. The intervals decrease with z, in the minimum the two points coincide (Fig. 1). +Corrolary (properties of the 1-dimensional minimal sections of the parameters) +The 1-dimensional minimal section F[[xi,y(xi)] is the smallest descent line with respect parameter +xi that proceed in the positive/negative xi directions toward the global minimizer pmin. The 1- +dimensional minimal section F[[xi,y(xi )] is a ‘deepest’ sensitivity section (ie., a section taken +along a simple curve passing through the global minimum, parametrised by xi). + + + + +5 + +Further comments on the hierarchical minimization +Some comments on the equivalence condition +We have found that if the analytic F is partially, globally strictly convex in the y direction, +then the implicit function of x globally and uniquely exists with a sub-minimum of F in the y +direction. The two kinds of minimisation (ie., the M-dimensional unconditional and the +hierarchical with a J dimensional sub-minimisation with respect y and the M-J dimensional +minimisation of the minimal section of x with respect x) have the same solution. +As a practical significance, it follows that only those parameters are important in the point of +view of non-linear minimisation, that influence the model non-linearly. The linear model-part can +be ‘eliminated’. In the case when only one single parameter influences the model non-linearly, the +non-linear minimisation can be solved even by one-dimensional bracketing since all parameters can +be eliminated by linear sub-minimisation automatically. +It can be noted that a global implicit function may exist for such partly convex function F that +has more than one minimum. In case of two distinct minima without common x coordinate value, +a global implicit function and the related minimal section with respect to x will cross both minima. +Algorithmic steps in a coordinate grid bracketing of convex functions +An implicit function point is a conditional minimum point of F and can be determined as +follows. Let us consider the orthogonal projection of the graph of F(x, y) onto the F - x coordinate +plan in the total Euclidean space generated by the graph. The critical values of the map - where the +fold of the projection is found - are related to the condition F y’ (x, y) = f(x, y) = 0 which defines +the implicit function. +Using this, an algorithm can be elaborated for the determination of the implicit function and +the hierarchical minimisation. The unique sub-minimum at the slice function Fx(y) : Rm → R for +fixed x can be determined by projection, minimum search with respect to y and from the +minimum, the graph point of the implicit function can be determined by inverse image operation. +A one-dimensional minimum can be bracketed only when there is a triplet of points a < b < c +(or c < b < a), such that f(b) is less than both f(a) and f(c). In this case we know that the function +(if it is non-singular) has a minimum in the interval (a; c). It is possible to bracket a one- +dimensional minimum but there is no analogous procedure in the general, multi-dimensional case +[1] unless the function is strictly convex, since hyper-planes generated by coordinate mesh points +can enclose decreasing convex level lines. +The global of minimum of strictly convex merit functions can be bracketed by triplets in every +coordinate direction, by using a coordinate mesh. Using the same coordinate mesh, by applying the +foregoing algorithmic step in each mesh point beyond the global minimum search, the +determination of the one-dimensional minimal sections is also possible. +Regularization +Some additional geometrical difficulties beyond the noise may arise when the merit function is +distorted being the minimum quasi-degenerated (eg., the testing time can be “too short” in the +measurements in case of the fitting of a model which is the injective solution of a PDE, depending +on the time variable). The problem can be treated by regularization ([1, 3]). Explicit regularization +is regularization whenever one explicitly adds a term to the optimization problem. These terms +could be priors, penalties, or constraints. Explicit regularization is commonly employed with ill- + + + + +6 + +posed optimization problems. +The regularization term, or penalty, imposes a cost on the optimization function to make the +optimal solution unique. Implicit regularization is all other forms of regularization. This includes, +for example, early stopping, using a robust loss function, and discarding outliers. +The one-dimensional minimal section related to a given parameter has some special geometry +features. It is a smallest descent line, it can be used to determine parameter sensitivity / error. +Being analytic, it can be used to solve quasi-degenerate minimum problems with implicit +regularization as follows. +In this work a new, implicit method is suggested, based on the analytic feature of the one- +dimensional implicit functions which may have injective coordinate functions (eg., the pmin can be +determined by an injective coordinate function of the one-dimensional implicit function related to +a given parameter using a known coordinate of this parameter of the parameter vector pmin). The +suggested method does not fail even in such a case when the convexity is slightly destroyed by +model perturbation, and there are two degenerated minimuma. +Acknowledgement +The discussion with András Stipsicz, András Szűcs, Endre Szabó, József Bodnár is greatly +acknowledged. +References +[1] Press, W.H.; Flannery, B.P.; Teukolsky, S.A.; Wetterling, W.T. (1986): Numerical +Recipes. Cambridge Univ. Press. +[2] Meier J, Rudolph, S,. Schanz T: Effective Algorithm for parameter back calculation - +Geotechnical Applications published in: Bautechnik Volume 86 Issue S1, 86 – 97 +[3] Hegedűs, Cs (2008): Numerical Analysis, ELTE, http://www.inf.elte.hu/Budapest, +Hungary +[4] Zhang, Fuzhen (2005). Zhang, Fuzhen (ed.) The Schur Complement and Its +Applications. +Numerical +Methods +and +Algorithms. +Vol. +4. +Springer. +doi:10.1007/b105056. ISBN 0-387-24271-6. +[5] Császár, Á. (1983) Real Analysis. Tankönyvkiadó, in Hungarian. +[6] Milnor, J.W. (1963) Morse theory. Princeton University Press. 153 p. +[7] Milnor, J.W. (1963) Topology from the Differentiable Viewpoint Princeton University +Press, Dec 14, 1997. 157 p. +[8] Hirsch, M. W. Differential Topology. Graduate Texts in Mathematics 1973 GTM, +volume 33 Hardcover ISBN: 978-0-387-90148-0 Copyright Information: Springer-Verlag +New York. + + diff --git a/ddE1T4oBgHgl3EQfyAUB/content/tmp_files/load_file.txt b/ddE1T4oBgHgl3EQfyAUB/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0f3184cd7a6520f4fbb2c8c458a257dc9ceb9e89 --- /dev/null +++ b/ddE1T4oBgHgl3EQfyAUB/content/tmp_files/load_file.txt @@ -0,0 +1,174 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf,len=173 +page_content='1 A comment on the combination of the implicit function theorem and the Morse lemma Emőke Imre Óbuda University, Hydro-Bio-Mechanical Systems Research Center, Budapest, Hungary Abstract The analytic implicit function theorem is extended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' The function f of the theorem is integrated with respect to the dependent variable of the implicit function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' A geometrical interpretation is given for the sub-geometry of the integral function F by using the Morse lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' The result is used in the analysis of the hierarchical technique related to the minimization inon-linear parameter identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' Keywords: Implicit, Morse, non-linear minimization, hierarchical parameter identification Introduction The hierarchical solution of non-linear inverse problems The solution of the non-linear inverse problems can be determined by non-linear minimization which is hindered by the fact that the LS merit function has an unimaginable complexity of the M- dimensional topography due to the noise, where M is the parameter number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' The minima can be global or local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' Two heuristics are used: (i) to find local extrema starting from varying initial values of the independent variables, and then pick the most extreme [1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' or (ii) to perturb a local extremum by taking a step away from it, and then see if the iteration returns to a better point, or to the same one [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' Another approach is to fit locally a nicer hyper-surface and descending along it [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' Some additional geometrical difficulties may arise in case of a quasi- degenerated global minimum, which can theoretically be treated by regularization ([3]) needing modified algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' No hierarchical solution method has been reported for the non-linear case despite of the fact that the number of the parameters in the non-linear algorithm can be decreased resulting in less critical points due to the noise and less numerical work besides other advantages (the one-dimensional sections of the merit functions so determined can be used for parameter error estimation and implicit regularization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' The aim of the paper The concept of the hierarchical method [4] in well-known for linear system of equation eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=', int he context of linear inverse problems and is started to be introduced into the solution of non-linear inverse problems here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' The use of the hierarchical solution is visualized as follows (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' We may assume that F : RM = Rn+m → {0U R+} is an analytic Least Squares merit function which has a single minimum pmin within the parameter domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' Let us assume that the parameter space is split into direct sum p=(x,y) and the solution of the inverse problem is split into two, smaller dimensional parts (in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' 1: x1, x2), accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' 2 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' The implicit function x2(x1) is defined by f(x , y)= f (x1, x2) =F’y (x1, x2)= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' The implicit function x2(x1) is the inverse image of the set of the conditional minimum points at fixed x=x1 values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' The minimum of F (x1, x2) with respect to the two parameters is the same as the minimum of the ‘minimal section’ F [x1, x2(x1)] if the sub-minima along each vertical (y=x2 directional) hyper-plane is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' One kind of minimization happens along the y directional parallel plane sections of the parameter space: the y2 solution part is searched in every fixed value of x by sub-minimization (using minimization method 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' The set of the so determined sub-minima are found along the graph of an implicit function y(x) related to the condition F’y = f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' The so called minimal section F[x,y(x)] is minimized (with method 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' The paper treats the condition of the equivalence of the original and the hierarchical inverse problem solutions, by the analysis of the geometry of the merit function along the y planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' It is found that for the equivalence, F is needed to be convex in the y direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' The condition is met in case of strictly convex merit functions, or if the model depends linearly on y when the merit function is partly strictly convex (ie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=', in the y direction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' Moreover, some consequences and inferences are mentioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' In the strictly convex case, the set of the graphs of the implicit functions related to the various subspaces is isomorphic with the algebraic lattice of the subspaces of the parameter space RM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' The geometry and the analytic nature of the one-dimensional minimal sections can be used in reliability testing and in regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' The extension of the implicit function theorem Assumptions, concepts, theorems The basic concepts used are as follows [5 to 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' For the smooth function F : RM → R the points where the first derivative vanishes are called critical points and their images are called critical values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' If at a critical point the matrix of second derivative (the Hessian matrix) is non- singular, then it is called a non-degenerate critical point;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' if the Hessian is singular then it is a degenerate critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' Some basic theorems are presented which are coupled in the extended implicit function theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' Morse lemma +2 hyper planes leval lines nyper planes Parameter 1 3 Let p be a non-degenerate critical point of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' Then there exists a chart in a neighborhood U of p such that for all xi(p) = 0 where xi constitute a local map around p and (1) throughout U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' Here is equal to the index of F at p (the number of the negative eigenvalues of the Hessian matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' As a corollary of the Morse lemma, one sees that non-degenerate critical points are isolated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' The last Morse inequality on Sn, relating the number of critical points with index k and the Euler characteristic of Sn – being an equality – is as follows: (2) From this, the equation for Dn (n-dimensional ball) can be derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' A 1 must be subtracted from both sides if a minimum (critical point with index 0) and (-1)n if a maximum (critical point with index n) is left out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' If the function "curls up" at the edge of the ball (the gradient is outward at the edge): (3) In the strict convex case, the edge Dn (n-dimensional ball, which is a half Sn) has a single critical point with index of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' The implicit function theorem Let f : Rn+m → Rm be an analytic function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' Rn+m is the direct sum of Rn and Rm, a point of this is (x, y) = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=', xn, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=', ym).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' Starting from the given function f, the goal is to construct a function g: Rn → Rm whose graph (x, g(x)) is precisely the set of all (x, y) such that f(x, y) =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' Fix a point (a,b) = (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=', an, b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=', bm) with f(a, b) = 0, where 0 is the element of Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' If the matrix [(∂fi/∂yj)(a, b)] is invertible, then there exists an open set U containing a, an open set V containing b, and a unique function g: U → V such that \uf07b \uf07d \uf07b \uf07d 0 y x, f x y x x x g x = \uf0ce = \uf0ce ) ( , ) , ( ) ( , U U (4) Whenever we have the hypothesis that f is analytic inside U × V, then the same holds true for the implicit function g inside U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' Extension Statement 1 (The extension of the implicit function theorem with geometrical interpretation) Let us assume that the parameter space is split into the direct sum p=(x,y) and the conditions of the analytic implicit function theorem are met in (a,b) = (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=', an, b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=', bm), being f(a, b) = 0, where 0 is the element of Rm and being f’’y regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' As a result, there exists an open set U containing a, and an open set V containing b, and a unique, analytic function g: U → V , y=g(x), g: Rn → Rm , such that f[x,g(x)] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' (i) Let us assume that the function f : Rn+m → Rm in the analytic implicit function theorem has an analytic partial integral function with respect to y, denoted by F : Rn+m → R such that f= F’y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' The F[x,g(x)] is called minimal section of F with respect to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' The points (a,b) or [x,g(x)] are non- 4 degenerate, “partial” critical points of F in the y direction, with local sub-geometry determined by the index of F’’yy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' (The points can be the degenerate critical points of F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' Explanation: Let us consider the point (a,b) and it environment U, V of the analytic implicit function theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' Let us introduce here the partial function Fx(y) : Rm → R1, describing the variation of y at fixed x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' Its first derivative is equal to F’y and its second derivative is equal to F’’yy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' The slice function Fa(y) : Rm → R is a Morse function, being its first derivative zero, its Hessian non-degenerate in b having an isolated, non-degenerate critical point there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' The same index can be expected along the implicit function due to the nice features of the function F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' Statement 2 (The application for the hierarchical minimization) If F has a non-degenerate minimum in the point (a,b), then for any direct sum decomposition p=(x,y), every possible implicit function locally exists with a local sub-minimum in terms of y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' However, the implicit function for a special direct sum decomposition p=(x,y) exists globally if F’’yy is positive definite globally on the whole parameter domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' Explanation: Since F’’yy is globally positive definite, the implicit function can locally be extended repetitively (for any x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' The index of Fx(y) is 0, having a non-degenerate local sub-minimum where F’y = f[x,g(x)] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' Moreover, F is strictly convex in the y direction since F’’yy is positive definite in the y direction globally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' Therefore, Fx(y) has only one critical point, the minimum (the statement also follows from the last Morse inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' Statement 3 (The case of hierarchical minimization of convex functions on RM) The conditions of the implicit function theorem are met in the non-degenerate minimum of F for any direct sum decomposition p=(x,y) and, therefore, the related implicit functions locally exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' (i) In case of a strictly convex function F, the partial derivative F’’yy is globally positive definite for any direct sum decomposition p=(x,y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' Therefore, every possible minimal section globally and uniquely exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' The set of these is isomorphic with the algebraic lattice of the sub-spaces of RM with partial ordering of the containing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' If x’ is contained by x then the smaller dimensional minimal section of x’ is contained by a larger dimensional minimal section of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' Therefore, the hierarchical technique can be applied in a repetitive way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' (ii) Let us consider the Euclidean space generated by the graph of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' For every z level line there are two coordinate hyper-planes being normal to the coordinate direction of parameter xi and tangent to the given z level line (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' Such hyper-plane pair series bracket decreasing z level lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' The tangent point pairs are the points of the 1-dimensional minimal section of xi, the interval defined by the point pairs [xi z-, xi z+] is the orthogonal projection of the sub-level set z of F onto the axis of the parameter xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' The intervals decrease with z, in the minimum the two points coincide (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' Corrolary (properties of the 1-dimensional minimal sections of the parameters) The 1-dimensional minimal section F[[xi,y(xi)] is the smallest descent line with respect parameter xi that proceed in the positive/negative xi directions toward the global minimizer pmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' The 1- dimensional minimal section F[[xi,y(xi )] is a ‘deepest’ sensitivity section (ie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=', a section taken along a simple curve passing through the global minimum, parametrised by xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' 5 Further comments on the hierarchical minimization Some comments on the equivalence condition We have found that if the analytic F is partially, globally strictly convex in the y direction, then the implicit function of x globally and uniquely exists with a sub-minimum of F in the y direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' The two kinds of minimisation (ie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=', the M-dimensional unconditional and the hierarchical with a J dimensional sub-minimisation with respect y and the M-J dimensional minimisation of the minimal section of x with respect x) have the same solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' As a practical significance, it follows that only those parameters are important in the point of view of non-linear minimisation, that influence the model non-linearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' The linear model-part can be ‘eliminated’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' In the case when only one single parameter influences the model non-linearly, the non-linear minimisation can be solved even by one-dimensional bracketing since all parameters can be eliminated by linear sub-minimisation automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' It can be noted that a global implicit function may exist for such partly convex function F that has more than one minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' In case of two distinct minima without common x coordinate value, a global implicit function and the related minimal section with respect to x will cross both minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' Algorithmic steps in a coordinate grid bracketing of convex functions An implicit function point is a conditional minimum point of F and can be determined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' Let us consider the orthogonal projection of the graph of F(x, y) onto the F - x coordinate plan in the total Euclidean space generated by the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' The critical values of the map - where the fold of the projection is found - are related to the condition F y’ (x, y) = f(x, y) = 0 which defines the implicit function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' Using this, an algorithm can be elaborated for the determination of the implicit function and the hierarchical minimisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' The unique sub-minimum at the slice function Fx(y) : Rm → R for fixed x can be determined by projection, minimum search with respect to y and from the minimum, the graph point of the implicit function can be determined by inverse image operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' A one-dimensional minimum can be bracketed only when there is a triplet of points a < b < c (or c < b < a), such that f(b) is less than both f(a) and f(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' In this case we know that the function (if it is non-singular) has a minimum in the interval (a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' It is possible to bracket a one- dimensional minimum but there is no analogous procedure in the general, multi-dimensional case [1] unless the function is strictly convex, since hyper-planes generated by coordinate mesh points can enclose decreasing convex level lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' The global of minimum of strictly convex merit functions can be bracketed by triplets in every coordinate direction, by using a coordinate mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' Using the same coordinate mesh, by applying the foregoing algorithmic step in each mesh point beyond the global minimum search, the determination of the one-dimensional minimal sections is also possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' Regularization Some additional geometrical difficulties beyond the noise may arise when the merit function is distorted being the minimum quasi-degenerated (eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=', the testing time can be “too short” in the measurements in case of the fitting of a model which is the injective solution of a PDE, depending on the time variable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' The problem can be treated by regularization ([1, 3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' Explicit regularization is regularization whenever one explicitly adds a term to the optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' These terms could be priors, penalties, or constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' Explicit regularization is commonly employed with ill- 6 posed optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' The regularization term, or penalty, imposes a cost on the optimization function to make the optimal solution unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' Implicit regularization is all other forms of regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' This includes, for example, early stopping, using a robust loss function, and discarding outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' The one-dimensional minimal section related to a given parameter has some special geometry features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' It is a smallest descent line, it can be used to determine parameter sensitivity / error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' Being analytic, it can be used to solve quasi-degenerate minimum problems with implicit regularization as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' In this work a new, implicit method is suggested, based on the analytic feature of the one- dimensional implicit functions which may have injective coordinate functions (eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=', the pmin can be determined by an injective coordinate function of the one-dimensional implicit function related to a given parameter using a known coordinate of this parameter of the parameter vector pmin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' The suggested method does not fail even in such a case when the convexity is slightly destroyed by model perturbation, and there are two degenerated minimuma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' Acknowledgement The discussion with András Stipsicz, András Szűcs, Endre Szabó, József Bodnár is greatly acknowledged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' References [1] Press, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' Flannery, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' Teukolsky, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' Wetterling, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' (1986): Numerical Recipes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' Cambridge Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' [2] Meier J, Rudolph, S,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' Schanz T: Effective Algorithm for parameter back calculation - Geotechnical Applications published in: Bautechnik Volume 86 Issue S1, 86 – 97 [3] Hegedűs, Cs (2008): Numerical Analysis, ELTE, http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content='inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content='elte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content='hu/Budapest, Hungary [4] Zhang, Fuzhen (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' Zhang, Fuzhen (ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=') The Schur Complement and Its Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' Numerical Methods and Algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content='1007/b105056.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' ISBN 0-387-24271-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' [5] Császár, Á.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' (1983) Real Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' Tankönyvkiadó, in Hungarian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' [6] Milnor, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' (1963) Morse theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' Princeton University Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' 153 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' [7] Milnor, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' (1963) Topology from the Differentiable Viewpoint Princeton University Press, Dec 14, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' 157 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' [8] Hirsch, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' Differential Topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} +page_content=' Graduate Texts in Mathematics 1973 GTM, volume 33 Hardcover ISBN: 978-0-387-90148-0 Copyright Information: Springer-Verlag New York.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE1T4oBgHgl3EQfyAUB/content/2301.03427v1.pdf'} diff --git a/edE0T4oBgHgl3EQfogFH/vector_store/index.faiss b/edE0T4oBgHgl3EQfogFH/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..0ade85fea46eb2a2c0d68981fec1b7ef43cb6227 --- /dev/null +++ b/edE0T4oBgHgl3EQfogFH/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:76d5476a20bfc6b56703dd4b788e9b47d8f5844e99cee1a46dd89c450b2b21f7 +size 4390957 diff --git a/edE1T4oBgHgl3EQfeQSt/content/tmp_files/2301.03205v1.pdf.txt b/edE1T4oBgHgl3EQfeQSt/content/tmp_files/2301.03205v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..cede3fb9d9842cb9de97cf67d3056e18158f0a7d --- /dev/null +++ b/edE1T4oBgHgl3EQfeQSt/content/tmp_files/2301.03205v1.pdf.txt @@ -0,0 +1,799 @@ +Enhanced photovoltaic effect in graphene-silicon Schottky junction under +mechanical manipulation +Dong Pu,1, 2, a) Muhammad Abid Anwar,1, 2, a) Jiachao Zhou,1, 2 Renwei Mao,3 Xin Pan,3 Jian Chai,1, 2 Feng +Tian,1, 2 Hua Wang,1, 2, b) Huan Hu,1, 3, c) and Yang Xu1, 2, 3, d) +1)School of Micro-Nano Electronics, Zhejiang University, Hangzhou, China +2)ZJU Global Scientific and Technological Innovation Center, Hangzhou, China +3)ZJU-UIUC Institute (ZJUI), Zhejiang University, Haining, China +(Dated: 10 January 2023) +Graphene-silicon Schottky junction (GSJ) which has the potential for large-scale manufacturing and integra- +tion can bring new opportunities to Schottky solar cells for photovoltaic (PV) power conversion. However, +the essential power conversion limitation for these devices lies in the small open-circuit voltage (Voc), which +depends on the Schottky barrier height (SBH). In this study, we introduce an electromechanical method based +on the flexoelectric effect to enhance the PV efficiency in GSJ. By atomic force microscope (AFM) tip-based +indentation and in situ current measurement, the current-voltage (I-V) responses under flexoelectric strain +gradient are obtained. The Voc is observed to increase for up to 20%, leading to an evident improvement of the +power conversion efficiency. Our studies suggest that strain gradient may offer unprecedented opportunities +for the development of GSJ based flexo-photovoltaic applications. +Due to its special two-dimensional structural proper- +ties and excellent performance1, graphene has the poten- +tial to be integrated into existing semiconductor tech- +nologies and used in next-generation electronics. +Re- +cent research has shown that the formation of junc- +tions between graphene and three-dimensional or two- +dimensional semiconductors2 can produce the rectifica- +tion effect of a typical Schottky junction. +Its tunable +Schottky barrier makes graphene junctions an excellent +platform for studying the transport properties of inter- +faces and has led to applications in scenarios such as +photodetection3, light-speed communication4, and chem- +ical and biological detection5. Solar energy collection and +conversion have attracted much attention in recent years. +The Schottky junction devices can naturally work as so- +lar cell or photovoltaic cell6, as the built-in electrical field +provides the voltage potential difference that drives the +current7. +Conventional metal/semiconductor Schottky +devices suffer from the contact instability, high cost and +high-temperature fabrication process. Graphene, which +has unique optical properties and excellent mechanical +properties, offers Schottky solar cells with low sheet resis- +tance, high optical transparency, large area growth, and +low-cost transferring. Over the last decade, many studies +have been focused on Gr/Si Schottky junction8–11 (GSJ) +for solar cell applications. An overall power conversion +efficiency (PCE) of 1-1.7% is achieved with open circuit +voltage (Voc) and short circuit current (JSC) linearly de- +pending on the intensity of incident light6. +By chem- +ical doping, PCE can be further increased for 8.5%12. +With systematical optimization including the number +of graphene layers, PCE can surpass 3%13. +Further- +a)These two authors contributed equally +b)Corresponding author; Electronic mail: daodaohw@zju.edu.cn +c)Corresponding author; Electronic mail: huanhu@intl.zju.edu.cn +d)Corresponding author; Electronic mail: yangxu-isee@zju.edu.cn +more, substrate metasurface14,15, such as Si nanowire, +Si nanohole array, and additional antireflection layer16 +are proposed for PCE enhancement. +However, one primary limitation for the GSJ Schot- +tky solar cell is the low open circuit voltage Voc, which +relates to the small Schottky barrier height (SBH). +Here, we introduce an electromechanical coupling, called +flexoelectricity17, to increase the SBH, thus enhancing +the performance of GSJ solar cell. The flexoelectric ef- +fect describes the linear coupling between electric polar- +ization and strain gradient in solid state materials18,19. +It suggests that the polarization can originate from the +strain gradient even in the centrosymmetric systems. +Compared with the piezoelectric effect that has been +studied extensively, the response of the flexoelectric ef- +fect is very weak, and remains underexplored17. +The +electric polarization induced by a strain gradient is typ- +ical of the order of 10−9 C/m217,20 at the macro scale. +As the geometric size scales down, the strain gradient +is inversely proportional to the spatial scale21. +Thus +micro-nano structure is able to achieve a large strain +gradient, the flexoelectric effect induced by a strain gra- +dient, and dominates over the piezoelectric effect at +nanoscale22,23. In recent years, By using an atomic force +microscope (AFM) to introduce large strain gradients, +a number of experimental studies concentrate on the +mechanism of flexoelectricity and its applications have +emerged24–26. Conductive AFM (CAFM) probe coated +with metal can introduce a strain gradient and simulta- +neously monitor the current flow through the junctions. +The strain gradient breaks the inversion symmetry of +centrosymmetric materials and induces electric field po- +larization, named flexoelectric effect20. +In contrast, a +uniform strain cannot induce dipoles in graphene27, and +it is difficult to change the Schottky barrier. The bar- +rier height of the Schottky junction interface between +the probe and silicon can be tuned by the flexoelec- +tric effect20,28. The flexo-photovoltaic effect (FPV) was +arXiv:2301.03205v1 [cond-mat.mtrl-sci] 9 Jan 2023 + +2 +0.24 +3.5 +I2D/IG +ID/IG +0 +2 +4 +6 +10-6 +0 +1 +2 +3 +4 +5 +6 +10-6 +-1 +0 +1 +2 +3 +10-8 +Position (m) +Position (m) +Height (m) +c +a +b +d +e +-2 +-1 +0 +1 +2 +Voltage (V) +-100 +0 +100 +200 +300 +400 +500 +Current ( A) +light on +light off +-2 +0 +2 +Voltage (V) +10-10 +10-5 +Current (A) +FIG. 1. Experimental characterization of the GSJ device. a. The Scanning electron micrograph (SEM) of the device. The +white bar is 100 µm. b. The surface topography of the device measured by AFM. c. Current distribution of Gr/Si junction +measured by AFM with bias voltage Vbias−1 V and applied force 0.26 µN. The white scale bar represents 600 pm. d. The +Raman mapping of the device. e. The current-voltage response curve. The inset picture shows the logarithmic current as a +function of the bias voltage. +found in perovskites29,30 and two-dimensional material +system31,32, and significantly improves the solar cell per- +formance. These motivate us to systematically investi- +gate the flexo-photovoltaic in GSJ. +In this study, we introduce the flexo-photovoltaic effect +in the Gr/Si Schottky junction. We find the GSJ perfor- +mance as a solar cell can be largely enhanced through +the flexoelectric effect by using AFM tip pressing. By +in situ adding mechanical force on the GSJ, the current +flows through the junction to the tip can be detected, +then being read out based on the CAFM module using +AFM. The current-voltage curves under different applied +forces are analyzed, and we obtained the corresponding +SBH variation as a function of applied force. Under il- +lumination, the GSJ device shows PV effect. +Voc can +be improved by this electromechanical effect. We finally +demonstrate the enhanced PV through the flexoelectric +effect in GSJ. +The graphene/silicon junction Schottky devices are +prepared with a single layer of CVD graphene being wet- +transferred to the lightly-doped n-type silicon layer form- +ing the two dimensional and three dimensional (2d-3d) +Schottky contact. An n-doped Si/SiO2 500 µm/100 µm +substrate with a resistivity of 1-10 Ω · cm correspond- +ing to a doping concentration of 4.5 × 1014 cm−3 to +4.94 × 1015 cm−3 is used. +The bottom of the silicon substrate is mechanically +scraped to remove the thin oxide layer and coated with +GaIn and copper to form an ohmic contact33. The scan- +ning electron micrograph (SEM) of the device is shown +in Fig.1a. The graphene covers the silicon window (dark +area) forming the Schottky contact, connecting with the +Au electrode (surrounding gray part), and forming an +ohmic contact with Au electrodes. +The graphene is +etched into ribbons. The surface morphology of the Gr +ribbon edge tested by AFM is shown in Fig.1b. +The +left-hand side flatten area is silicon covered by graphene, +while the other is bare silicon after the graphene is etched. +With the tip contacting the GSJ area, the current can be +read out with the CAFM module. Figure.1c shows the +high spatial resolution current distribution of the junc- +tion with Vbias = −1 V. Figure.1d shows the correspond- +ing Raman mapping on the graphene ribbon device. The +intensity of the 2D peak versus the G peak and the in- +tensity of the D peak versus the G peak are shown to +illustrate the clear graphene ribbon areas. According to +the contour scale bar, the 2D peak has a larger peak, +while the D peak has a smaller peak, indicating the high +quality of the graphene in the sample. With the probe +connecting the bottom copper and the top-surface elec- + +nA +-7.77 +-8.00 +600pm +-8.223 +0 +-4 +-8 +0 +4 +8 +x (nm) +z (nm) +0.03 +-0.03 +0 +Tip +Gr +Si +Electrode +hν +EF +hν +EF +Tip +P +Graphene +Silicon +Interface +Graphene +Silicon +Interface +Ec +Ev +Ec +Ev +Vbias +Igsj +z +x +Ohmic contact +a +b +c +d +FIG. 2. a. Schematic of the CAFM experimental setup. b. Finite element analysis (FEA) of strain distribution in the substrate +under pressing. c. Energy bandgap structure of the GSJ with laser on. d. The bandgap structure bending by tip-pressing +induced flexoelectricity modulation. +trode separately, we obtain the basic current-voltage (I- +V) response of the device using a semiconductor analyzer +(Agilent B1500), shown in Fig.1e. The red line shows I- +V response with the light turned off while the blue line +represents the light-on case. In the light-off case, the I- +V curves show the typical Schottky diode properties of +GSJ. The center wavelength is 532 nm and the approx- +imate power density used is 50 µW. +The reverse cur- +rent is on the order of 10−10 A, showing a good diode +characteristic. When the light turns on, the reverse pho- +tocurrent exceeds 10 µA. The device performs a typical +photoresponse of GSJ, and the inset picture shows the +half-logarithmic curve of the I-V results. In the reverse +bias region, the photocurrent reaches nearly 5 orders of +magnitude under the effect of the photoresponse. In the +following experiments, we use AFM to do further tests to +study the electromechanical effects on the responses by +in situ exerting mechanical stress. Note that due to the +current-limiting protection of the instruments (20 nA), +we focus on the results in the range of ±20 nA in the +following tests. +Considering the graphene massless carrier property +and the inhomogeneity of the Gr/Si contact, the mod- +ified equation for GSJ based on thermal emission theory +can be expressed as the following formula7,34, +J = A∗T 3 exp +� +�− +¯φB − +δ2 +P +2kBT +kBT +� +� [exp +� qV +ηkBT − 1 +� +] (1) +where A∗ = 0.011 58 A/cm2/K3 and δP = 135 meV, and +η, kB, T and q are the ideal factor, the Boltzmann con- +stant, temperature and elementary charge, respectively. +The effective working area A of GSJ for our device here +is 1.25 × 10−3 cm2. +Fitting Fig.1e with Eq.1, we then +obtain the Schottky barrier height (SBH) ¯φB = 0.6 eV +before applying mechanical stress. To find the effect of +electrical polarization on the Schottky junction induced +by strain gradient, we use the Conductive-AFM tip to +exert stress on the graphene surface and in situ read-out +the current (Igsj) by sweeping the bias voltage Vbias, the +schematic of the experimental setup is shown in Fig.2a. +The tip coated with conductive diamond (Adama- +AD40) is directly pressed onto the CVD graphene layer. +Note that, in these AFM based experiments, Vbias is ap- +plied to the bottom ohmic electrode and the current re- +versely flows through the Gr/Si Schottky junction to the +probe for current detection. Thus the forward bias re- +sponse is under the condition of negative bias voltage +(Vbias < 0). +Directly experimental calibration of the +tip-induced strain distribution is still challenging24. In + +4 +-1 +-0.5 +0 +0.5 +1 +Bias Voltage (V) +-20 +-10 +0 +10 +20 +Current (nA) +Laser-on +Laser-off +-1 +-0.5 +0 +0.5 +1 +Bias Voltage (V) +10-14 +10-12 +10-10 +10-8 +Current (A) +Laser-on +Laser-off +-1 +-0.5 +0 +0.5 +1 +Bias Voltage (V) +-25 +-20 +-15 +-10 +-5 +0 +5 +Current (nA) +2.07 N +3.44 N +4.82 N +6.2 N +7.58 N +-1 +-0.5 +0 +0.5 +1 +Bias Voltage (V) +-20 +-10 +0 +10 +20 +Current (nA) +2.07 N +3.44 N +4.82 N +6.2 N +7.58 N +a +b +c +d +Laser o! +Laser on +FIG. 3. +GSJ I-V responses under tip-pressing obtained by AFM. Photo-response effects on the GSJ performance with fixed +applied force (a. Linear ordinate coordinates; b. Logarithmic ordinate coordinates). c. GSJ I-V responses of increased applied +forces with laser turned off. d. GSJ I-V responses of increased applied forces with laser turned on. +the following, we estimate the strain distribution based +on finite element simulation via COMSOL. As shown in +Fig.1, the graphene is etched into ribbons. Due to the +atomic thickness of graphene and the free-standing edge +of the ribbon, the graphene layer has negligible effects on +the out-of-plane deformation of the junction. The finite +element analysis on the strain distribution under press- +ing thus only takes the tip and the silicon substrate into +consideration for simplification. We set the tip radius to +10 nm according to the SEM imaging measurement. The +bottom of the substrate is set to be fixed according to +the actual situation of the experiment. The tip is set us- +ing diamond material with density 3515 kg/m3, Young’s +modulus 105 × 1010 Pa and Poisson’s ratio 0.1 while the +silicon substrate with density 2329 kg/m3, Young’s mod- +ulus 170 × 109 Pa and Poisson’s ratio 0.28. +The half- +cross-section of the substrate is shown in Fig.2b, as the +contact area has rotation symmetry of z-axis. While at +an applied force on the order of ∼5 µN, the corresponding +strain is about ±0.3. For the vertical direction (z-axis), +the strain gradient is estimated to be on the order of +107 m−1. +For a typical graphene low-doped n-type silicon junc- +tion, the electronic band structure and energy-band dia- +gram are shown in Fig.2c. While under illumination, part +of the photo-generated carriers moves to the graphene +thus the barrier height slightly changes. +Note that in +our following experiments, the laser central wavelength +from AFM is 840 nm. +In Fig.2d, we show the further +changes (dashed blue lines) of the barrier height. While +under tip pressing, the strain gradient in the substrate +silicon breaks the inverse symmetry of the centrosym- +metric material, which results in a flexoelectric effect20,29 +(the blue arrow represents the polarization). The corre- +sponding built-in electric field thus increases the SBH. +This leads to the flexoelectricity induced Schottky bar- +rier height tuning of the device, which will consequently +affect the photovoltaic effect in this junction. +Figure 3 shows the GSJ I-V responses obtained directly +by AFM while applying forces. While the laser is turned +off, the GSJ shows a typical Schottky diode feature (the +orange line in Fig.3a). As aforementioned, the forward +bias range is when Vbias < 0, while the reverse bias area +Vbias > 0. While Vbias >−0.3 V, the response current + +5 +-1.5 +-1 +-0.5 +0 +0.5 +1 +1.5 +2 +Bias Voltage (V) +10-12 +10-10 +10-8 +Current (A) +2.07 N +3.44 N +4.82 N +6.2 N +7.58 N +2.07 N +3.44 N +4.82 N +6.2 N +7.58 N +2 +4 +6 +8 +Force ( N) +-0.5 +-0.45 +-0.4 +-0.35 +Voc (V) +0.05 +0.1 +0.15 +0.2 +Voc (V) +0 +2 +4 +6 +8 +Force ( N) +0.62 +0.64 +0.66 +0.68 +0.7 +0.72 +a +b +c +Laser on +Laser o! +Laser on +Laser o! +FIG. 4. +a. The GSJ current as a function of bias voltage under different applied forces. The orange curves show the laser-off +cases while the blue curves show the laser-on cases. b. The open-circuit voltage as a function of applied forces. The blue points +represent the laser-on cases while the orange points represent the laser-off cases. c. The extracted Schottky barrier height as a +function of applied forces. +reaches the upper testing limit of our AFM instrument. +The open-circuit voltage (Voc) increases to ∼0.4 V, shown +clearly in Fig.3b. The short-circuit current (Jsc) can not +be directly obtained, because of the instrument current +limits. Nevertheless, the photoresponse (in Fig.3a and +Fig.3b) implies that this GSJ device has an evident pho- +tovoltaic effect. +To further investigate the mechanical +manipulation of the PV effect, we applied forces by AFM +nanoindentation on the GSJ. We first turn the laser off to +study the force induced effects on the GSJ. The response +current (absolute value) in the forward bias area can be +enlarged by increasing the applied force (Fig.3c) from +2.07 µN to 7.58 µN, while the response current in reverse +bias area maintains being cut off. Similarly, for each ap- +plied force, we then collect the current data by sweeping +the bias voltage Vbias when turning the laser on, shown in +Fig.3d. In the forward bias area, the GSJ current (IGSJ) +maintained being enlarged (absolute value) when the ap- +plied force is increased in contrast to the Fig.3c. Due +to the laser-induced photoresponse, the current dramati- +cally changed. There exists an evident photovoltaic area. +By directly contrasting the I-V responses under different +applied forces, the half-logarithmic graph of the absolute +GSJ current as a function of the bias voltage is shown in +Fig.4a. For the laser-off cases (orange lines), the open- +circuit voltage Voc remains stable under different applied +forces. Remarkably, for the laser-on cases (blue lines), +due to the existing photovoltaic effects, Voc varies from +near zero to around −0.5 V in contrast to the laser-off +cases. The absolute value of Voc is increased from 0.38 V +to 0.46 V for more than 20%, as shown in Fig.4b (the +blue circles). Fitting the laser-off cases with Eq.1, the +Schottky barrier height as a function of applied forces is +obtained, as shown in Fig.4c. The strain gradient in the +substrate originates from the contacts between the tip +and the substrate surface24, which breaks the inversion +symmetry in the silicon20,28 and causes an extra flexo- +electricity induced built-in potential in addition to the +native depletion layer in the GSJ. The SBH can be addi- +tionally increased by the forces. + +6 +For this GSJ, the open-circuit voltage can be obtained +for I = 0 while short-circuiting current for V = 0. When +V = 0, the simple expression can be obtained7 Isc ≈ +−Iph, where Iph is the photogenerated current. The Voc +can be expressed as, +Voc ≈ ηkBT +q +ln(Iph +I0 +) ≈ η +q φB + Const. +(2) +The Voc has approximately a linear relation with the +Schottky barrier height, and agrees well with experi- +ments. +Considering the photoresponse of Gr/Si Schottky junc- +tion, we need the short-circuit current Isc and fill factor +FF for estimation. However, because of the limitation of +our instrument, the short-circuit current can not be pre- +cisely measured experimentally. +Here, our laser power +has not been changed as an AFM detection light source +for the laser-on cases, i.e. photogenerated current Iph is +fixed. We then assume that the short-circuit current for +different applied forces are equal, i.e. for each case, when +Vbias = 0, the GSJ reaches the reverse saturated state. +In addition, when Vbias > Voc, the current rises sharply +to the upper limit in a series of parabolic shapes. Con- +sidering the actual Isc ≫20 nA, the fill factors FF are +assumed to be approximately equal for the cases under +different forces. Consequently, the power conversion ef- +ficiency (PCE) can be expressed as PCE = IscVocFF +Pin +, +which can be improved for about at least 20% attributed +to the enhancement of Voc under strain gradient. +In this article, we investigate the flexoelectricity- +enhanced photovoltaic effect in GSJ. Using the AFM, +we apply different forces to generate the strain gradient +in the junction and in situ obtain the response current +of GSJ under these conditions. +By experimental vali- +dation, we find that the open-circuit voltage (Voc) can +be enhanced from 0.38 V to 0.46 V by the applied forces +when the laser is turned on and the Schottky barrier +height can be increased from 0.65 eV to 0.7 eV, respec- +tively. As a consequence, the power conversion efficiency +can be enhanced for more than 20% with the assumption +that the short-circuit current Isc is identical and similar +fill factor FF. Our work shed light on the GSJ devices +for PV effect enhancement with mechanical manipulation +approach and pave the way for the PV enhancements +2D heterostructure based on the similar mechanism. At +present, our research is still in the early stage of explo- +ration, and combining this technology with integration is +still a challenge for future applications. +ACKNOWLEDGMENTS +This work was supported in part by the National Natu- +ral Science Foundation of China under Grants 92164106 +and 61874094, China Postdoctoral Science Foundation +(2021M692789), and in part by the Fundamental Re- +search Funds for the Central Universities under Grants +K20200060 and 2021FZZX001-17. We also thank Prof. +Chunli Zhang for his valuable discussions and comments. +1A. K. Geim and K. S. Novoselov. +The rise of graphene. +Na- +ture Materials, 6(3):183–191, mar 2007. ISSN 1476-1122. doi: +10.1038/nmat1849. +2Yang Xu, Cheng Cheng, Sichao Du, Jianyi Yang, Bin Yu, Jack +Luo, Wenyan Yin, Erping Li, Shurong Dong, Peide Ye, et al. +Contacts between two-and three-dimensional materials: ohmic, +schottky, and p–n heterojunctions. ACS nano, 10(5):4895–4919, +2016. +3Xiaomu Wang, Zhenzhou Cheng, Ke Xu, Hon Ki Tsang, and +Jian-Bin Xu. High-responsivity graphene/silicon-heterostructure +waveguide photodetectors. Nature Photonics, 7(11):888–891, nov +2013. ISSN 1749-4885. doi:10.1038/nphoton.2013.241. +4Quan Li, Zhen Tian, Xueqian Zhang, Ranjan Singh, Liangliang +Du, Jianqiang Gu, Jiaguang Han, and Weili Zhang. +Active +graphene-silicon hybrid diode for terahertz waves. Nature Com- +munications, 6(May), 2015. ISSN 20411723. +5Rong Xiang He, +Peng Lin, +Zhi Ke Liu, +Hong Wei Zhu, +Xing Zhong Zhao, Helen L.W. Chan, and Feng Yan. Solution- +gated graphene field effect transistors integrated in microfluidic +systems and used for flow velocity detection. Nano Letters, 12 +(3):1404–1409, 2012. ISSN 15306984. doi:10.1021/nl2040805. +6Xinming Li, Hongwei Zhu, Kunlin Wang, Anyuan Cao, Jin- +quan Wei, Chunyan Li, Yi Jia, Zhen Li, Xiao Li, and Dehai +Wu. +Graphene-on-silicon schottky junction solar cells. +Ad- +vanced Materials, 22(25):2743–2748, 2010. ISSN 09359648. doi: +10.1002/adma.200904383. +7Antonio Di Bartolomeo. Graphene Schottky diodes: An experi- +mental review of the rectifying graphene/semiconductor hetero- +junction. Physics Reports, 606:1–58, 2016. ISSN 03701573. +8S Doukas, P Mensz, N Myoung, AC Ferrari, I Goykhman, and +E Lidorikis. Thermionic graphene/silicon schottky infrared pho- +todetectors. Physical Review B, 105(11):115417, 2022. +9Yuzuki Ono and Hojun Im. +Thermal annealing effects on +graphene/n-si schottky junction solar cell: +Removal of pmma +residues. arXiv preprint arXiv:2209.08365, 2022. +10Fan Wu, Zheng-Qiang Zhu, He Tian, Zhaoyi Yan, Yanming +Liu, Yang Xu, Chao-Yang Xing, and Tian-ling Ren. +Vertical +wse2/bp/mos2 heterostructures with tunneling behaviors and +photodetection. Applied Physics Letters, 121(11):113508, 2022. +11Xiangshun Geng, Peigen Zhang, Jun Ren, Guan-Hua Dun, +Yuanyuan Li, Jialun Jin, Chaolun Wang, Xing Wu, Dan Xie, +He Tian, et al. Directly integrated mixed-dimensional van der +waals graphene/perovskite heterojunction for fast photodetec- +tion. InfoMat, 4(8):e12347, 2022. +12Chun-Chung Chen, Mehmet Aykol, Chia-Chi Chang, A. F. J. +Levi, and Stephen B. Cronin. Graphene-Silicon Schottky Diodes. +Nano Letters, 11(11):5097–5097, 2011. ISSN 1530-6984. +13Xiaohong An, Fangze Liu, and Swastik Kar. +Optimizing per- +formance parameters of graphene-silicon and thin transparent +graphite-silicon heterojunction solar cells. Carbon, 57:329–337, +2013. ISSN 00086223. +14Xiaozhen Zhang, Chao Xie, Jiansheng Jie, Xiwei Zhang, Yiming +Wu, and Wenjun Zhang. +High-efficiency graphene/si nanoar- +ray schottky junction solar cells via surface modification and +graphene doping. Journal of Materials Chemistry A, 1(22):6593– +6601, 2013. +15Yiming Wu, Xiaozhen Zhang, Jiansheng Jie, Chao Xie, Xiwei +Zhang, Baoquan Sun, Yan Wang, and Peng Gao. +Graphene +transparent conductive electrodes for highly efficient silicon +nanostructures-based hybrid heterojunction solar cells. +The +Journal of Physical Chemistry C, 117(23):11968–11976, 2013. +16Enzheng Shi, Hongbian Li, Long Yang, Luhui Zhang, Zhen Li, +Peixu Li, Yuanyuan Shang, Shiting Wu, Xinming Li, Jinquan +Wei, et al. Colloidal antireflection coating improves graphene– +silicon solar cells. Nano letters, 13(4):1776–1781, 2013. +17Bo Wang, Yijia Gu, Shujun Zhang, and Long Qing Chen. Flex- + +7 +oelectricity in solids: +Progress, challenges, and perspectives. +Progress in Materials Science, 106(May), 2019. ISSN 00796425. +18Sh M Kogan. Piezoelectric effect during inhomogeneous defor- +mation and acoustic scattering of carriers in crystals. +Soviet +Physics-Solid State, 5(10):2069–2070, 1964. +19A. K. Tagantsev. +Piezoelectricity and flexoelectricity in crys- +talline dielectrics. +Physical Review B, 34(8):5883–5889, 1986. +ISSN 01631829. +20Longfei Wang, Shuhai Liu, Xiaolong Feng, Chunli Zhang, Laipan +Zhu, Junyi Zhai, Yong Qin, and Zhong Lin Wang. Flexoelectron- +ics of centrosymmetric semiconductors. Nature Nanotechnology, +15(8):661–667, 2020. ISSN 17483395. +21Sung Min Park, Bo Wang, Long Qing Chen, Tae Won Noh, +Sang Mo Yang, and Daesu Lee. Flexoelectric control of physical +properties by atomic force microscopy. Applied Physics Reviews, +8(4), 2021. ISSN 19319401. +22Ivan Naumov, Alexander M. Bratkovsky, and V. Ranjan. Un- +usual flexoelectric effect in two-dimensional noncentrosymmetric +sp2-bonded crystals. Physical Review Letters, 102(21):2–5, 2009. +ISSN 00319007. doi:10.1103/PhysRevLett.102.217601. +23Matteo Springolo, Miquel Royo, and Massimiliano Stengel. Di- +rect and Converse Flexoelectricity in Two-Dimensional Materi- +als. Physical Review Letters, 127(21):216801, nov 2021. ISSN +0031-9007. doi:10.1103/PhysRevLett.127.216801. +24Sung Min Park, Bo Wang, Long-Qing Chen, Tae Won Noh, +Sang Mo Yang, and Daesu Lee. Flexoelectric control of physical +properties by atomic force microscopy. Applied Physics Reviews, +8(4):041327, 2021. +25Xingan Jiang, Xueyun Wang, Xiaolei Wang, Xiangping Zhang, +Ruirui Niu, Jianming Deng, Sheng Xu, Yingzhuo Lun, Yanyu +Liu, Tianlong Xia, Jianming Lu, and Jiawang Hong. Manipula- +tion of current rectification in van der Waals ferroionic CuInP2S6. +Nature Communications, 13(1):574, dec 2022. ISSN 2041-1723. +26Xiang Wang, Xin Zhou, Anyang Cui, Menghan Deng, Xionghu +Xu, Liping Xu, Yan Ye, Kai Jiang, Liyan Shang, Liangqing Zhu, +Jinzhong Zhang, Yawei Li, Zhigao Hu, and Junhao Chu. Flexo- +photoelectronic effect in n-type/p-type two-dimensional semicon- +ductors and a deriving light-stimulated artificial synapse. Mate- +rials Horizons, 8(7):1985–1997, 2021. ISSN 20516355. +27Xiaomu Wang, He Tian, Weiguang Xie, Yi Shu, Wen-Tian Mi, +Mohammad Ali Mohammad, Qian-Yi Xie, Yi Yang, Jian-Bin +Xu, and Tian-Ling Ren. Observation of a giant two-dimensional +band-piezoelectric effect on biaxial-strained graphene. NPG Asia +Materials, 7(1):e154–e154, 2015. +28Liang Sun, +Lifeng Zhu, +Chunli Zhang, +Weiqiu Chen, +and +Zhonglin Wang. Mechanical Manipulation of Silicon-based Schot- +tky Diodes via Flexoelectricity. +Nano Energy, 83(December +2020):105855, 2021. ISSN 22112855. +29Ming-min Yang, Dong Jik Kim, and Marin Alexe. +Flexo- +photovoltaic effect. Science, 360(6391):904–907, may 2018. ISSN +0036-8075. +30Longlong Shu, Shanming Ke, Linfeng Fei, Wenbin Huang, Zhiguo +Wang, Jinhui Gong, Xiaoning Jiang, Li Wang, Fei Li, Shuijin +Lei, Zhenggang Rao, Yangbo Zhou, Ren Kui Zheng, Xi Yao, +Yu Wang, Massimiliano Stengel, and Gustau Catalan. Photoflex- +oelectric effect in halide perovskites. +Nature Materials, 19(6): +605–609, 2020. ISSN 14764660. +31Y. J. Zhang, T. Ideue, M. Onga, F. Qin, R. Suzuki, A. Zak, +R. Tenne, J. H. Smet, and Y. Iwasa. Enhanced intrinsic photo- +voltaic effect in tungsten disulfide nanotubes. Nature, 570(7761): +349–353, 2019. ISSN 14764687. +32Jie Jiang, Zhizhong Chen, Yang Hu, Yu Xiang, Lifu Zhang, Yip- +ing Wang, Gwo-ching Wang, and Jian Shi. Flexo-photovoltaic +effect in MoS2. Nature Nanotechnology, 16(8):894–901, aug 2021. +ISSN 1748-3387. doi:10.1038/s41565-021-00919-y. +33Li Peng, Lixiang Liu, Sichao Du, Srikrishna Chanakya Bodepudi, +Lingfei Li, Wei Liu, Runchen Lai, Xiaoxue Cao, Wenzhang Fang, +Yingjun Liu, Xinyu Liu, Jianhang Lv, Muhammad Abid, Junxue +Liu, Shengye Jin, Kaifeng Wu, Miao-Ling Lin, Xin Cong, Ping- +Heng Tan, Haiming Zhu, Qihua Xiong, Xiaomu Wang, Weida +Hu, Xiangfeng Duan, Bin Yu, Zhen Xu, Yang Xu, and Chao Gao. +Macroscopic assembled graphene nanofilms based room temper- +ature ultrafast mid-infrared photodetectors. +InfoMat, 4(6):1– +12, jun 2022. +ISSN 2567-3165. +doi:10.1002/inf2.12309. +URL +https://onlinelibrary.wiley.com/doi/10.1002/inf2.12309. +34Shi Jun Liang, Wei Hu, A. Di Bartolomeo, Shaffique Adam, +and Lay Kee Ang. +A modified Schottky model for graphene- +semiconductor (3D/2D) contact: A combined theoretical and ex- +perimental study. Technical Digest - International Electron De- +vices Meeting, IEDM, pages 14.4.1–14.4.4, 2017. ISSN 01631918. + diff --git a/edE1T4oBgHgl3EQfeQSt/content/tmp_files/load_file.txt b/edE1T4oBgHgl3EQfeQSt/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2a4bdba3fe92f2a2c370577ae6f7fdb6d8d9cbf1 --- /dev/null +++ b/edE1T4oBgHgl3EQfeQSt/content/tmp_files/load_file.txt @@ -0,0 +1,510 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf,len=509 +page_content='Enhanced photovoltaic effect in graphene-silicon Schottky junction under mechanical manipulation Dong Pu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' a) Muhammad Abid Anwar,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' a) Jiachao Zhou,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' 2 Renwei Mao,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='3 Xin Pan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='3 Jian Chai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' 2 Feng Tian,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' 2 Hua Wang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' b) Huan Hu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' c) and Yang Xu1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' d) 1)School of Micro-Nano Electronics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Zhejiang University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Hangzhou,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' China 2)ZJU Global Scientific and Technological Innovation Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Hangzhou,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' China 3)ZJU-UIUC Institute (ZJUI),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Zhejiang University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Haining,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' China (Dated: 10 January 2023) Graphene-silicon Schottky junction (GSJ) which has the potential for large-scale manufacturing and integra- tion can bring new opportunities to Schottky solar cells for photovoltaic (PV) power conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' However, the essential power conversion limitation for these devices lies in the small open-circuit voltage (Voc), which depends on the Schottky barrier height (SBH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' In this study, we introduce an electromechanical method based on the flexoelectric effect to enhance the PV efficiency in GSJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' By atomic force microscope (AFM) tip-based indentation and in situ current measurement, the current-voltage (I-V) responses under flexoelectric strain gradient are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' The Voc is observed to increase for up to 20%, leading to an evident improvement of the power conversion efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Our studies suggest that strain gradient may offer unprecedented opportunities for the development of GSJ based flexo-photovoltaic applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Due to its special two-dimensional structural proper- ties and excellent performance1, graphene has the poten- tial to be integrated into existing semiconductor tech- nologies and used in next-generation electronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Re- cent research has shown that the formation of junc- tions between graphene and three-dimensional or two- dimensional semiconductors2 can produce the rectifica- tion effect of a typical Schottky junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Its tunable Schottky barrier makes graphene junctions an excellent platform for studying the transport properties of inter- faces and has led to applications in scenarios such as photodetection3, light-speed communication4, and chem- ical and biological detection5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Solar energy collection and conversion have attracted much attention in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' The Schottky junction devices can naturally work as so- lar cell or photovoltaic cell6, as the built-in electrical field provides the voltage potential difference that drives the current7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Conventional metal/semiconductor Schottky devices suffer from the contact instability, high cost and high-temperature fabrication process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Graphene, which has unique optical properties and excellent mechanical properties, offers Schottky solar cells with low sheet resis- tance, high optical transparency, large area growth, and low-cost transferring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Over the last decade, many studies have been focused on Gr/Si Schottky junction8–11 (GSJ) for solar cell applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' An overall power conversion efficiency (PCE) of 1-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='7% is achieved with open circuit voltage (Voc) and short circuit current (JSC) linearly de- pending on the intensity of incident light6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' By chem- ical doping, PCE can be further increased for 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='5%12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' With systematical optimization including the number of graphene layers, PCE can surpass 3%13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Further- a)These two authors contributed equally b)Corresponding author;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Electronic mail: daodaohw@zju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='cn c)Corresponding author;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Electronic mail: huanhu@intl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='zju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='cn d)Corresponding author;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Electronic mail: yangxu-isee@zju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='cn more, substrate metasurface14,15, such as Si nanowire, Si nanohole array, and additional antireflection layer16 are proposed for PCE enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' However, one primary limitation for the GSJ Schot- tky solar cell is the low open circuit voltage Voc, which relates to the small Schottky barrier height (SBH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Here, we introduce an electromechanical coupling, called flexoelectricity17, to increase the SBH, thus enhancing the performance of GSJ solar cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' The flexoelectric ef- fect describes the linear coupling between electric polar- ization and strain gradient in solid state materials18,19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' It suggests that the polarization can originate from the strain gradient even in the centrosymmetric systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Compared with the piezoelectric effect that has been studied extensively, the response of the flexoelectric ef- fect is very weak, and remains underexplored17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' The electric polarization induced by a strain gradient is typ- ical of the order of 10−9 C/m217,20 at the macro scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' As the geometric size scales down, the strain gradient is inversely proportional to the spatial scale21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Thus micro-nano structure is able to achieve a large strain gradient, the flexoelectric effect induced by a strain gra- dient, and dominates over the piezoelectric effect at nanoscale22,23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' In recent years, By using an atomic force microscope (AFM) to introduce large strain gradients, a number of experimental studies concentrate on the mechanism of flexoelectricity and its applications have emerged24–26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Conductive AFM (CAFM) probe coated with metal can introduce a strain gradient and simulta- neously monitor the current flow through the junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' The strain gradient breaks the inversion symmetry of centrosymmetric materials and induces electric field po- larization, named flexoelectric effect20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' In contrast, a uniform strain cannot induce dipoles in graphene27, and it is difficult to change the Schottky barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' The bar- rier height of the Schottky junction interface between the probe and silicon can be tuned by the flexoelec- tric effect20,28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' The flexo-photovoltaic effect (FPV) was arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='03205v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='mtrl-sci] 9 Jan 2023 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='24 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='5 I2D/IG ID/IG 0 2 4 6 10-6 0 1 2 3 4 5 6 10-6 1 0 1 2 3 10-8 Position (m) Position (m) Height (m) c a b d e 2 1 0 1 2 Voltage (V) 100 0 100 200 300 400 500 Current ( A) light on light off 2 0 2 Voltage (V) 10-10 10-5 Current (A) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Experimental characterization of the GSJ device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' The Scanning electron micrograph (SEM) of the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' The white bar is 100 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' The surface topography of the device measured by AFM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Current distribution of Gr/Si junction measured by AFM with bias voltage Vbias−1 V and applied force 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='26 µN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' The white scale bar represents 600 pm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' The Raman mapping of the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' The current-voltage response curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' The inset picture shows the logarithmic current as a function of the bias voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' found in perovskites29,30 and two-dimensional material system31,32, and significantly improves the solar cell per- formance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' These motivate us to systematically investi- gate the flexo-photovoltaic in GSJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' In this study, we introduce the flexo-photovoltaic effect in the Gr/Si Schottky junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' We find the GSJ perfor- mance as a solar cell can be largely enhanced through the flexoelectric effect by using AFM tip pressing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' By in situ adding mechanical force on the GSJ, the current flows through the junction to the tip can be detected, then being read out based on the CAFM module using AFM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' The current-voltage curves under different applied forces are analyzed, and we obtained the corresponding SBH variation as a function of applied force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Under il- lumination, the GSJ device shows PV effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Voc can be improved by this electromechanical effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' We finally demonstrate the enhanced PV through the flexoelectric effect in GSJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' The graphene/silicon junction Schottky devices are prepared with a single layer of CVD graphene being wet- transferred to the lightly-doped n-type silicon layer form- ing the two dimensional and three dimensional (2d-3d) Schottky contact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' An n-doped Si/SiO2 500 µm/100 µm substrate with a resistivity of 1-10 Ω · cm correspond- ing to a doping concentration of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='5 × 1014 cm−3 to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='94 × 1015 cm−3 is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' The bottom of the silicon substrate is mechanically scraped to remove the thin oxide layer and coated with GaIn and copper to form an ohmic contact33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' The scan- ning electron micrograph (SEM) of the device is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' The graphene covers the silicon window (dark area) forming the Schottky contact, connecting with the Au electrode (surrounding gray part), and forming an ohmic contact with Au electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' The graphene is etched into ribbons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' The surface morphology of the Gr ribbon edge tested by AFM is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' The left-hand side flatten area is silicon covered by graphene, while the other is bare silicon after the graphene is etched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' With the tip contacting the GSJ area, the current can be read out with the CAFM module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='1c shows the high spatial resolution current distribution of the junc- tion with Vbias = −1 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='1d shows the correspond- ing Raman mapping on the graphene ribbon device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' The intensity of the 2D peak versus the G peak and the in- tensity of the D peak versus the G peak are shown to illustrate the clear graphene ribbon areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' According to the contour scale bar, the 2D peak has a larger peak, while the D peak has a smaller peak, indicating the high quality of the graphene in the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' With the probe connecting the bottom copper and the top-surface elec- nA 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='77 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='00 600pm 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='223 0 4 8 0 4 8 x (nm) z (nm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='03 0 Tip Gr Si Electrode hν EF hν EF Tip P Graphene Silicon Interface Graphene Silicon Interface Ec Ev Ec Ev Vbias Igsj z x Ohmic contact a b c d FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Schematic of the CAFM experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Finite element analysis (FEA) of strain distribution in the substrate under pressing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Energy bandgap structure of the GSJ with laser on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' The bandgap structure bending by tip-pressing induced flexoelectricity modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' trode separately, we obtain the basic current-voltage (I- V) response of the device using a semiconductor analyzer (Agilent B1500), shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='1e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' The red line shows I- V response with the light turned off while the blue line represents the light-on case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' In the light-off case, the I- V curves show the typical Schottky diode properties of GSJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' The center wavelength is 532 nm and the approx- imate power density used is 50 µW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' The reverse cur- rent is on the order of 10−10 A, showing a good diode characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' When the light turns on, the reverse pho- tocurrent exceeds 10 µA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' The device performs a typical photoresponse of GSJ, and the inset picture shows the half-logarithmic curve of the I-V results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' In the reverse bias region, the photocurrent reaches nearly 5 orders of magnitude under the effect of the photoresponse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' In the following experiments, we use AFM to do further tests to study the electromechanical effects on the responses by in situ exerting mechanical stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Note that due to the current-limiting protection of the instruments (20 nA), we focus on the results in the range of ±20 nA in the following tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Considering the graphene massless carrier property and the inhomogeneity of the Gr/Si contact, the mod- ified equation for GSJ based on thermal emission theory can be expressed as the following formula7,34, J = A∗T 3 exp � �− ¯φB − δ2 P 2kBT kBT � � [exp � qV ηkBT − 1 � ] (1) where A∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='011 58 A/cm2/K3 and δP = 135 meV, and η, kB, T and q are the ideal factor, the Boltzmann con- stant, temperature and elementary charge, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' The effective working area A of GSJ for our device here is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='25 × 10−3 cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Fitting Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='1e with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='1, we then obtain the Schottky barrier height (SBH) ¯φB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='6 eV before applying mechanical stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' To find the effect of electrical polarization on the Schottky junction induced by strain gradient, we use the Conductive-AFM tip to exert stress on the graphene surface and in situ read-out the current (Igsj) by sweeping the bias voltage Vbias, the schematic of the experimental setup is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' The tip coated with conductive diamond (Adama- AD40) is directly pressed onto the CVD graphene layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Note that, in these AFM based experiments, Vbias is ap- plied to the bottom ohmic electrode and the current re- versely flows through the Gr/Si Schottky junction to the probe for current detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Thus the forward bias re- sponse is under the condition of negative bias voltage (Vbias < 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Directly experimental calibration of the tip-induced strain distribution is still challenging24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' In 4 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='5 1 Bias Voltage (V) 20 10 0 10 20 Current (nA) Laser-on Laser-off 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='5 1 Bias Voltage (V) 10-14 10-12 10-10 10-8 Current (A) Laser-on Laser-off 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='5 1 Bias Voltage (V) 25 20 15 10 5 0 5 Current (nA) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='07 N 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='44 N 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='82 N 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='2 N 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='58 N 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='5 1 Bias Voltage (V) 20 10 0 10 20 Current (nA) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='07 N 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='44 N 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='82 N 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='2 N 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='58 N a b c d Laser o!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Laser on FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' GSJ I-V responses under tip-pressing obtained by AFM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Photo-response effects on the GSJ performance with fixed applied force (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Linear ordinate coordinates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Logarithmic ordinate coordinates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' GSJ I-V responses of increased applied forces with laser turned off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' GSJ I-V responses of increased applied forces with laser turned on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' the following, we estimate the strain distribution based on finite element simulation via COMSOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='1, the graphene is etched into ribbons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Due to the atomic thickness of graphene and the free-standing edge of the ribbon, the graphene layer has negligible effects on the out-of-plane deformation of the junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' The finite element analysis on the strain distribution under press- ing thus only takes the tip and the silicon substrate into consideration for simplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' We set the tip radius to 10 nm according to the SEM imaging measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' The bottom of the substrate is set to be fixed according to the actual situation of the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' The tip is set us- ing diamond material with density 3515 kg/m3, Young’s modulus 105 × 1010 Pa and Poisson’s ratio 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='1 while the silicon substrate with density 2329 kg/m3, Young’s mod- ulus 170 × 109 Pa and Poisson’s ratio 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' The half- cross-section of the substrate is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='2b, as the contact area has rotation symmetry of z-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' While at an applied force on the order of ∼5 µN, the corresponding strain is about ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' For the vertical direction (z-axis), the strain gradient is estimated to be on the order of 107 m−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' For a typical graphene low-doped n-type silicon junc- tion, the electronic band structure and energy-band dia- gram are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' While under illumination, part of the photo-generated carriers moves to the graphene thus the barrier height slightly changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Note that in our following experiments, the laser central wavelength from AFM is 840 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='2d, we show the further changes (dashed blue lines) of the barrier height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' While under tip pressing, the strain gradient in the substrate silicon breaks the inverse symmetry of the centrosym- metric material, which results in a flexoelectric effect20,29 (the blue arrow represents the polarization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' The corre- sponding built-in electric field thus increases the SBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' This leads to the flexoelectricity induced Schottky bar- rier height tuning of the device, which will consequently affect the photovoltaic effect in this junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Figure 3 shows the GSJ I-V responses obtained directly by AFM while applying forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' While the laser is turned off, the GSJ shows a typical Schottky diode feature (the orange line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' As aforementioned, the forward bias range is when Vbias < 0, while the reverse bias area Vbias > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' While Vbias >−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='3 V, the response current 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='5 2 Bias Voltage (V) 10-12 10-10 10-8 Current (A) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='07 N 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='44 N 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='82 N 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='2 N 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='58 N 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='07 N 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='44 N 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='82 N 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='2 N 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='58 N 2 4 6 8 Force ( N) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='35 Voc (V) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='2 Voc (V) 0 2 4 6 8 Force ( N) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='72 a b c Laser on Laser o!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Laser on Laser o!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' The GSJ current as a function of bias voltage under different applied forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' The orange curves show the laser-off cases while the blue curves show the laser-on cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' The open-circuit voltage as a function of applied forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' The blue points represent the laser-on cases while the orange points represent the laser-off cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' The extracted Schottky barrier height as a function of applied forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' reaches the upper testing limit of our AFM instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' The open-circuit voltage (Voc) increases to ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='4 V, shown clearly in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' The short-circuit current (Jsc) can not be directly obtained, because of the instrument current limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Nevertheless, the photoresponse (in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='3a and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='3b) implies that this GSJ device has an evident pho- tovoltaic effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' To further investigate the mechanical manipulation of the PV effect, we applied forces by AFM nanoindentation on the GSJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' We first turn the laser off to study the force induced effects on the GSJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' The response current (absolute value) in the forward bias area can be enlarged by increasing the applied force (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='3c) from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='07 µN to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='58 µN, while the response current in reverse bias area maintains being cut off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Similarly, for each ap- plied force, we then collect the current data by sweeping the bias voltage Vbias when turning the laser on, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='3d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' In the forward bias area, the GSJ current (IGSJ) maintained being enlarged (absolute value) when the ap- plied force is increased in contrast to the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Due to the laser-induced photoresponse, the current dramati- cally changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' There exists an evident photovoltaic area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' By directly contrasting the I-V responses under different applied forces, the half-logarithmic graph of the absolute GSJ current as a function of the bias voltage is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' For the laser-off cases (orange lines), the open- circuit voltage Voc remains stable under different applied forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Remarkably, for the laser-on cases (blue lines), due to the existing photovoltaic effects, Voc varies from near zero to around −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='5 V in contrast to the laser-off cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' The absolute value of Voc is increased from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='38 V to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='46 V for more than 20%, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='4b (the blue circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Fitting the laser-off cases with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='1, the Schottky barrier height as a function of applied forces is obtained, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='4c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' The strain gradient in the substrate originates from the contacts between the tip and the substrate surface24, which breaks the inversion symmetry in the silicon20,28 and causes an extra flexo- electricity induced built-in potential in addition to the native depletion layer in the GSJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' The SBH can be addi- tionally increased by the forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' 6 For this GSJ, the open-circuit voltage can be obtained for I = 0 while short-circuiting current for V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' When V = 0, the simple expression can be obtained7 Isc ≈ −Iph, where Iph is the photogenerated current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' The Voc can be expressed as, Voc ≈ ηkBT q ln(Iph I0 ) ≈ η q φB + Const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' (2) The Voc has approximately a linear relation with the Schottky barrier height, and agrees well with experi- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Considering the photoresponse of Gr/Si Schottky junc- tion, we need the short-circuit current Isc and fill factor FF for estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' However, because of the limitation of our instrument, the short-circuit current can not be pre- cisely measured experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Here, our laser power has not been changed as an AFM detection light source for the laser-on cases, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' photogenerated current Iph is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' We then assume that the short-circuit current for different applied forces are equal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' for each case, when Vbias = 0, the GSJ reaches the reverse saturated state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' In addition, when Vbias > Voc, the current rises sharply to the upper limit in a series of parabolic shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Con- sidering the actual Isc ≫20 nA, the fill factors FF are assumed to be approximately equal for the cases under different forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Consequently, the power conversion ef- ficiency (PCE) can be expressed as PCE = IscVocFF Pin , which can be improved for about at least 20% attributed to the enhancement of Voc under strain gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' In this article, we investigate the flexoelectricity- enhanced photovoltaic effect in GSJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Using the AFM, we apply different forces to generate the strain gradient in the junction and in situ obtain the response current of GSJ under these conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' By experimental vali- dation, we find that the open-circuit voltage (Voc) can be enhanced from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='38 V to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='46 V by the applied forces when the laser is turned on and the Schottky barrier height can be increased from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='65 eV to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='7 eV, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' As a consequence, the power conversion efficiency can be enhanced for more than 20% with the assumption that the short-circuit current Isc is identical and similar fill factor FF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Our work shed light on the GSJ devices for PV effect enhancement with mechanical manipulation approach and pave the way for the PV enhancements 2D heterostructure based on the similar mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' At present, our research is still in the early stage of explo- ration, and combining this technology with integration is still a challenge for future applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work was supported in part by the National Natu- ral Science Foundation of China under Grants 92164106 and 61874094, China Postdoctoral Science Foundation (2021M692789), and in part by the Fundamental Re- search Funds for the Central Universities under Grants K20200060 and 2021FZZX001-17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' We also thank Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Chunli Zhang for his valuable discussions and comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' 1A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Geim and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Novoselov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' The rise of graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Na- ture Materials, 6(3):183–191, mar 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' ISSN 1476-1122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='1038/nmat1849.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' 2Yang Xu, Cheng Cheng, Sichao Du, Jianyi Yang, Bin Yu, Jack Luo, Wenyan Yin, Erping Li, Shurong Dong, Peide Ye, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Contacts between two-and three-dimensional materials: ohmic, schottky, and p–n heterojunctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' ACS nano, 10(5):4895–4919, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' 3Xiaomu Wang, Zhenzhou Cheng, Ke Xu, Hon Ki Tsang, and Jian-Bin Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' High-responsivity graphene/silicon-heterostructure waveguide photodetectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Nature Photonics, 7(11):888–891, nov 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' ISSN 1749-4885.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='1038/nphoton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='241.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' 4Quan Li, Zhen Tian, Xueqian Zhang, Ranjan Singh, Liangliang Du, Jianqiang Gu, Jiaguang Han, and Weili Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Active graphene-silicon hybrid diode for terahertz waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Nature Com- munications, 6(May), 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' ISSN 20411723.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' 5Rong Xiang He, Peng Lin, Zhi Ke Liu, Hong Wei Zhu, Xing Zhong Zhao, Helen L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Chan, and Feng Yan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Solution- gated graphene field effect transistors integrated in microfluidic systems and used for flow velocity detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Nano Letters, 12 (3):1404–1409, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' ISSN 15306984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='1021/nl2040805.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' 6Xinming Li, Hongwei Zhu, Kunlin Wang, Anyuan Cao, Jin- quan Wei, Chunyan Li, Yi Jia, Zhen Li, Xiao Li, and Dehai Wu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Graphene-on-silicon schottky junction solar cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Ad- vanced Materials, 22(25):2743–2748, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' ISSN 09359648.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='1002/adma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='200904383.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' 7Antonio Di Bartolomeo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Graphene Schottky diodes: An experi- mental review of the rectifying graphene/semiconductor hetero- junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Physics Reports, 606:1–58, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' ISSN 03701573.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' 8S Doukas, P Mensz, N Myoung, AC Ferrari, I Goykhman, and E Lidorikis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Thermionic graphene/silicon schottky infrared pho- todetectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Physical Review B, 105(11):115417, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' 9Yuzuki Ono and Hojun Im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Thermal annealing effects on graphene/n-si schottky junction solar cell: Removal of pmma residues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' arXiv preprint arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='08365, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' 10Fan Wu, Zheng-Qiang Zhu, He Tian, Zhaoyi Yan, Yanming Liu, Yang Xu, Chao-Yang Xing, and Tian-ling Ren.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Vertical wse2/bp/mos2 heterostructures with tunneling behaviors and photodetection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Applied Physics Letters, 121(11):113508, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' 11Xiangshun Geng, Peigen Zhang, Jun Ren, Guan-Hua Dun, Yuanyuan Li, Jialun Jin, Chaolun Wang, Xing Wu, Dan Xie, He Tian, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Directly integrated mixed-dimensional van der waals graphene/perovskite heterojunction for fast photodetec- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' InfoMat, 4(8):e12347, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' 12Chun-Chung Chen, Mehmet Aykol, Chia-Chi Chang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Levi, and Stephen B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Cronin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Graphene-Silicon Schottky Diodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Nano Letters, 11(11):5097–5097, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' ISSN 1530-6984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' 13Xiaohong An, Fangze Liu, and Swastik Kar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Optimizing per- formance parameters of graphene-silicon and thin transparent graphite-silicon heterojunction solar cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Carbon, 57:329–337, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' ISSN 00086223.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' 14Xiaozhen Zhang, Chao Xie, Jiansheng Jie, Xiwei Zhang, Yiming Wu, and Wenjun Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' High-efficiency graphene/si nanoar- ray schottky junction solar cells via surface modification and graphene doping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Journal of Materials Chemistry A, 1(22):6593– 6601, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' 15Yiming Wu, Xiaozhen Zhang, Jiansheng Jie, Chao Xie, Xiwei Zhang, Baoquan Sun, Yan Wang, and Peng Gao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Graphene transparent conductive electrodes for highly efficient silicon nanostructures-based hybrid heterojunction solar cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' The Journal of Physical Chemistry C, 117(23):11968–11976, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' 16Enzheng Shi, Hongbian Li, Long Yang, Luhui Zhang, Zhen Li, Peixu Li, Yuanyuan Shang, Shiting Wu, Xinming Li, Jinquan Wei, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Colloidal antireflection coating improves graphene– silicon solar cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Nano letters, 13(4):1776–1781, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' 17Bo Wang, Yijia Gu, Shujun Zhang, and Long Qing Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Flex- 7 oelectricity in solids: Progress, challenges, and perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Progress in Materials Science, 106(May), 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' ISSN 00796425.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' 18Sh M Kogan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Piezoelectric effect during inhomogeneous defor- mation and acoustic scattering of carriers in crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Soviet Physics-Solid State, 5(10):2069–2070, 1964.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' 19A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Tagantsev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Piezoelectricity and flexoelectricity in crys- talline dielectrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Physical Review B, 34(8):5883–5889, 1986.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' ISSN 01631829.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' 20Longfei Wang, Shuhai Liu, Xiaolong Feng, Chunli Zhang, Laipan Zhu, Junyi Zhai, Yong Qin, and Zhong Lin Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Flexoelectron- ics of centrosymmetric semiconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Nature Nanotechnology, 15(8):661–667, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' ISSN 17483395.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' 21Sung Min Park, Bo Wang, Long Qing Chen, Tae Won Noh, Sang Mo Yang, and Daesu Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Flexoelectric control of physical properties by atomic force microscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Applied Physics Reviews, 8(4), 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' ISSN 19319401.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' 22Ivan Naumov, Alexander M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Bratkovsky, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Ranjan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Un- usual flexoelectric effect in two-dimensional noncentrosymmetric sp2-bonded crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Physical Review Letters, 102(21):2–5, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' ISSN 00319007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='217601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' 23Matteo Springolo, Miquel Royo, and Massimiliano Stengel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Di- rect and Converse Flexoelectricity in Two-Dimensional Materi- als.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Physical Review Letters, 127(21):216801, nov 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' ISSN 0031-9007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='216801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' 24Sung Min Park, Bo Wang, Long-Qing Chen, Tae Won Noh, Sang Mo Yang, and Daesu Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Flexoelectric control of physical properties by atomic force microscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Applied Physics Reviews, 8(4):041327, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' 25Xingan Jiang, Xueyun Wang, Xiaolei Wang, Xiangping Zhang, Ruirui Niu, Jianming Deng, Sheng Xu, Yingzhuo Lun, Yanyu Liu, Tianlong Xia, Jianming Lu, and Jiawang Hong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Manipula- tion of current rectification in van der Waals ferroionic CuInP2S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Nature Communications, 13(1):574, dec 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' ISSN 2041-1723.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' 26Xiang Wang, Xin Zhou, Anyang Cui, Menghan Deng, Xionghu Xu, Liping Xu, Yan Ye, Kai Jiang, Liyan Shang, Liangqing Zhu, Jinzhong Zhang, Yawei Li, Zhigao Hu, and Junhao Chu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Flexo- photoelectronic effect in n-type/p-type two-dimensional semicon- ductors and a deriving light-stimulated artificial synapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Mate- rials Horizons, 8(7):1985–1997, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' ISSN 20516355.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' 27Xiaomu Wang, He Tian, Weiguang Xie, Yi Shu, Wen-Tian Mi, Mohammad Ali Mohammad, Qian-Yi Xie, Yi Yang, Jian-Bin Xu, and Tian-Ling Ren.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Observation of a giant two-dimensional band-piezoelectric effect on biaxial-strained graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' NPG Asia Materials, 7(1):e154–e154, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' 28Liang Sun, Lifeng Zhu, Chunli Zhang, Weiqiu Chen, and Zhonglin Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Mechanical Manipulation of Silicon-based Schot- tky Diodes via Flexoelectricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Nano Energy, 83(December 2020):105855, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' ISSN 22112855.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' 29Ming-min Yang, Dong Jik Kim, and Marin Alexe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Flexo- photovoltaic effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Science, 360(6391):904–907, may 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' ISSN 0036-8075.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' 30Longlong Shu, Shanming Ke, Linfeng Fei, Wenbin Huang, Zhiguo Wang, Jinhui Gong, Xiaoning Jiang, Li Wang, Fei Li, Shuijin Lei, Zhenggang Rao, Yangbo Zhou, Ren Kui Zheng, Xi Yao, Yu Wang, Massimiliano Stengel, and Gustau Catalan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Photoflex- oelectric effect in halide perovskites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Nature Materials, 19(6): 605–609, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' ISSN 14764660.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' 31Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Zhang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Ideue, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Onga, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Qin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Suzuki, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Zak, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Tenne, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Smet, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Iwasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Enhanced intrinsic photo- voltaic effect in tungsten disulfide nanotubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Nature, 570(7761): 349–353, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' ISSN 14764687.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' 32Jie Jiang, Zhizhong Chen, Yang Hu, Yu Xiang, Lifu Zhang, Yip- ing Wang, Gwo-ching Wang, and Jian Shi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Flexo-photovoltaic effect in MoS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Nature Nanotechnology, 16(8):894–901, aug 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' ISSN 1748-3387.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='1038/s41565-021-00919-y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' 33Li Peng, Lixiang Liu, Sichao Du, Srikrishna Chanakya Bodepudi, Lingfei Li, Wei Liu, Runchen Lai, Xiaoxue Cao, Wenzhang Fang, Yingjun Liu, Xinyu Liu, Jianhang Lv, Muhammad Abid, Junxue Liu, Shengye Jin, Kaifeng Wu, Miao-Ling Lin, Xin Cong, Ping- Heng Tan, Haiming Zhu, Qihua Xiong, Xiaomu Wang, Weida Hu, Xiangfeng Duan, Bin Yu, Zhen Xu, Yang Xu, and Chao Gao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Macroscopic assembled graphene nanofilms based room temper- ature ultrafast mid-infrared photodetectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' InfoMat, 4(6):1– 12, jun 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' ISSN 2567-3165.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='1002/inf2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='12309.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' URL https://onlinelibrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='wiley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='com/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='1002/inf2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='12309.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' 34Shi Jun Liang, Wei Hu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Di Bartolomeo, Shaffique Adam, and Lay Kee Ang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' A modified Schottky model for graphene- semiconductor (3D/2D) contact: A combined theoretical and ex- perimental study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' Technical Digest - International Electron De- vices Meeting, IEDM, pages 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='1–14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content='4, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} +page_content=' ISSN 01631918.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE1T4oBgHgl3EQfeQSt/content/2301.03205v1.pdf'} diff --git a/etE5T4oBgHgl3EQfgw9z/content/tmp_files/2301.05636v1.pdf.txt b/etE5T4oBgHgl3EQfgw9z/content/tmp_files/2301.05636v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b7b8a8b59138cbdd9900cba285ef74af617e8f77 --- /dev/null +++ b/etE5T4oBgHgl3EQfgw9z/content/tmp_files/2301.05636v1.pdf.txt @@ -0,0 +1,1560 @@ +Improving Power by Conditioning on Less in Post-selection +Inference for Changepoints +Rachel Carrington and Paul Fearnhead +January 2023 +Abstract +Post-selection inference has recently been proposed as a way of quantifying uncertainty about detected +changepoints. The idea is to run a changepoint detection algorithm, and then re-use the same data to +perform a test for a change near each of the detected changes. +By defining the p-value for the test +appropriately, so that it is conditional on the information used to choose the test, this approach will +produce valid p-values. We show how to improve the power of these procedures by conditioning on less +information. This gives rise to an ideal selective p-value that is intractable but can be approximated by +Monte Carlo. We show that for any Monte Carlo sample size, this procedure produces valid p-values, and +empirically that noticeable increase in power is possible with only very modest Monte Carlo sample sizes. +Our procedure is easy to implement given existing post-selection inference methods, as we just need to +generate perturbations of the data set and re-apply the post-selection method to each of these. On genomic +data consisting of human GC content, our procedure increases the number of significant changepoints that +are detected from e.g. 17 to 27, when compared to the method of Jewell et al. (2021). +Keywords: Binary segmentation; Breakpoint; Fused Lasso; Penalised likelihood; Selective +p-value; +1 +Introduction +Detecting abrupt changes in time-series, or other ordered, data has been one of the most active +research areas of the past decade. It has applications in bioinformatics (e.g. Braun et al., 2000; +Olshen et al., 2004), computer performance (Barrett et al., 2017), climate science (Reeves et al., +2007; Shi et al., 2022a), cyber security (Heard and Turcotte, 2014; Fearnhead and Rigaill, 2019), +neuroscience (Aston and Kirch, 2012; Jewell et al., 2020), and industrial process monitoring (Maleki +et al., 2016) amongst many others. +There has been a wide range of methods that have been +1 +arXiv:2301.05636v1 [stat.ME] 13 Jan 2023 + +proposed, dealing with detecting different types of change, such as change in mean, variance or +slope; different algorithms for searching for multiple changepoints, including binary segmentation +and its variants (Olshen et al., 2004; Fryzlewicz, 2014; Baranowski et al., 2019), moving window +methods (Hao et al., 2013; Eichinger and Kirch, 2018; Meier et al., 2021), L1 penalised regression +methods (Kim et al., 2009; Tibshirani, 2014), and dynamic programming approaches to maximising +an L0 penalised likelihood (e.g. Killick et al., 2012; Maidstone et al., 2017); and for different types +of data, such as high-dimensional data (Wang and Samworth, 2018), network data (Wang et al., +2021), and general non-Euclidean data (Song and Chen, 2022; Dubey and M¨uller, 2020). +See +Truong et al. (2020), Fearnhead and Rigaill (2020) and Shi et al. (2022b) for an overview of this +area. +There has been much less work looking at quantifying the uncertainty of estimated change- +points. Whilst Bayesian methods (e.g. Fearnhead, 2006) that sample from a posterior over the +number and location of the changepoints naturally give measures of uncertainty, assessing un- +certainty for non-Bayesian methods is more challenging. Current work in this area includes the +SMUCE method (Frick et al., 2014; Li et al., 2016; Pein et al., 2017), and global methods that try +to give regions that produce sets of intervals, all of which must include a change at a pre-specified +significance level (Fryzlewicz, 2020, 2021). +A different approach is to try and assign a measure of significance to each detected changepoint. +The challenge here is to avoid so-called double peeking at the data (Zhao et al., 2021), where you use +the same data both to detect a change and then to test for the change, as a naive implementation +of test based on using the same data twice will be invalid. This is because, in the absence of any +change, the detection process will bias you to performing tests that are more likely to have small +p-values. This results in tests where the p-values are neither uniform, nor stochastically bounded +below by a uniform distribution (see e.g. Jewell et al., 2021). +One simple approach to circumvent this is sample splitting (Rinaldo et al., 2019), where you +use a proportion of the data to detect changes and the other other part to perform a test for +each detected change. However using only part of the data for each of detection and testing is +sub-optimal. Instead post-selection inference ideas for regression (Berk et al., 2013; Fithian et al., +2014; Kuchibhotla et al., 2022) have recently been applied to the changepoint setting. These allow +the same data to be used for detection and testing, but with the p-values for each change being +calculated conditional on information from the data that includes whatever information is used to +choose the test that is being performed. These are called selective p-values. +Methods for calculating selective p-values have been developed for the change in mean problem +with Gaussian noise and for a range of detection algorithms. +Hyun et al. (2021) develop an +2 + +approach for binary segmentation and its variants, and for the fused lasso; while Jewell et al. +(2021) and Duy and Takeuchi (2021) propose methods that work if changes are detected using an +L0 penalised likelihood. Furthermore Jewell et al. (2021) show how to improve on the method of +Hyun et al. (2021) by conditioning on less information when defining the selective p-value, and +show that conditioning on less information can lead to a substantial increase in power. There has +also been recent work on post-selection inference beyond the change in mean problem (e.g Chen +et al., 2021). +Our work is motivated by further wanting to reduce the information that one conditions on +when calculating the selective p-value. Current methods condition on the projection of the data +that is orthogonal to the test statistic. If, as is common, our test has a null hypothesis where, say, +the mean of the data does not change within a region about the tested changepoint, then we can +reduce this to conditioning on the data outside the region and an appropriate sufficient statistic, +such as the sample mean, within the region. This, together with whatever aspect of the detected +changes is used to pick the test, is the minimum amount of information that we need to condition +on to make the selective p-value well-defined. In Section 3.2 we show that, for a natural class of +distributions for the data under the alternative, the resulting selective p-value is optimal. +Unfortunately we cannot directly calculate this p-value. Instead we propose a simple Monte +Carlo approximation. This is based on simulating new data within the region around the change- +point that is being tested, applying the existing post-selection inference methodology to each such +data set, and then calculating a weighted average of the selective p-values for each data set. Im- +portantly, we show that if one of the data sets we average over is the observed data, then this leads +to a valid selective p-value, in that its distribution is uniform on [0, 1] under the null, regardless +of the Monte Carlo sample size. Furthermore, it is simple to calculate provided we have a method +that calculates a selective p-value based on conditioning on the projection of the data orthogonal +to the test statistic. As such, our method applies to all changepoint scenarios considered in Hyun +et al. (2018), Hyun et al. (2021), Jewell et al. (2021) and Chen et al. (2021). We present empirical +results that show one can obtain a noticeable improvement in power even with modest Monte +Carlo sample sizes, say of the order of 10. For a data set of GC content on human chromosome 1, +this increased power leads to the number of significant changepoints that are detected increasing +from 17 to 27, as compared to the method of Jewell et al. (2021). Whilst our method has been +developed for the changepoint problem, the underlying ideas apply more widely, see Section 6. All +proofs are given in the Appendix. +3 + +2 +Background +2.1 +Selective p-values for changepoints +Suppose we have a dataset X = (X1, . . . , XT ) and we fit a changepoint model which consists of +K changepoints M(X) = {ˆτ1, . . . , ˆτK}. We are interested in quantifying the level of uncertainty +associated with these changepoints: how confident can we be that the changepoints we have found +correspond to real changes and not false discoveries? One approach is to compute p-values for each +changepoint of interest. +One aspect in quantifying uncertainty in this way is deciding what we mean by τ being a +changepoint. Or more specifically, what null hypothesis do we want to test? In many applications +we say that τ is a changepoint providing some aspect of the data (that we are interested in) +changes at or close to τ. That is, the null hypothesis would be that there is no change in some +region centered on τ. +Even once we have decided on the null hypothesis, naively applying a test for a change at each +of ˆτ1, . . . , ˆτK is not possible, as we have already used the data to detect changepoints. If the data +contains no changes, we would expect any detected changepoint locations to be where, by chance, +the patterns of the data are similar to patterns produced by a change. This will bias the p-values +(see Jewell et al., 2021, for examples of this). +To overcome this, we can correct the naive test to take account of the fact that we are using +the data twice (Fithian et al., 2014) . This can be done by calculating a selective p-value, which +uses the distribution of the test statistic under the null but also conditional on any information +used to choose the test that we are performing. +To make the idea concrete let F(X) denote the information we want to condition on. We have +freedom over the choice of F(X), except that it must contain the information from the data that is +used to choose the test we performing. So, for example, if we choose to test that τ is a changepoint +based only on the property that τ is one of the estimated changepoints, then F(X) must include +the information τ ∈ M(X). The selective p-value is then +Pr(T ≥ Tobs | F (X) = F (Xobs) ), +for some test statistic T , and where we use Xobs and Tobs to denote the observed data and test +statistic respectively. +The challenge is then how to calculate this selective p-value. Often this will require a careful +choice of the information we condition on, both to make the selective p-value well defined, and also +possible to calculate. +4 + +2.2 +Selective p-value for change in mean +For ease of presentation it is helpful to consider a specific example. We will consider the univariate +change in mean model, for which methods for calculating selective p-values have been developed +by Hyun et al. (2018), Hyun et al. (2021), Jewell et al. (2021) and Duy et al. (2020). However, the +ideas we introduce for increasing the power of post-selection inference can apply more widely (e.g. +to the scenarios considered in Chen et al., 2021). +For the change in mean model, we assume the data is of the form +Xt = µt + ϵt, +t = 1, . . . , T, +where µt is piecewise constant, with µt+1 ̸= µt only at K changepoints τ1, . . . , τK. We assume +that ϵt ∼iid N(0, σ2), with σ known. +We run a changepoint algorithm – for example binary segmentation (Scott and Knott, 1974), +wild binary segmentation (Fryzlewicz, 2014), narrowest-over-threshold (Baranowski et al., 2019), +fused lasso (Tibshirani et al., 2005), or a penalised likelihood approach (Maidstone et al., 2017) – +and detect a set of changepoints {ˆτ1, . . . , ˆτK}. We now want to test for a change at a particular +estimated changepoint, which for simplicity we will denote ˆτ. +As mentioned above, for many applications a natural null hypothesis is that there is no change +in mean close to ˆτ. There are various possible choices for what we mean by “close to”, but here +we will assume that there is a pre-determined distance h that is appropriate for our application. +Our null hypothesis is therefore +H0 : µˆτ−h+1 = · · · = µˆτ = µˆτ+1 = · · · = µˆτ+h, +with the alternative hypothesis being that there is at least one inequality. (Extensions to other +choices of null hypothesis will be discussed in Section 3.4.) +Let ν ˆτ to be a T-dimensional vector whose tth entry is +(vˆτ)t = +� +� +� +� +� +� +� +� +� +� +� +� +� +1 +h +if ˆτ − h < t ≤ ˆτ +− 1 +h +if ˆτ < t ≤ ˆτ + h +0 +if t ≤ ˆτ − h or t > ˆτ + h. +Then, under H0, and without conditioning on any information in the data, νT +ˆτ X ∼ N +� +0, 2σ2 +h +� +. +Under H1, where there is a changepoint at or near ˆτ, we would expect the mean of νT +ˆτ X to be +non-zero. We can therefore take the test statistic to be T = |νT +ˆτ X|. +As above let M(X) = {ˆτ1, . . . , ˆτK}. The information used to choose the null hypothesis to test +is that ˆτ ∈ M(X). Thus for our selective p-value we need a conditioning event that includes this +5 + +information. Unfortunately it is not possible to just choose F(X) to be ˆτ ∈ M(X), because the +probability of this event depends on parameters that are unknown under the null hypothesis. +To deal with this, current approaches (Jewell et al., 2021; Hyun et al., 2021) condition also on +the projection of the data that is orthogonal to ν ˆτ. Denote this orthogonal projection by Πˆτ, then +this leads to the selective p-value that is +Pr(|νT +ˆτ X| > |νT +ˆτ Xobs| | ˆτ ∈ M(X), ΠˆτX = ΠˆτXobs). +(1) +While this is well-defined, calculating the required conditional distribution of νT +ˆτ X is non- +trivial. Hyun et al. (2021) show that for binary segmentation or the fused lasso, if you condition +on further information, namely the order in which the changepoints are detected and the esti- +mated sign of the changepoint, then the conditional distribution will be a truncated Gaussian. +Furthermore the truncation region can be calculated be solving a series of linear equations. +Motivated by intuition that conditioning on less information will improve power (Fithian et al., +2014; Liu et al., 2018), Jewell et al. (2021) shows how to reduce the amount of information condi- +tioned on, by avoiding having to condition on the order and signs of the changepoints. As we are +conditioning on the projection of the data orthogonal to νT +ˆτ , X will be uniquely determined if, in +addition, we known νT +ˆτ X. Let φ = νT +ˆτ X, then we can define the set of possible data sets that are +possible as we vary φ by +X′(φ) = Xobs − +1 +||ν ˆτ||2 ν ˆτνT +ˆτ Xobs + +1 +||ν ˆτ||2 ν ˆτφ. +(2) +If we define S = {φ : ˆτ ∈ M(X′(φ))}, then the p-value in Equation 1 is equal to +Pr +� +|φ| ≥ |νT +ˆτ Xobs| | ˆτ ∈ M(X′(φ)) +� += Pr +� +|φ| ≥ |νT +ˆτ Xobs| | φ ∈ S +� +, +where, unconditionally, φ ∼ N(0, σ2||ν ˆτ||2). +Jewell et al. (2021) shows how the set S can be +efficiently computed for changepoint methods including binary segmentation, L0 segmentation and +the fused lasso; in each case S is a union of intervals. Their methods can also be extended to other +similar changepoint algorithms such as wild binary segmentation and narrowest-over-threshold. +The method of Jewell et al. (2021) leads to an increase in power compared to the approach of +Hyun et al. (2021), as it requires conditioning on less information. However, we still condition on +T − 1 parameters that are orthogonal to ν ˆτ. The method we propose further reduces the amount +of information we need to condition on, leading to greater power to detect changepoints. +6 + +3 +Conditioning on less information +3.1 +The ideal selective p-value +Instead of conditioning on ΠˆτX, we could consider just conditioning on the minimum amount of +information to make the selective p-value well defined. Our null hypothesis fixes that µˆτ−h+1 = +· · · = µˆτ+h, so this contains no information about the mean for data points outside of {ˆτ − h + +1, . . . , ˆτ + h}, nor does it specify the mean within this window. +Hence, as a minimum we need to condition on +Xt = Xobs,t for t ∈ {1, . . . , ˆτ − h, ˆτ + h + 1, . . . , T} +1 +2h +ˆτ+h +� +i=ˆτ−h+1 +Xt = 1 +2h +ˆτ+h +� +i=ˆτ−h+1 +Xobs,t, +(3) +that is the data outside of {ˆτ − h + 1, . . . , ˆτ + h}, and the sample mean of the data in this window +(which is a sufficient statistic for the unknown, constant mean in the window). These have total +dimension T − 2h + 1, so we gain an additional 2h − 2 degrees of freedom compared to the method +in Jewell et al. (2021). +Let B be the T × (T − 2h) matrix obtained by removing the columns corresponding to {ˆτ − +h + 1, . . . , ˆτ + h} from the T × T identity matrix, and let a be a T-dimensional vector such that +at = +� +� +� +� +� +1 +2h +if t ∈ {ˆτ − h + 1, . . . , ˆτ + h} +0 +otherwise. +The conditions in (3) are equivalent to: +BT X = BT Xobs +aT X = aT Xobs. +We can rewrite X as +X = X − +� +BBT + +1 +||a||2 +2 +aaT + +1 +||ν ˆτ||2 +2 +ν ˆτνT +ˆτ +� +X + +� +BBT + +1 +||a||2 +2 +aaT + +1 +||ν ˆτ||2 +2 +ν ˆτνT +ˆτ +� +X += +� +I − +� +BBT + +1 +||a||2 +2 +aaT + +1 +||ν ˆτ||2 +2 +ν ˆτνT +ˆτ +�� +X + +1 +||ν ˆτ||2 +2 +ν ˆτνT +ˆτ X + +� +BBT + +1 +||a||2 +2 +aaT +� +X += ZX + +1 +||ν ˆτ||2 +2 +ν ˆτνT +ˆτ X + +� +BBT + +1 +||a||2 +2 +aaT +� +X, +where Z = I − +� +BBT + +1 +||a||2 +2 aaT + +1 +||ν ˆτ ||2 +2 ν ˆτνT +ˆτ +� +. +Z is a T × T matrix with rank 2h − 2, so ZX follows a degenerate multivariate normal +distribution. However, since Z is a symmetric matrix with all its non-zero eigenvalues equal to 1, +we can write Z = UU T , where U is a T × (2h − 2) matrix with orthonormal columns. Under H0, +7 + +U T X ∼ N(U T µ, σ2U T U) = N(0, σ2I). The matrix U is not uniquely defined, but the choice of +basis is arbitrary. It can be found, for example, using the Singular Value Decomposition. +Let ψ = U T X and, as before, let φ = νT +ˆτ X. Then, given the information we are conditioning +on, as we vary ψ and φ we get data +X = X′(φ, ψ) = Uψ + +1 +||ν ˆτ||2 +2 +ν ˆτνT +ˆτ φ + +� +1 +||a||2 +2 +aaT + BBT +� +Xobs. +Furthermore, under the null and without conditioning on further aspects of the data, such as the +estimated changepoints, φ ∼ N(0, 2σ2 +h ) and ψi ∼iid N(0, σ2). +The resulting selective p-value is +Pφ,ψ +� +|φ| ≥ |νT +ˆτ Xobs| | ˆτ ∈ M(X′(φ, ψ)) +� +. +(4) +As this p-value is obtained by conditioning on the least amount of information needed for it to be +well-defined, we will call it the ideal p-value. +3.2 +Intuition behind new selective p-value +To understand the difference between the ideal selective p-value (4) and the p-value of Jewell +et al. (2021), we give a schematic comparison in Figure 1. To enable us to present a plot we have +assumed that ψ is scalar, and have also used the probability inverse mapping to transform (φ, ψ) +from independent normal to independent uniform on [0, 1]. +With this mapping, and given the conditioning in (3), data sets correspond to points in (φ, ψ)- +space, and under the null such points are uniform on the unit square. The conditioning on ˆτ being +a detected changepoint corresponds to restricting the possible set of (φ, ψ) values – to the non-grey +area in Figure 1. We have plotted the (φ, ψ) value for the observed data by an a cross in the +top row of Figure 1. The p-value of Jewell et al. (2021) then fixes the ψ value so the conditional +distribution of φ is uniform on the coloured line – i.e. all values that are consistent with detecting +a change at ˆτ for that value of ψ. The p-value is the probability of observing a more extreme value +than that for the data – which is the proportion of the line that is red in the top left plot of Figure +1. +By comparison, the p-value of (4) allows ψ to vary. It is thus the probability of observing a +more extreme value of φ than that for the data over all possible (φ, ψ) values that are consistent +with ˆτ being a detected change. This is the proportion of the non-grey area that is red in the top +left plot of Figure 1. If we generate the data by simulating a (φ, ψ) point uniformly in the non-grey +region, then it is simple to show that the distribution of either p-value will be uniform on [0, 1]. +To see why the ideal p-value (4) is to be preferred, we plot the set of (φ, ψ) values that would +correspond to data with a selective p-value of less than 0.2 in the bottom row of Figure 1. For +8 + +φ +ψ +x +φ +ψ +x +φ +ψ +φ +ψ +Figure 1: Comparison of the p-value of Jewell et al. (2021) (left-hand column) and the ideal p-value (right- +hand column) for the case of a univariate ψ parameter. We have used the probability inverse mapping to +transform φ and ψ so that they are uniformly and independently distributed on [0, 1] under the prior. We +view data sets as being a function of (φ, ψ), and the selective event – which corresponds to the information +in the data used to choose the test – corresponds to a region of (φ, ψ) values (non-grey region in all plots). +The observed data corresponds to a specific (φ, ψ) value shown by a cross (top-row plots). For the method +of Jewell et al. (2021), the p-value is the probability of observing a more extreme value of φ conditional +on the observed ψ-value. This is the proportion of the coloured line that is red in the top-right plot. For +our method, the p-value is the (unconditional) probability of observing a more extreme value of φ: the +proportion of the non-grey area that is red (top-right plot). In the bottom row we show the data-sets, as +represented by their (φ, ψ) value, that would give a selective p-value that is 0.2 or lower (red region in both +plots). +9 + +both p-values these give regions whose area is 0.2 of the non-grey area. The difference is the shape +of the regions, with the ideal p-value consisting of requiring just φ greater than some constant, +whereas the p-value of Jewell et al. (2021) has different regions for φ as we vary ψ. The former will +have more power if we have alternative hypotheses that, compared to the null, place increasing +probability on larger values of |φ|. +To make this precise, let A be the projection of the data we condition on (3). Under the null, +and conditional on A denote the density for (φ, ψ) as +f(φ) +2h−2 +� +i=1 +g(ψi). +Consider alternative hypotheses that correspond to a density of (φ, ψ) of the form +k(φ)f(φ) +2h−2 +� +i=1 +g(ψi), +(5) +for some function k(φ). That is, under the alternative hypothesis the distribution of φ is altered, +and k(φ) represents the ratio of density between the alternative and the null. +Theorem 1. Conditional on A and (φ, ψ) ∈ S, define PI to be the p-value given by (4), and P ∗ +be any other valid p-value, i.e. that satisfies that under the null +Pr(P ∗ ≤ α) ≤ α, for all α ∈ [0, 1]. +Then under an alternative with density of the form (5) for a function k(φ) = ˜k(|φ|) with ˜k increas- +ing, +Pr(PI ≤ α) ≥ Pr(P ∗ ≤ α), for all α ∈ [0, 1]. +3.3 +Estimating p-values with sampling +Unfortunately it is not possible to analytically calculate the ideal selective p-value (4). Instead +we will resort to using Monte Carlo to estimate it, under the assumption that we have a method +for calculating the null distribution of φ given ψ – this would refer to any combination of type +of change, choice of null hypothesis and method for detecting the changepoints for which current +post-selection inference methods exist. +Let +S = {(φ, ψ) : ˆτ ∈ M +� +X′(φ, ψ) +� +}, +so the conditioning event for (4) corresponds to (φ, ψ). Furthermore, define +Sψ = {φ : ˆτ ∈ M +� +X′(φ, ψ) +� +}, +10 + +0.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +P−value +U(0,1) +(a) +h = 10 +0.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +P−value +U(0,1) +(b) +h = 20 +0.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +P−value +U(0,1) +(c) +h = 10 +0.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +P−value +U(0,1) +(d) +h = 20 +Figure 2: QQ plots of p-value estimates, simulated under H0 with T = 1000, for different values of h and +N. On each plot the ordered p-values obtained using different values of N (N = 1, 5, 10, 50) are plotted +against theoretical quantiles from U(0, 1). In (a) and (b) the p-values are calculated as in Equation 6, +where all ψ(j)’s are simulated randomly. In (c) and (d), we take ψ(1) = U T Xobs. If p-values are valid, +the points should lie approximately along the line y = x. +the set of φ values corresponding to data where we estimate ˆτ as a changepoint, for a given value +of ψ. These regions will depend on the algorithm used to estimate the changepoints, and we are +assuming that for any given ψ value we can calculate Sψ. For example, Jewell et al. (2021) show +how to calculate these regions for the change in mean problem with changepoints estimated by +binary segmentation, wild binary segmentation and L0 penalised likelihood methods. +Now that we have 2h − 2 additional parameters in ψ, the truncation region S becomes much +more complicated to calculate explicitly, as the values of φ that yield ˆτ ∈ M(X′(φ)) depend on ψ. +However, for a given ψ∗, by replacing X with Xψ∗ = UΨ∗+ +1 +||ν ˆτ ||2 +2 ν ˆτνT +ˆτ X+ +1 +||a||2 +2 aaT X+BBT X, +we can calculate Sψ∗ using the method of Jewell et al. (2021). We can then calculate a p-value +conditional on ψ∗: +pψ∗ = Pr(|φ| ≥ |νT +ˆτ Xobs| ∩ φ ∈ Sψ∗) +Pr(φ ∈ Sψ∗) +. +To estimate the overall p-value, we take N samples, {ψ(1), . . . , ψ(N)}, and calculate Sψ(j) for +each ψ(j). We then estimate the p-value as +Pr(|φ| ≥ |νT +ˆτ Xobs| ∩ φ ∈ S) +Pr(φ ∈ S) +≈ +1 +N +�N +j=1 Pr(|φ| ≥ |νT +ˆτ Xobs| ∩ φ ∈ Sψ(j)) +1 +N +�N +j=1 Pr(φ ∈ Sψ(j)) += ˆpN. +(6) +This can also be written as a weighted average of individual p-value estimates +ˆpN = +1 +�N +j=1 wj +N +� +j=1 +wjpψ(j), +(7) +where wj = Pr(φ ∈ Sψ(j)). +As N → ∞ this Monte Carlo estimate will converge to the ideal selective p-value (4). However +for finite N it will not necessarily be a valid p-value, in that there is no guarantee that under the +11 + +null, and conditional on choosing to test the null, that the p-value will be uniformly distributed +on [0, 1]. +To see this, we simulated this Monte Carlo p-value for different values of N: see Figure 2(a) +and (b). We see that, in particular, there is a non-trivial probability that some ˆpψ(j) = 1: for some +values of ψ we have Sψ ⊂ {φ : |φ| ≥ |νT +ˆτ Xobs|}. Hence, we often get ˆpN = 1 if N is not sufficiently +large. +Remarkably, we can overcome these issues by just setting one of the ψ values to be the value for +the observed data. Remember that ψ = U T X. Let ψ(1) = U T Xobs, the value of ψ corresponding +to the observed data. Simulate ψ(2), . . . , ψ(N) independently from the null distribution for ψ, and +calculate the p-value as ˆpN in (6). +The following theorem shows that the resulting selective p-value will be distributed uniformly +on [0, 1] under the null, for any value of N. +Theorem 2. Let +ˆpN = +1 +�N +j=1 wj +N +� +j=1 +wjpψ(j), +where wj = Pr(φ ∈ Sψ(j)). Given that there is one j∗ ∈ {1, . . . , N} such that ψ(j∗) corresponds to +the observed data, and that other ψ(j) are drawn independently from their distribution under the +null, then under H0, ˆpN ∼ U(0, 1). +Figure 2 (c) and (d) show empirical validation of this result. An important consequence of this +result is that ˆpN is a valid p-value for any value of N, even if computational constraints limit N +to be small. Below, in Section 4, we show that even small to moderate values of N can lead to a +substantial increase in power. +3.4 +Extension to other null hypotheses +So far, we have calculated p-values for the change in mean model, based on the assumption that +there are no other changepoints within a fixed window h of ˆτ. +However, it is straightforward +to extend our method to cover a range of other scenarios, such as different null hypotheses and +different types of changepoint model. This will lead to different choices for φ and F(X), but as +long as we can define ψ and have a method for calculating Sψ, we can still use this method. We +outline some examples below. +For example, consider the null hypothesis that there are no changepoints between the detected +changepoints on either side of ˆτ. If we take ˆτj to be the changepoint of interest (for some j ∈ +{1, . . . , K}), then the null hypothesis is +H0 : +µˆτj−1+1 = · · · = µˆτj = µˆτj+1 = · · · = µˆτj+1, +12 + +where we take ˆτ0 = 0 and ˆτK+1 = T, and the alternative hypothesis is that there is at least one +inequality. The test statistic is νT +ˆτjX, where here ν ˆτj is defined as +� +ν ˆτj +� +t = +� +� +� +� +� +� +� +� +� +� +� +� +� +1 +ˆτj−ˆτj−1 +if ˆτj−1 < t ≤ ˆτj +− +1 +ˆτj+1−ˆτj +if ˆτj < t ≤ ˆτj+1 +0 +if t ≤ ˆτj−1 or t > ˆτj+1. +In this case, we fix the values of X outside of the window {ˆτj−1 + 1, . . . , ˆτj+1}, and the mean +of X within this window, and calculate U as before. The dimension of Ψ is (ˆτj+1 − ˆτj−1 −2). The +main difference is in the choice of F: since H0 depends on other changepoints as well as ˆτj, we +condition on all the changepoints in the model, not just ˆτ. So we replace the condition ˆτ ∈ M(X) +with M(X) = M(Xobs). The p-value is +p = Pr +φ +� +|φ| ≥ |νT +ˆτ Xobs| | M(X′(φ, ψ)) = M(Xobs) +� +. +This can be calculated in the same way as previously. +As another example, we consider the model of Chen et al. (2021). They use a model of the +form +Xt = ct + ϵt, t = 1, . . . , T, +where ϵt ∼ N(0, σ2), and ct = γct−1 + zt, with zt = 0 except at changepoints, and γ is assumed to +be known. +In the paper they develop a selective inference procedure similar to the methods in Jewell et al. +(2021), where they fix a window of size h around an estimated change ˆτ, and take as the null +hypothesis that there are no changes within this window: i.e. +(cˆτ−h+1, cˆτ−h+2, . . . , cˆτ, . . . , cˆτ+h) = +� +γ−h+1, γ−h+2, . . . , γ0, . . . , γh� +cˆτ. +This leads to a test statistic νT X, where ν is defined as +ν = +� +� +� +� +� +� +� +� +� +� +� +� +� +− +γ(γ2−1) +γ2−γ−2h+2 γt−ˆτ +if ˆτ − h < t ≤ ˆτ +γ2−1 +γ2h−1γt−ˆτ−1 +if ˆτ < t ≤ ˆτ + h +0 +if t ≤ ˆτ − h or t > ˆτ + h. +Letting φ = νT X as before and conditioning on ΠˆτX = ΠˆτXobs, they show how to calculate the +set S = {φ : ˆτ ∈ M(X′(φ))}, and so to calculate the p-value +Pr +� +φ ≥ νT Xobs|φ ∈ S +� +, +where X′(φ) is defined as in Equation 2. +13 + +As before, in our approach we condition on the values of X outside of the window (ˆτ −h+1, ˆτ + +h); this is equivalent to fixing the value of BT X where B is the T × (T − 2h) matrix obtained by +removing the columns corresponding to {ˆτ − h + 1, . . . , ˆτ + h} from the T × T identity matrix. We +also need to account for the unknown parameter cˆτ within this window. Letting a be defined as +at = +� +� +� +� +� +γt−ˆτ +if ˆτ − h + 1 ≤ t ≤ ˆτ + h +0 +otherwise, +we condition on ˆcˆτ = +1 +||a||2 aT X, which is a sufficient statistic for cˆτ. (It can be shown that a and +ν as defined here are orthogonal.) +Having defined ν, a and B, we can then write, as in Section 3.1), +X = ZX + +1 +||ν||2 +2 +ννT X + +� +1 +||a||2 +2 +aaT + BBT +� +Xobs, +where Z = I − +� +BBT + +1 +||a||2 +2 aaT + +1 +||ν ˆτ ||2 +2 ν ˆτνT +ˆτ +� +, and hence +X′(φ, ψ) = Uψ + +1 +||ν||2 +2 +νφ + +� +1 +||a||2 +2 +aaT + BBT +� +Xobs, +with the p-value +p = Pr +φ +� +φ ≥ νT +ˆτ Xobs|M(X′(φ, ψ)) = M(Xobs) +� +. +Since Chen et al. (2021) provides a method for computing S given ψ, it is then straightforward to +simulate ψ and calculate the estimated p-value as in Equation 6. +Algorithm 1 gives the algorithm to calculate p-values, as in Equation 6, for a general case. +To implement the algorithm, we first must select a changepoint algorithm and a null hypothesis +(e.g. that there are no changepoints within a window h of the estimated changepoint), and have a +method for calculating Sψ given both of these. The algorithm is then straightforward to implement. +4 +Power simulations +In Section 3.3, we showed that our p-value estimates were valid p-values under H0. In this section +we show that, under H1, our method has greater power to detect changepoints than that of +Jewell et al. (2021) for binary segmentation, wild binary segmentation, and L0 segmentation, +and we investigate how the power changes with window size h, size of change δ, and number of +samples N. All simulations are conducted in R; the code is available at https://github.com/ +rachelcarrington/changepointsR. +In each case, we set the number of data points T = 1000, and take σ2 = 1. For Figures 3 to 6, +we simulate from a model with a single change at t = 500, where the change is of size δ: we consider +14 + +Algorithm 1 General algorithm for calculating p-values. +Implement changepoint algorithm to obtain M(X). +j = 1. +ψ(1) = ψobs +Calculate Sψ(1). +Calculate p(1) = Pr(|φ| ≥ |νT +ˆτ Xobs| | φ ∈ Sψ(1)) and w(1) = Pr(φ ∈ Sψ(1)). +while j < N do +j = j + 1. +Sample ψ(j) ∼ N2h−2(0, σ2I). +Calculate Sψ(j). +Calculate p(j) and w(j). +end while +Calculate p-value estimate: +ˆpN = +1 +N +�N +j=1 w(j)p(j) +1 +N +�N +j=1 w(j) +. +δ = 1, 2, 3. In each case, we run a changepoint algorithm (binary segmentation or L0 segmentation) +to estimate changepoints, and calculate the p-value for the first detected changepoint. Simulations +where the changepoint algorithm returns no changepoints are discarded. In Figure 7 we simulate +from a model with 4 changes, at t = 100, 400, 500, 700, and test for changes at the first 4 estimated +changepoints. +Figure 3 shows QQ plots of p-values for binary segmentation, where we simulate from a model +with a single change of size δ. Figure 3 (a) shows that our test has power when we simulate from +H1, and the power increases with the number of samples N. In Figure 3 (b)-(d), p-values from +the method of Jewell et al. (2021) (equivalent to N = 1) are plotted against p-values from our +method with different values of N. The p-values generated from our method are generally smaller, +indicating increased power, particularly so as h and δ increase. Figure 4 shows plots of the power +for different values of h and δ: we see that initially increasing the number of Monte Carlo samples +N leads to substantial increases in power, but this levels off as N continues to increase, particularly +when δ is larger. +Figures 5 and 6 show equivalent plots for L0 segmentation, from which we make similar obser- +vations. +Figure 7 shows plots of the power when we simulate from a model with 4 changepoints, and fit a +changepoint model using binary segmentation and wild binary segmentation. The power increases +with N in a similar fashion for most changepoints. +15 + +0.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +P−value +U(0,1) +(a) +h = 10, δ = 1 +0.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +P−value +P−value (N = 1) +(b) +h = 10, δ = 1 +0.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +p−value +P−value (N = 1) +(c) +h = 50, δ = 1 +0.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +p−value +P−value (N = 1) +(d) +h = 10, δ = 2 +Figure 3: QQ plots of p-values for changepoints obtained using binary segmentation: h is the window +size and δ the size of the change in the model from which we simulate. In (a), p-values from our method +are plotted against theoretical quantiles from U(0, 1) for N = 1, 2, 5, 10, 20, 50. (b), (c) and (d) show QQ +plots of p-values calculated using our method (with N = 2, 5, 10, 20, 50) against p-values from the method +of Jewell et al. (2021) (equivalent to N = 1). +0.4 +0.6 +0.8 +1.0 +0 +5 +10 +15 +20 +Number of samples +Power +h = 10 +0.4 +0.6 +0.8 +1.0 +0 +5 +10 +15 +20 +Number of samples +Power +h = 20 +0.4 +0.6 +0.8 +1.0 +0 +5 +10 +15 +20 +Number of samples +Power +h = 50 +Figure 4: Rejection rates of H0 for binary segmentation, plotted against N. On each plot the three lines +show the proportion of samples where the p-value was below 0.05, leading H0 to be rejected. Each line +corresponds to a different size of change δ: green corresponds to δ = 3, blue to δ = 2, and red to δ = 1. +0.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +P−value +U(0,1) +(a) +h = 10, δ = 1 +0.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +P−value +P−value (N = 1) +(b) +h = 10, δ = 1 +0.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +P−value +P−value (N = 1) +(c) +h = 30, δ = 1 +0.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +P−value +P−value (N = 1) +(d) +h = 10, δ = 2 +Figure 5: QQ plot of p-values for L0 segmentation; h is the window size and δ the size of the change in the +model from which we simulate. In (a), p-values from our method are plotted against theoretical quantiles +from U(0, 1) for N = 1, 2, 5, 10, 20, 50. (b), (c) and (d) show QQ plots of p-values calculated using our +method (with N = 2, 5, 10, 20, 50) against p-values from the method of Jewell et al. (2021) (equivalent to +N = 1). +16 + +0.4 +0.6 +0.8 +1.0 +0 +5 +10 +15 +20 +Number of samples +Power +h = 10 +0.4 +0.6 +0.8 +1.0 +0 +5 +10 +15 +20 +Number of samples +Power +h = 20 +0.4 +0.6 +0.8 +1.0 +0 +5 +10 +15 +20 +Number of samples +Power +h = 30 +Figure 6: Rejection rates of H0 for L0 segmentation. On each plot the three lines show the proportion +of samples (of 1000 total) where the p-value was below 0.05, resulting in H0 being rejected, for different +values of N. Each line corresponds to a different size of change δ: green corresponds to δ = 3, blue to +δ = 2, and red to δ = 1. +0.25 +0.50 +0.75 +1.00 +0 +10 +20 +30 +40 +50 +Number of samples +Power +h = 10 +0.2 +0.4 +0.6 +0.8 +1.0 +0 +10 +20 +30 +40 +50 +Number of samples +Power +h = 20 +0.6 +0.8 +1.0 +0 +10 +20 +30 +40 +50 +Number of samples +Power +h = 30 +0.2 +0.4 +0.6 +0.8 +1.0 +0 +10 +20 +30 +40 +50 +Number of samples +Power +0.4 +0.6 +0.8 +1.0 +0 +10 +20 +30 +40 +50 +Number of samples +Power +0.7 +0.8 +0.9 +1.0 +0 +10 +20 +30 +40 +50 +Number of samples +Power +Figure 7: Rejection rates of H0 for different values of h, δ, and N, when we simulate from a model with 4 +changepoints, and apply binary segmentation (top row) or wild binary segmentation (bottom row) with 4 +changepoints. For each h, the process is run three times with changes of size δ = 1, 2, 3, which are shown +on the plots as red, blue, and green lines respectively. Each line corresponds to a changepoint. +17 + +6 +8 +10 +0 +500 +1000 +1500 +2000 +Position +GC content +6 +8 +10 +0 +500 +1000 +1500 +2000 +Position +GC content +Figure 8: Estimated changepoints in GC content data. Binary segmentation was used to estimate 38 +changepoints, and we set h = 10. Each vertical line corresponds to an estimated changepoint; changepoints +found to be significant at significance level α = 0.05 are shown in red, with others shown in grey. In the +top panel, we used N = 1 (equivalent to the method of Jewell et al. (2021)) to calculate p-values; in the +bottom panel, we used N = 10. (N = 50 is not shown as it gives the same results.) +5 +Application to genomic data +We now apply our method to genomic data consisting of GC content in 3kb windows along the +human chromosome. The data is available in the R package changepoint. As in Jewell et al. +(2021), only the first 2000 data points are used, and we set the number of changepoints to detect +to K = 38. For each changepoint, we calculate a p-value using both the method of Jewell et al. +(2021) (equivalent to our method with N = 1) and our method (with N = 10 and N = 50), using +a window size of h = 10. Figures 8 and 9 show plots of the data and estimated changepoints for +binary segmentation and L0 segmentation respectively, where changepoints with p-values smaller +than 0.05 are shown in red, and those with p-values above 0.05 in grey. In both cases, a greater +proportion of changepoints are deemed to be significant using our method, indicating that it has +greater power to detect changes. +Table 1 gives the number of significant changepoints found in each case. +For both binary +segmentation and L0 segmentation, we get a greater number of significant changepoints when +N = 10 than when N = 1. However, increasing N more than this does not lead to improved power +for binary segmentation. So in practice having only a moderate number of Monte Carlo samples +may be sufficient. +18 + +6 +8 +10 +0 +500 +1000 +1500 +2000 +Position +GC content +6 +8 +10 +0 +500 +1000 +1500 +2000 +Position +GC content +6 +8 +10 +0 +500 +1000 +1500 +2000 +Position +GC content +Figure 9: Estimated changepoints in GC content data. L0 segmentation was used to estimate 38 change- +points, and we set h = 10. Each vertical line corresponds to an estimated changepoint; changepoints found +to be significant at significance level α = 0.05 are shown in red, with others shown in grey. In the top +panel, we used N = 1 (equivalent to the method of Jewell et al. (2021)) to calculate p-values. The middle +and bottom panels show results obtained using N = 10 and N = 50 respectively. +Number of p-values < 0.05 +Number of p-values < 0.01 +N = 1 +N = 10 +N = 50 +N = 1 +N = 10 +N = 50 +BS +16 +26 +26 +10 +23 +23 +L0 +17 +23 +27 +12 +21 +22 +Table 1: Number of changepoints (out of 38) in GC content data with p-values below α = 0.05 and +α = 0.01. N = 1 is equivalent to the method of Jewell et al. (2021). +19 + +6 +Discussion +We have introduced a method for increasing the power of changepoint inference procedures, by +reducing the amount of information we condition on. We have shown that this method is effective +in increasing power compared to existing methods, both in simulated and real-world datasets. +Whilst our approach has been developed for changepoint problems, the general idea can be +applied to other scenarios such as clustering (Gao et al., 2022; Chen and Witten, 2022) or re- +gression tress (Neufeld et al., 2022). For example, current methods for post-selection inference +after clustering are based on a test statistic that compares the mean of the cluster, and fixed the +projection of the data that is orthogonal to this. However we could reduce this to conditioning +just on the sample mean of one of the clusters, and the data in the clusters that are not being +combined. Our method would then re-simulate the perturbations of each data point about its +clustered mean, apply the existing post-selection inference method to each dataset, and calculate +the weighted average as we do in this paper. We believe that this approach would have similar +properties, of being a valid p-value regardless of the Monte Carlo sample size, and of having larger +power as the Monte Carlo sample size increases. +Our approach for constructing valid p-values when using Monte Carlo to estimate selective +p-values (e.g. Saha et al., 2022) may also be applicable more widely. +References +Aston, J. A. and Kirch, C. (2012). +Evaluating stationarity via change-point alternatives with +applications to fMRI data. The Annals of Applied Statistics, 6(4):1906–1948. +Baranowski, R., Chen, Y., and Fryzlewicz, P. (2019). Narrowest-over-threshold detection of multi- +ple change points and change-point-like features. Journal of the Royal Statistical Society: Series +B (Statistical Methodology), 81(3):649–672. +Barrett, E., Bolz-Tereick, C. F., Killick, R., Mount, S., and Tratt, L. (2017). +Virtual ma- +chine warmup blows hot and cold. +Proceedings of the ACM on Programming Languages, +1(OOPSLA):1–27. +Berk, R., Brown, L., Buja, A., Zhang, K., and Zhao, L. (2013). Valid post-selection inference. The +Annals of Statistics, 41:802–837. +Braun, J. V., Braun, R., and M¨uller, H.-G. (2000). Multiple changepoint fitting via quasilikelihood, +with application to DNA sequence segmentation. Biometrika, 87(2):301–314. +20 + +Chen, Y. T., Jewell, S. W., and Witten, D. M. (2021). Quantifying uncertainty in spikes estimated +from calcium imaging data. arXiv:2103.07818. +Chen, +Y. +T. +and +Witten, +D. +M. +(2022). +Selective +inference +for +k-means +clustering. +arXiv:2203.15267. +Dubey, P. and M¨uller, H.-G. (2020). Fr´echet change-point detection. The Annals of Statistics, +48(6):3312–3335. +Duy, V. N. L. and Takeuchi, I. (2021). More powerful conditional selective inference for generalized +lasso by parametric programming. arXiv:2105.04920. +Duy, V. N. L., Toda, H., Sugiyama, R., and Takeuchi, I. (2020). Computing valid p-value for +optimal changepoint by selective inference using dynamic programming. Advances in Neural +Information Processing Systems, 33:11356–11367. +Eichinger, B. and Kirch, C. (2018). A mosum procedure for the estimation of multiple random +change points. Bernoulli, 24(1):526–564. +Fearnhead, P. (2006). Exact and efficient Bayesian inference for multiple changepoint problems. +Statistics and Computing, 16(2):203–213. +Fearnhead, P. and Rigaill, G. (2019). Changepoint detection in the presence of outliers. Journal +of the American Statistical Association, 114(525):169–183. +Fearnhead, P. and Rigaill, G. (2020). Relating and comparing methods for detecting changes in +mean. Stat, 9(1):e291. +Fithian, W., Sun, D., and Taylor, J. (2014). +Optimal inference after model selection. +arXiv:1410.2597. +Frick, K., Munk, A., and Sieling, H. (2014). Multiscale change point inference. Journal of the +Royal Statistical Society: Series B (Statistical Methodology), 76(3):495–580. +Fryzlewicz, P. (2014). Wild binary segmentation for multiple change-point detection. The Annals +of Statistics, 42(6):2243–2281. +Fryzlewicz, P. (2020). Narrowest significance pursuit: inference for multiple change-points in linear +models. arXiv:2009.05431. +Fryzlewicz, P. (2021). Robust narrowest significance pursuit: inference for multiple change-points +in the median. arXiv:2109.02487. +21 + +Gao, L. L., Bien, J., and Witten, D. (2022). Selective inference for hierarchical clustering. Journal +of the American Statistical Association, 00(0):1–27. +Hao, N., Niu, Y. S., and Zhang, H. (2013). Multiple change-point detection via a screening and +ranking algorithm. Statistica Sinica, 23(4):1553. +Heard, N. A. and Turcotte, M. J. (2014). Monitoring a device in a communication network. In +Data Analysis for Network Cyber-security, pages 151–188. World Scientific. +Hyun, S., G’Sell, M., and Tibshirani, R. J. (2018). Exact post-selection inference for the generalized +lasso path. Electronic Journal of Statistics, 12(1):1053–1097. +Hyun, S., Lin, K. Z., G’Sell, M., and Tibshirani, R. J. (2021). +Post-selection inference for +changepoint detection algorithms with application to copy number variation data. Biometrics, +77(3):1037–1049. +Jewell, S., Fearnhead, P., and Witten, D. (2021). Testing for a change in mean after changepoint +detection. To appear in Journal of the Royal Statistical Society, Series B (Statistical Methodol- +ogy). +Jewell, S. W., Hocking, T. D., Fearnhead, P., and Witten, D. M. (2020). Fast nonconvex deconvo- +lution of calcium imaging data. Biostatistics, 21(4):709–726. +Killick, R., Fearnhead, P., and Eckley, I. A. (2012). Optimal detection of changepoints with a +linear computational cost. Journal of the American Statistical Association, 107(500):1590–1598. +Kim, S.-J., Koh, K., Boyd, S., and Gorinevsky, D. (2009). +l1 trend filtering. +SIAM review, +51(2):339–360. +Kuchibhotla, A. K., Kolassa, J. E., and Kuffner, T. A. (2022). Post-selection inference. Annual +Review of Statistics and Its Application, 9:505–527. +Li, H., Munk, A., and Sieling, H. (2016). FDR-control in multiscale change-point segmentation. +Electronic Journal of Statistics, 10(1):918–959. +Liu, K., Markovic, J., and Tibshirani, R. (2018). More powerful post-selection inference, with +application to the lasso. arXiv:1801.09037. +Maidstone, R., Hocking, T., Rigaill, G., and Fearnhead, P. (2017). On optimal multiple changepoint +algorithms for large data. Statistics and Computing, 27(2):519–533. +22 + +Maleki, S., Bingham, C., and Zhang, Y. (2016). Development and realization of changepoint anal- +ysis for the detection of emerging faults on industrial systems. IEEE Transactions on Industrial +Informatics, 12(3):1180–1187. +Meier, A., Kirch, C., and Cho, H. (2021). mosum: A package for moving sums in change-point +analysis. Journal of Statistical Software, 97:1–42. +Neufeld, A. C., Gao, L. L., and Witten, D. M. (2022). Tree-values: selective inference for regression +trees. Journal of Machine Learning Research, 23(305):1–43. +Olshen, A. B., Venkatraman, E. S., Lucito, R., and Wigler, M. (2004). Circular binary segmentation +for the analysis of array-based DNA copy number data. Biostatistics, 5(4):557–572. +Pein, F., Sieling, H., and Munk, A. (2017). Heterogeneous change point inference. Journal of the +Royal Statistical Society: Series B (Statistical Methodology), 79(4):1207–1227. +Reeves, J., Chen, J., Wang, X. L., Lund, R., and Lu, Q. Q. (2007). +A review and compari- +son of changepoint detection techniques for climate data. Journal of Applied Meteorology and +Climatology, 46(6):900–915. +Rinaldo, A., Wasserman, L., and G’Sell, M. (2019). Bootstrapping and sample splitting for high- +dimensional, assumption-lean inference. The Annals of Statistics, 47(6):3438–3469. +Saha, A., Witten, D., and Bien, J. (2022). Inferring independent sets of Gaussian variables after +thresholding correlations. arXiv:2211.01521. +Scott, A. J. and Knott, M. (1974). A cluster analysis method for grouping means in the analysis +of variance. Biometrics, 30:507–512. +Shi, X., Beaulieu, C., Killick, R., and Lund, R. (2022a). Changepoint detection: An analysis of +the central England temperature series. Journal of Climate, 35(19):2729–2742. +Shi, X., Gallagher, C., Lund, R., and Killick, R. (2022b). +A comparison of single and multi- +ple changepoint techniques for time series data. +Computational Statistics & Data Analysis, +170:107433. +Song, H. and Chen, H. (2022). Asymptotic distribution-free changepoint detection for data with +repeated observations. Biometrika, 109(3):783–798. +Tibshirani, R., Saunders, M., Rosset, S., Zhu, J., and Knight, K. (2005). Sparsity and smoothness +via the fused lasso. Journal of the Royal Statistical Society: Series B (Statistical Methodology), +67(1):91–108. +23 + +Tibshirani, R. J. (2014). Adaptive piecewise polynomial estimation via trend filtering. The Annals +of Statistics, 42(1):285–323. +Truong, C., Oudre, L., and Vayatis, N. (2020). Selective review of offline change point detection +methods. Signal Processing, 167:107299. +Wang, D., Yu, Y., and Rinaldo, A. (2021). Optimal change point detection and localization in +sparse dynamic networks. The Annals of Statistics, 49(1):203–232. +Wang, T. and Samworth, R. J. (2018). +High dimensional change point estimation via sparse +projection. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 80(1):57– +83. +Zhao, S., Witten, D., and Shojaie, A. (2021). In defense of the indefensible: A very naive approach +to high-dimensional inference. Statistical Science, 36(4):562–577. +24 + +A +Proofs +A.1 +Proof of Theorem 1 +Proof. Fix α, and consider maximising Pr(P ∗ ≤ α). This corresponds to choosing a rejection +region Rα for (φ, ψ) that maximises +� +Rα +˜k(|φ|)f(φ) +2h−2 +� +i=1 +g(ψi)dφdψ, +subject to +� +Rα +f(φ) +2h−2 +� +i=1 +g(ψi)dφdψ ≤ α +� +S +f(φ) +2h−2 +� +i=1 +g(ψi)dφdψ. +As ˜k is an increasing, it is straightforward that this is achieved for the region +Rα = {(φ, ψ) : |φ| ≥ cα}, +with cα defined by +� +Rα +f(φ) +2h−2 +� +i=1 +g(ψi)dφdψ = α +� +S +f(φ) +2h−2 +� +i=1 +g(ψi)dφdψ. +This is precisely the form of PI, as PI ≤ α corresponds to |φ| ≥ cα. The result follows directly. +A.2 +Proof of Theorem 2 +Proof. The p-value, ˆpN, is invariant to shuffling the labels of the ψ(j)s. Let ψ(1:N) denote the set +of ψ(j) values after shuffling, and I the label of ψ(j) that corresponds to the observed data. +The proof follows by calculating Pr(ˆpN > α|ψ(1:N)). This requires calculating the distribution +of φ given ψ(1:N). +As before, let f and g represent the pdfs under the null of φ and each component of ψ, +respectively. Then +f(φ, ψ(1:N), I|S) ∝ f(φ) +� N +� +i=1 +g(ψ(i)) +� +I{φ∈Sψ(I)}. +If we condition on ψ(1:N), then we get +f(φ, I|ψ(1:N), S) ∝ f(φ)I{φ∈Sψ(I)}. +To normalize this, evaluate +W = +N +� +i=1 +� +φ +f(φ)I{φ∈Sψ(I)}dφ = +N +� +i=1 +Pr(φ ∈ Sψ(i)) = +N +� +i=1 +wi, +where wi is as defined above. Thus, +f(φ, I|ψ(1:N), S) = 1 +W f(φ)I{φ∈Sψ(I)}. +25 + +We can now marginalise out I to get +f(φ|ψ(1:N), S) = +N +� +I=1 +f(φ, I|ψ(1:N), S). +So, +f(φ|ψ(1:N), S) = 1 +W +N +� +I=1 +f(φ)I{φ∈Sψ(I)} = 1 +W +N +� +I=1 +wI +f(φ)I{φ∈Sψ(I)} +wI += 1 +W +N +� +I=1 +wIf(φ|φ ∈ Sψ(I)). +So +Pr(|φ| > α|ψ(1:N), S) = 1 +W +N +� +I=1 +wi Pr(|φ| > α|ψ(I), S). +This is the form of ˆpN, but with α replaced by φobs. So by the probability inverse transform, ˆpN +will have a uniform distribution on [0, 1]. +26 + diff --git a/etE5T4oBgHgl3EQfgw9z/content/tmp_files/load_file.txt b/etE5T4oBgHgl3EQfgw9z/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..89d43e0bceab877639887b584975a090f40c8dc3 --- /dev/null +++ b/etE5T4oBgHgl3EQfgw9z/content/tmp_files/load_file.txt @@ -0,0 +1,1042 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf,len=1041 +page_content='Improving Power by Conditioning on Less in Post-selection Inference for Changepoints Rachel Carrington and Paul Fearnhead January 2023 Abstract Post-selection inference has recently been proposed as a way of quantifying uncertainty about detected changepoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' The idea is to run a changepoint detection algorithm, and then re-use the same data to perform a test for a change near each of the detected changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' By defining the p-value for the test appropriately, so that it is conditional on the information used to choose the test, this approach will produce valid p-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' We show how to improve the power of these procedures by conditioning on less information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' This gives rise to an ideal selective p-value that is intractable but can be approximated by Monte Carlo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' We show that for any Monte Carlo sample size, this procedure produces valid p-values, and empirically that noticeable increase in power is possible with only very modest Monte Carlo sample sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Our procedure is easy to implement given existing post-selection inference methods, as we just need to generate perturbations of the data set and re-apply the post-selection method to each of these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' On genomic data consisting of human GC content, our procedure increases the number of significant changepoints that are detected from e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' 17 to 27, when compared to the method of Jewell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Keywords: Binary segmentation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Breakpoint;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Fused Lasso;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Penalised likelihood;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Selective p-value;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' 1 Introduction Detecting abrupt changes in time-series, or other ordered, data has been one of the most active research areas of the past decade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' It has applications in bioinformatics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Braun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Olshen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', 2004), computer performance (Barrett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', 2017), climate science (Reeves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', 2022a), cyber security (Heard and Turcotte, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Fearnhead and Rigaill, 2019), neuroscience (Aston and Kirch, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Jewell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', 2020), and industrial process monitoring (Maleki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', 2016) amongst many others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' There has been a wide range of methods that have been 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='05636v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='ME] 13 Jan 2023 proposed, dealing with detecting different types of change, such as change in mean, variance or slope;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' different algorithms for searching for multiple changepoints, including binary segmentation and its variants (Olshen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Fryzlewicz, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Baranowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', 2019), moving window methods (Hao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Eichinger and Kirch, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Meier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', 2021), L1 penalised regression methods (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Tibshirani, 2014), and dynamic programming approaches to maximising an L0 penalised likelihood (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Killick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Maidstone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', 2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' and for different types of data, such as high-dimensional data (Wang and Samworth, 2018), network data (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', 2021), and general non-Euclidean data (Song and Chen, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Dubey and M¨uller, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' See Truong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2020), Fearnhead and Rigaill (2020) and Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2022b) for an overview of this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' There has been much less work looking at quantifying the uncertainty of estimated change- points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Whilst Bayesian methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Fearnhead, 2006) that sample from a posterior over the number and location of the changepoints naturally give measures of uncertainty, assessing un- certainty for non-Bayesian methods is more challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Current work in this area includes the SMUCE method (Frick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Pein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', 2017), and global methods that try to give regions that produce sets of intervals, all of which must include a change at a pre-specified significance level (Fryzlewicz, 2020, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' A different approach is to try and assign a measure of significance to each detected changepoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' The challenge here is to avoid so-called double peeking at the data (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', 2021), where you use the same data both to detect a change and then to test for the change, as a naive implementation of test based on using the same data twice will be invalid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' This is because, in the absence of any change, the detection process will bias you to performing tests that are more likely to have small p-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' This results in tests where the p-values are neither uniform, nor stochastically bounded below by a uniform distribution (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Jewell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' One simple approach to circumvent this is sample splitting (Rinaldo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', 2019), where you use a proportion of the data to detect changes and the other other part to perform a test for each detected change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' However using only part of the data for each of detection and testing is sub-optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Instead post-selection inference ideas for regression (Berk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Fithian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Kuchibhotla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', 2022) have recently been applied to the changepoint setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' These allow the same data to be used for detection and testing, but with the p-values for each change being calculated conditional on information from the data that includes whatever information is used to choose the test that is being performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' These are called selective p-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Methods for calculating selective p-values have been developed for the change in mean problem with Gaussian noise and for a range of detection algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Hyun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2021) develop an 2 approach for binary segmentation and its variants, and for the fused lasso;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' while Jewell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2021) and Duy and Takeuchi (2021) propose methods that work if changes are detected using an L0 penalised likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Furthermore Jewell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2021) show how to improve on the method of Hyun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2021) by conditioning on less information when defining the selective p-value, and show that conditioning on less information can lead to a substantial increase in power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' There has also been recent work on post-selection inference beyond the change in mean problem (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='g Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Our work is motivated by further wanting to reduce the information that one conditions on when calculating the selective p-value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Current methods condition on the projection of the data that is orthogonal to the test statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' If, as is common, our test has a null hypothesis where, say, the mean of the data does not change within a region about the tested changepoint, then we can reduce this to conditioning on the data outside the region and an appropriate sufficient statistic, such as the sample mean, within the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' This, together with whatever aspect of the detected changes is used to pick the test, is the minimum amount of information that we need to condition on to make the selective p-value well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='2 we show that, for a natural class of distributions for the data under the alternative, the resulting selective p-value is optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Unfortunately we cannot directly calculate this p-value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Instead we propose a simple Monte Carlo approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' This is based on simulating new data within the region around the change- point that is being tested, applying the existing post-selection inference methodology to each such data set, and then calculating a weighted average of the selective p-values for each data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Im- portantly, we show that if one of the data sets we average over is the observed data, then this leads to a valid selective p-value, in that its distribution is uniform on [0, 1] under the null, regardless of the Monte Carlo sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Furthermore, it is simple to calculate provided we have a method that calculates a selective p-value based on conditioning on the projection of the data orthogonal to the test statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' As such, our method applies to all changepoint scenarios considered in Hyun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2018), Hyun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2021), Jewell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2021) and Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' We present empirical results that show one can obtain a noticeable improvement in power even with modest Monte Carlo sample sizes, say of the order of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' For a data set of GC content on human chromosome 1, this increased power leads to the number of significant changepoints that are detected increasing from 17 to 27, as compared to the method of Jewell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Whilst our method has been developed for the changepoint problem, the underlying ideas apply more widely, see Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' All proofs are given in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' 3 2 Background 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='1 Selective p-values for changepoints Suppose we have a dataset X = (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' , XT ) and we fit a changepoint model which consists of K changepoints M(X) = {ˆτ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' , ˆτK}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' We are interested in quantifying the level of uncertainty associated with these changepoints: how confident can we be that the changepoints we have found correspond to real changes and not false discoveries?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' One approach is to compute p-values for each changepoint of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' One aspect in quantifying uncertainty in this way is deciding what we mean by τ being a changepoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Or more specifically, what null hypothesis do we want to test?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' In many applications we say that τ is a changepoint providing some aspect of the data (that we are interested in) changes at or close to τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' That is, the null hypothesis would be that there is no change in some region centered on τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Even once we have decided on the null hypothesis, naively applying a test for a change at each of ˆτ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' , ˆτK is not possible, as we have already used the data to detect changepoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' If the data contains no changes, we would expect any detected changepoint locations to be where, by chance, the patterns of the data are similar to patterns produced by a change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' This will bias the p-values (see Jewell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', 2021, for examples of this).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' To overcome this, we can correct the naive test to take account of the fact that we are using the data twice (Fithian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', 2014) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' This can be done by calculating a selective p-value, which uses the distribution of the test statistic under the null but also conditional on any information used to choose the test that we are performing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' To make the idea concrete let F(X) denote the information we want to condition on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' We have freedom over the choice of F(X), except that it must contain the information from the data that is used to choose the test we performing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' So, for example, if we choose to test that τ is a changepoint based only on the property that τ is one of the estimated changepoints, then F(X) must include the information τ ∈ M(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' The selective p-value is then Pr(T ≥ Tobs | F (X) = F (Xobs) ), for some test statistic T , and where we use Xobs and Tobs to denote the observed data and test statistic respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' The challenge is then how to calculate this selective p-value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Often this will require a careful choice of the information we condition on, both to make the selective p-value well defined, and also possible to calculate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='2 Selective p-value for change in mean For ease of presentation it is helpful to consider a specific example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' We will consider the univariate change in mean model, for which methods for calculating selective p-values have been developed by Hyun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2018), Hyun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2021), Jewell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2021) and Duy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' However, the ideas we introduce for increasing the power of post-selection inference can apply more widely (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' to the scenarios considered in Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' For the change in mean model, we assume the data is of the form Xt = µt + ϵt, t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' , T, where µt is piecewise constant, with µt+1 ̸= µt only at K changepoints τ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' , τK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' We assume that ϵt ∼iid N(0, σ2), with σ known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' We run a changepoint algorithm – for example binary segmentation (Scott and Knott, 1974), wild binary segmentation (Fryzlewicz, 2014), narrowest-over-threshold (Baranowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', 2019), fused lasso (Tibshirani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', 2005), or a penalised likelihood approach (Maidstone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', 2017) – and detect a set of changepoints {ˆτ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' , ˆτK}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' We now want to test for a change at a particular estimated changepoint, which for simplicity we will denote ˆτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' As mentioned above, for many applications a natural null hypothesis is that there is no change in mean close to ˆτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' There are various possible choices for what we mean by “close to”, but here we will assume that there is a pre-determined distance h that is appropriate for our application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Our null hypothesis is therefore H0 : µˆτ−h+1 = · · · = µˆτ = µˆτ+1 = · · · = µˆτ+h, with the alternative hypothesis being that there is at least one inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (Extensions to other choices of null hypothesis will be discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=') Let ν ˆτ to be a T-dimensional vector whose tth entry is (vˆτ)t = � � � � � � � � � � � � � 1 h if ˆτ − h < t ≤ ˆτ − 1 h if ˆτ < t ≤ ˆτ + h 0 if t ≤ ˆτ − h or t > ˆτ + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Then, under H0, and without conditioning on any information in the data, νT ˆτ X ∼ N � 0, 2σ2 h � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Under H1, where there is a changepoint at or near ˆτ, we would expect the mean of νT ˆτ X to be non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' We can therefore take the test statistic to be T = |νT ˆτ X|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' As above let M(X) = {ˆτ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' , ˆτK}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' The information used to choose the null hypothesis to test is that ˆτ ∈ M(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Thus for our selective p-value we need a conditioning event that includes this 5 information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Unfortunately it is not possible to just choose F(X) to be ˆτ ∈ M(X), because the probability of this event depends on parameters that are unknown under the null hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' To deal with this, current approaches (Jewell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Hyun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', 2021) condition also on the projection of the data that is orthogonal to ν ˆτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Denote this orthogonal projection by Πˆτ, then this leads to the selective p-value that is Pr(|νT ˆτ X| > |νT ˆτ Xobs| | ˆτ ∈ M(X), ΠˆτX = ΠˆτXobs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (1) While this is well-defined, calculating the required conditional distribution of νT ˆτ X is non- trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Hyun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2021) show that for binary segmentation or the fused lasso, if you condition on further information, namely the order in which the changepoints are detected and the esti- mated sign of the changepoint, then the conditional distribution will be a truncated Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Furthermore the truncation region can be calculated be solving a series of linear equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Motivated by intuition that conditioning on less information will improve power (Fithian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', 2018), Jewell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2021) shows how to reduce the amount of information condi- tioned on, by avoiding having to condition on the order and signs of the changepoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' As we are conditioning on the projection of the data orthogonal to νT ˆτ , X will be uniquely determined if, in addition, we known νT ˆτ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Let φ = νT ˆτ X, then we can define the set of possible data sets that are possible as we vary φ by X′(φ) = Xobs − 1 ||ν ˆτ||2 ν ˆτνT ˆτ Xobs + 1 ||ν ˆτ||2 ν ˆτφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2) If we define S = {φ : ˆτ ∈ M(X′(φ))}, then the p-value in Equation 1 is equal to Pr � |φ| ≥ |νT ˆτ Xobs| | ˆτ ∈ M(X′(φ)) � = Pr � |φ| ≥ |νT ˆτ Xobs| | φ ∈ S � , where, unconditionally, φ ∼ N(0, σ2||ν ˆτ||2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Jewell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2021) shows how the set S can be efficiently computed for changepoint methods including binary segmentation, L0 segmentation and the fused lasso;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' in each case S is a union of intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Their methods can also be extended to other similar changepoint algorithms such as wild binary segmentation and narrowest-over-threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' The method of Jewell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2021) leads to an increase in power compared to the approach of Hyun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2021), as it requires conditioning on less information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' However, we still condition on T − 1 parameters that are orthogonal to ν ˆτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' The method we propose further reduces the amount of information we need to condition on, leading to greater power to detect changepoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' 6 3 Conditioning on less information 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='1 The ideal selective p-value Instead of conditioning on ΠˆτX, we could consider just conditioning on the minimum amount of information to make the selective p-value well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Our null hypothesis fixes that µˆτ−h+1 = · · = µˆτ+h, so this contains no information about the mean for data points outside of {ˆτ − h + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' , ˆτ + h}, nor does it specify the mean within this window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Hence, as a minimum we need to condition on Xt = Xobs,t for t ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' , ˆτ − h, ˆτ + h + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' , T} 1 2h ˆτ+h � i=ˆτ−h+1 Xt = 1 2h ˆτ+h � i=ˆτ−h+1 Xobs,t, (3) that is the data outside of {ˆτ − h + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' , ˆτ + h}, and the sample mean of the data in this window (which is a sufficient statistic for the unknown, constant mean in the window).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' These have total dimension T − 2h + 1, so we gain an additional 2h − 2 degrees of freedom compared to the method in Jewell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Let B be the T × (T − 2h) matrix obtained by removing the columns corresponding to {ˆτ − h + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' , ˆτ + h} from the T × T identity matrix, and let a be a T-dimensional vector such that at = � � � � � 1 2h if t ∈ {ˆτ − h + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' , ˆτ + h} 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' The conditions in (3) are equivalent to: BT X = BT Xobs aT X = aT Xobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' We can rewrite X as X = X − � BBT + 1 ||a||2 2 aaT + 1 ||ν ˆτ||2 2 ν ˆτνT ˆτ � X + � BBT + 1 ||a||2 2 aaT + 1 ||ν ˆτ||2 2 ν ˆτνT ˆτ � X = � I − � BBT + 1 ||a||2 2 aaT + 1 ||ν ˆτ||2 2 ν ˆτνT ˆτ �� X + 1 ||ν ˆτ||2 2 ν ˆτνT ˆτ X + � BBT + 1 ||a||2 2 aaT � X = ZX + 1 ||ν ˆτ||2 2 ν ˆτνT ˆτ X + � BBT + 1 ||a||2 2 aaT � X, where Z = I − � BBT + 1 ||a||2 2 aaT + 1 ||ν ˆτ ||2 2 ν ˆτνT ˆτ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Z is a T × T matrix with rank 2h − 2, so ZX follows a degenerate multivariate normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' However, since Z is a symmetric matrix with all its non-zero eigenvalues equal to 1, we can write Z = UU T , where U is a T × (2h − 2) matrix with orthonormal columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Under H0, 7 U T X ∼ N(U T µ, σ2U T U) = N(0, σ2I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' The matrix U is not uniquely defined, but the choice of basis is arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' It can be found, for example, using the Singular Value Decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Let ψ = U T X and, as before, let φ = νT ˆτ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Then, given the information we are conditioning on, as we vary ψ and φ we get data X = X′(φ, ψ) = Uψ + 1 ||ν ˆτ||2 2 ν ˆτνT ˆτ φ + � 1 ||a||2 2 aaT + BBT � Xobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Furthermore, under the null and without conditioning on further aspects of the data, such as the estimated changepoints, φ ∼ N(0, 2σ2 h ) and ψi ∼iid N(0, σ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' The resulting selective p-value is Pφ,ψ � |φ| ≥ |νT ˆτ Xobs| | ˆτ ∈ M(X′(φ, ψ)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (4) As this p-value is obtained by conditioning on the least amount of information needed for it to be well-defined, we will call it the ideal p-value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='2 Intuition behind new selective p-value To understand the difference between the ideal selective p-value (4) and the p-value of Jewell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2021), we give a schematic comparison in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' To enable us to present a plot we have assumed that ψ is scalar, and have also used the probability inverse mapping to transform (φ, ψ) from independent normal to independent uniform on [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' With this mapping, and given the conditioning in (3), data sets correspond to points in (φ, ψ)- space, and under the null such points are uniform on the unit square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' The conditioning on ˆτ being a detected changepoint corresponds to restricting the possible set of (φ, ψ) values – to the non-grey area in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' We have plotted the (φ, ψ) value for the observed data by an a cross in the top row of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' The p-value of Jewell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2021) then fixes the ψ value so the conditional distribution of φ is uniform on the coloured line – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' all values that are consistent with detecting a change at ˆτ for that value of ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' The p-value is the probability of observing a more extreme value than that for the data – which is the proportion of the line that is red in the top left plot of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' By comparison, the p-value of (4) allows ψ to vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' It is thus the probability of observing a more extreme value of φ than that for the data over all possible (φ, ψ) values that are consistent with ˆτ being a detected change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' This is the proportion of the non-grey area that is red in the top left plot of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' If we generate the data by simulating a (φ, ψ) point uniformly in the non-grey region, then it is simple to show that the distribution of either p-value will be uniform on [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' To see why the ideal p-value (4) is to be preferred, we plot the set of (φ, ψ) values that would correspond to data with a selective p-value of less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='2 in the bottom row of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' For 8 φ ψ x φ ψ x φ ψ φ ψ Figure 1: Comparison of the p-value of Jewell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2021) (left-hand column) and the ideal p-value (right- hand column) for the case of a univariate ψ parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' We have used the probability inverse mapping to transform φ and ψ so that they are uniformly and independently distributed on [0, 1] under the prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' We view data sets as being a function of (φ, ψ), and the selective event – which corresponds to the information in the data used to choose the test – corresponds to a region of (φ, ψ) values (non-grey region in all plots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' The observed data corresponds to a specific (φ, ψ) value shown by a cross (top-row plots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' For the method of Jewell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2021), the p-value is the probability of observing a more extreme value of φ conditional on the observed ψ-value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' This is the proportion of the coloured line that is red in the top-right plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' For our method, the p-value is the (unconditional) probability of observing a more extreme value of φ: the proportion of the non-grey area that is red (top-right plot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' In the bottom row we show the data-sets, as represented by their (φ, ψ) value, that would give a selective p-value that is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='2 or lower (red region in both plots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' 9 both p-values these give regions whose area is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='2 of the non-grey area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' The difference is the shape of the regions, with the ideal p-value consisting of requiring just φ greater than some constant, whereas the p-value of Jewell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2021) has different regions for φ as we vary ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' The former will have more power if we have alternative hypotheses that, compared to the null, place increasing probability on larger values of |φ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' To make this precise, let A be the projection of the data we condition on (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Under the null, and conditional on A denote the density for (φ, ψ) as f(φ) 2h−2 � i=1 g(ψi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Consider alternative hypotheses that correspond to a density of (φ, ψ) of the form k(φ)f(φ) 2h−2 � i=1 g(ψi), (5) for some function k(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' That is, under the alternative hypothesis the distribution of φ is altered, and k(φ) represents the ratio of density between the alternative and the null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Conditional on A and (φ, ψ) ∈ S, define PI to be the p-value given by (4), and P ∗ be any other valid p-value, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' that satisfies that under the null Pr(P ∗ ≤ α) ≤ α, for all α ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Then under an alternative with density of the form (5) for a function k(φ) = ˜k(|φ|) with ˜k increas- ing, Pr(PI ≤ α) ≥ Pr(P ∗ ≤ α), for all α ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='3 Estimating p-values with sampling Unfortunately it is not possible to analytically calculate the ideal selective p-value (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Instead we will resort to using Monte Carlo to estimate it, under the assumption that we have a method for calculating the null distribution of φ given ψ – this would refer to any combination of type of change, choice of null hypothesis and method for detecting the changepoints for which current post-selection inference methods exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Let S = {(φ, ψ) : ˆτ ∈ M � X′(φ, ψ) � }, so the conditioning event for (4) corresponds to (φ, ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Furthermore, define Sψ = {φ : ˆτ ∈ M � X′(φ, ψ) � }, 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='00 P−value U(0,1) (a) h = 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='00 P−value U(0,1) (b) h = 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='00 P−value U(0,1) (c) h = 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='00 P−value U(0,1) (d) h = 20 Figure 2: QQ plots of p-value estimates, simulated under H0 with T = 1000, for different values of h and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' On each plot the ordered p-values obtained using different values of N (N = 1, 5, 10, 50) are plotted against theoretical quantiles from U(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' In (a) and (b) the p-values are calculated as in Equation 6, where all ψ(j)’s are simulated randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' In (c) and (d), we take ψ(1) = U T Xobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' If p-values are valid, the points should lie approximately along the line y = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' the set of φ values corresponding to data where we estimate ˆτ as a changepoint, for a given value of ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' These regions will depend on the algorithm used to estimate the changepoints, and we are assuming that for any given ψ value we can calculate Sψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' For example, Jewell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2021) show how to calculate these regions for the change in mean problem with changepoints estimated by binary segmentation, wild binary segmentation and L0 penalised likelihood methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Now that we have 2h − 2 additional parameters in ψ, the truncation region S becomes much more complicated to calculate explicitly, as the values of φ that yield ˆτ ∈ M(X′(φ)) depend on ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' However, for a given ψ∗, by replacing X with Xψ∗ = UΨ∗+ 1 ||ν ˆτ ||2 2 ν ˆτνT ˆτ X+ 1 ||a||2 2 aaT X+BBT X, we can calculate Sψ∗ using the method of Jewell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' We can then calculate a p-value conditional on ψ∗: pψ∗ = Pr(|φ| ≥ |νT ˆτ Xobs| ∩ φ ∈ Sψ∗) Pr(φ ∈ Sψ∗) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' To estimate the overall p-value, we take N samples, {ψ(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' , ψ(N)}, and calculate Sψ(j) for each ψ(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' We then estimate the p-value as Pr(|φ| ≥ |νT ˆτ Xobs| ∩ φ ∈ S) Pr(φ ∈ S) ≈ 1 N �N j=1 Pr(|φ| ≥ |νT ˆτ Xobs| ∩ φ ∈ Sψ(j)) 1 N �N j=1 Pr(φ ∈ Sψ(j)) = ˆpN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (6) This can also be written as a weighted average of individual p-value estimates ˆpN = 1 �N j=1 wj N � j=1 wjpψ(j), (7) where wj = Pr(φ ∈ Sψ(j)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' As N → ∞ this Monte Carlo estimate will converge to the ideal selective p-value (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' However for finite N it will not necessarily be a valid p-value, in that there is no guarantee that under the 11 null, and conditional on choosing to test the null, that the p-value will be uniformly distributed on [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' To see this, we simulated this Monte Carlo p-value for different values of N: see Figure 2(a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' We see that, in particular, there is a non-trivial probability that some ˆpψ(j) = 1: for some values of ψ we have Sψ ⊂ {φ : |φ| ≥ |νT ˆτ Xobs|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Hence, we often get ˆpN = 1 if N is not sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Remarkably, we can overcome these issues by just setting one of the ψ values to be the value for the observed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Remember that ψ = U T X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Let ψ(1) = U T Xobs, the value of ψ corresponding to the observed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Simulate ψ(2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' , ψ(N) independently from the null distribution for ψ, and calculate the p-value as ˆpN in (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' The following theorem shows that the resulting selective p-value will be distributed uniformly on [0, 1] under the null, for any value of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Let ˆpN = 1 �N j=1 wj N � j=1 wjpψ(j), where wj = Pr(φ ∈ Sψ(j)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Given that there is one j∗ ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' , N} such that ψ(j∗) corresponds to the observed data, and that other ψ(j) are drawn independently from their distribution under the null, then under H0, ˆpN ∼ U(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Figure 2 (c) and (d) show empirical validation of this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' An important consequence of this result is that ˆpN is a valid p-value for any value of N, even if computational constraints limit N to be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Below, in Section 4, we show that even small to moderate values of N can lead to a substantial increase in power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='4 Extension to other null hypotheses So far, we have calculated p-values for the change in mean model, based on the assumption that there are no other changepoints within a fixed window h of ˆτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' However, it is straightforward to extend our method to cover a range of other scenarios, such as different null hypotheses and different types of changepoint model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' This will lead to different choices for φ and F(X), but as long as we can define ψ and have a method for calculating Sψ, we can still use this method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' We outline some examples below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' For example, consider the null hypothesis that there are no changepoints between the detected changepoints on either side of ˆτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' If we take ˆτj to be the changepoint of interest (for some j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' , K}), then the null hypothesis is H0 : µˆτj−1+1 = · · · = µˆτj = µˆτj+1 = · · · = µˆτj+1, 12 where we take ˆτ0 = 0 and ˆτK+1 = T, and the alternative hypothesis is that there is at least one inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' The test statistic is νT ˆτjX, where here ν ˆτj is defined as � ν ˆτj � t = � � � � � � � � � � � � � 1 ˆτj−ˆτj−1 if ˆτj−1 < t ≤ ˆτj − 1 ˆτj+1−ˆτj if ˆτj < t ≤ ˆτj+1 0 if t ≤ ˆτj−1 or t > ˆτj+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' In this case, we fix the values of X outside of the window {ˆτj−1 + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' , ˆτj+1}, and the mean of X within this window, and calculate U as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' The dimension of Ψ is (ˆτj+1 − ˆτj−1 −2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' The main difference is in the choice of F: since H0 depends on other changepoints as well as ˆτj, we condition on all the changepoints in the model, not just ˆτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' So we replace the condition ˆτ ∈ M(X) with M(X) = M(Xobs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' The p-value is p = Pr φ � |φ| ≥ |νT ˆτ Xobs| | M(X′(φ, ψ)) = M(Xobs) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' This can be calculated in the same way as previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' As another example, we consider the model of Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' They use a model of the form Xt = ct + ϵt, t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' , T, where ϵt ∼ N(0, σ2), and ct = γct−1 + zt, with zt = 0 except at changepoints, and γ is assumed to be known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' In the paper they develop a selective inference procedure similar to the methods in Jewell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2021), where they fix a window of size h around an estimated change ˆτ, and take as the null hypothesis that there are no changes within this window: i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (cˆτ−h+1, cˆτ−h+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' , cˆτ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' , cˆτ+h) = � γ−h+1, γ−h+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' , γ0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' , γh� cˆτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' This leads to a test statistic νT X, where ν is defined as ν = � � � � � � � � � � � � � − γ(γ2−1) γ2−γ−2h+2 γt−ˆτ if ˆτ − h < t ≤ ˆτ γ2−1 γ2h−1γt−ˆτ−1 if ˆτ < t ≤ ˆτ + h 0 if t ≤ ˆτ − h or t > ˆτ + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Letting φ = νT X as before and conditioning on ΠˆτX = ΠˆτXobs, they show how to calculate the set S = {φ : ˆτ ∈ M(X′(φ))}, and so to calculate the p-value Pr � φ ≥ νT Xobs|φ ∈ S � , where X′(φ) is defined as in Equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' 13 As before, in our approach we condition on the values of X outside of the window (ˆτ −h+1, ˆτ + h);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' this is equivalent to fixing the value of BT X where B is the T × (T − 2h) matrix obtained by removing the columns corresponding to {ˆτ − h + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' , ˆτ + h} from the T × T identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' We also need to account for the unknown parameter cˆτ within this window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Letting a be defined as at = � � � � � γt−ˆτ if ˆτ − h + 1 ≤ t ≤ ˆτ + h 0 otherwise, we condition on ˆcˆτ = 1 ||a||2 aT X, which is a sufficient statistic for cˆτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (It can be shown that a and ν as defined here are orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=') Having defined ν, a and B, we can then write, as in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='1), X = ZX + 1 ||ν||2 2 ννT X + � 1 ||a||2 2 aaT + BBT � Xobs, where Z = I − � BBT + 1 ||a||2 2 aaT + 1 ||ν ˆτ ||2 2 ν ˆτνT ˆτ � , and hence X′(φ, ψ) = Uψ + 1 ||ν||2 2 νφ + � 1 ||a||2 2 aaT + BBT � Xobs, with the p-value p = Pr φ � φ ≥ νT ˆτ Xobs|M(X′(φ, ψ)) = M(Xobs) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Since Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2021) provides a method for computing S given ψ, it is then straightforward to simulate ψ and calculate the estimated p-value as in Equation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Algorithm 1 gives the algorithm to calculate p-values, as in Equation 6, for a general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' To implement the algorithm, we first must select a changepoint algorithm and a null hypothesis (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' that there are no changepoints within a window h of the estimated changepoint), and have a method for calculating Sψ given both of these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' The algorithm is then straightforward to implement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' 4 Power simulations In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='3, we showed that our p-value estimates were valid p-values under H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' In this section we show that, under H1, our method has greater power to detect changepoints than that of Jewell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2021) for binary segmentation, wild binary segmentation, and L0 segmentation, and we investigate how the power changes with window size h, size of change δ, and number of samples N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' All simulations are conducted in R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' the code is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='com/ rachelcarrington/changepointsR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' In each case, we set the number of data points T = 1000, and take σ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' For Figures 3 to 6, we simulate from a model with a single change at t = 500, where the change is of size δ: we consider 14 Algorithm 1 General algorithm for calculating p-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Implement changepoint algorithm to obtain M(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' j = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' ψ(1) = ψobs Calculate Sψ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Calculate p(1) = Pr(|φ| ≥ |νT ˆτ Xobs| | φ ∈ Sψ(1)) and w(1) = Pr(φ ∈ Sψ(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' while j < N do j = j + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Sample ψ(j) ∼ N2h−2(0, σ2I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Calculate Sψ(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Calculate p(j) and w(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' end while Calculate p-value estimate: ˆpN = 1 N �N j=1 w(j)p(j) 1 N �N j=1 w(j) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' δ = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' In each case, we run a changepoint algorithm (binary segmentation or L0 segmentation) to estimate changepoints, and calculate the p-value for the first detected changepoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Simulations where the changepoint algorithm returns no changepoints are discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' In Figure 7 we simulate from a model with 4 changes, at t = 100, 400, 500, 700, and test for changes at the first 4 estimated changepoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Figure 3 shows QQ plots of p-values for binary segmentation, where we simulate from a model with a single change of size δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Figure 3 (a) shows that our test has power when we simulate from H1, and the power increases with the number of samples N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' In Figure 3 (b)-(d), p-values from the method of Jewell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2021) (equivalent to N = 1) are plotted against p-values from our method with different values of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' The p-values generated from our method are generally smaller, indicating increased power, particularly so as h and δ increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Figure 4 shows plots of the power for different values of h and δ: we see that initially increasing the number of Monte Carlo samples N leads to substantial increases in power, but this levels off as N continues to increase, particularly when δ is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Figures 5 and 6 show equivalent plots for L0 segmentation, from which we make similar obser- vations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Figure 7 shows plots of the power when we simulate from a model with 4 changepoints, and fit a changepoint model using binary segmentation and wild binary segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' The power increases with N in a similar fashion for most changepoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='00 P−value U(0,1) (a) h = 10, δ = 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='00 P−value P−value (N = 1) (b) h = 10, δ = 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='00 p−value P−value (N = 1) (c) h = 50, δ = 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='00 p−value P−value (N = 1) (d) h = 10, δ = 2 Figure 3: QQ plots of p-values for changepoints obtained using binary segmentation: h is the window size and δ the size of the change in the model from which we simulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' In (a), p-values from our method are plotted against theoretical quantiles from U(0, 1) for N = 1, 2, 5, 10, 20, 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (b), (c) and (d) show QQ plots of p-values calculated using our method (with N = 2, 5, 10, 20, 50) against p-values from the method of Jewell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2021) (equivalent to N = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='0 0 5 10 15 20 Number of samples Power h = 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='0 0 5 10 15 20 Number of samples Power h = 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='0 0 5 10 15 20 Number of samples Power h = 50 Figure 4: Rejection rates of H0 for binary segmentation, plotted against N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' On each plot the three lines show the proportion of samples where the p-value was below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='05, leading H0 to be rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Each line corresponds to a different size of change δ: green corresponds to δ = 3, blue to δ = 2, and red to δ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='00 P−value U(0,1) (a) h = 10, δ = 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='00 P−value P−value (N = 1) (b) h = 10, δ = 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='00 P−value P−value (N = 1) (c) h = 30, δ = 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='00 P−value P−value (N = 1) (d) h = 10, δ = 2 Figure 5: QQ plot of p-values for L0 segmentation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' h is the window size and δ the size of the change in the model from which we simulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' In (a), p-values from our method are plotted against theoretical quantiles from U(0, 1) for N = 1, 2, 5, 10, 20, 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (b), (c) and (d) show QQ plots of p-values calculated using our method (with N = 2, 5, 10, 20, 50) against p-values from the method of Jewell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2021) (equivalent to N = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='0 0 5 10 15 20 Number of samples Power h = 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='0 0 5 10 15 20 Number of samples Power h = 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='0 0 5 10 15 20 Number of samples Power h = 30 Figure 6: Rejection rates of H0 for L0 segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' On each plot the three lines show the proportion of samples (of 1000 total) where the p-value was below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='05, resulting in H0 being rejected, for different values of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Each line corresponds to a different size of change δ: green corresponds to δ = 3, blue to δ = 2, and red to δ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='00 0 10 20 30 40 50 Number of samples Power h = 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='0 0 10 20 30 40 50 Number of samples Power h = 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='0 0 10 20 30 40 50 Number of samples Power h = 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='0 0 10 20 30 40 50 Number of samples Power 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='0 0 10 20 30 40 50 Number of samples Power 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='0 0 10 20 30 40 50 Number of samples Power Figure 7: Rejection rates of H0 for different values of h, δ, and N, when we simulate from a model with 4 changepoints, and apply binary segmentation (top row) or wild binary segmentation (bottom row) with 4 changepoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' For each h, the process is run three times with changes of size δ = 1, 2, 3, which are shown on the plots as red, blue, and green lines respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Each line corresponds to a changepoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' 17 6 8 10 0 500 1000 1500 2000 Position GC content 6 8 10 0 500 1000 1500 2000 Position GC content Figure 8: Estimated changepoints in GC content data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Binary segmentation was used to estimate 38 changepoints, and we set h = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Each vertical line corresponds to an estimated changepoint;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' changepoints found to be significant at significance level α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='05 are shown in red, with others shown in grey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' In the top panel, we used N = 1 (equivalent to the method of Jewell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2021)) to calculate p-values;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' in the bottom panel, we used N = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (N = 50 is not shown as it gives the same results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=') 5 Application to genomic data We now apply our method to genomic data consisting of GC content in 3kb windows along the human chromosome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' The data is available in the R package changepoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' As in Jewell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2021), only the first 2000 data points are used, and we set the number of changepoints to detect to K = 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' For each changepoint, we calculate a p-value using both the method of Jewell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2021) (equivalent to our method with N = 1) and our method (with N = 10 and N = 50), using a window size of h = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Figures 8 and 9 show plots of the data and estimated changepoints for binary segmentation and L0 segmentation respectively, where changepoints with p-values smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='05 are shown in red, and those with p-values above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='05 in grey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' In both cases, a greater proportion of changepoints are deemed to be significant using our method, indicating that it has greater power to detect changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Table 1 gives the number of significant changepoints found in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' For both binary segmentation and L0 segmentation, we get a greater number of significant changepoints when N = 10 than when N = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' However, increasing N more than this does not lead to improved power for binary segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' So in practice having only a moderate number of Monte Carlo samples may be sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' 18 6 8 10 0 500 1000 1500 2000 Position GC content 6 8 10 0 500 1000 1500 2000 Position GC content 6 8 10 0 500 1000 1500 2000 Position GC content Figure 9: Estimated changepoints in GC content data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' L0 segmentation was used to estimate 38 change- points, and we set h = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Each vertical line corresponds to an estimated changepoint;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' changepoints found to be significant at significance level α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='05 are shown in red, with others shown in grey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' In the top panel, we used N = 1 (equivalent to the method of Jewell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2021)) to calculate p-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' The middle and bottom panels show results obtained using N = 10 and N = 50 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Number of p-values < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='05 Number of p-values < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='01 N = 1 N = 10 N = 50 N = 1 N = 10 N = 50 BS 16 26 26 10 23 23 L0 17 23 27 12 21 22 Table 1: Number of changepoints (out of 38) in GC content data with p-values below α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='05 and α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' N = 1 is equivalent to the method of Jewell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' 19 6 Discussion We have introduced a method for increasing the power of changepoint inference procedures, by reducing the amount of information we condition on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' We have shown that this method is effective in increasing power compared to existing methods, both in simulated and real-world datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Whilst our approach has been developed for changepoint problems, the general idea can be applied to other scenarios such as clustering (Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Chen and Witten, 2022) or re- gression tress (Neufeld et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' For example, current methods for post-selection inference after clustering are based on a test statistic that compares the mean of the cluster, and fixed the projection of the data that is orthogonal to this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' However we could reduce this to conditioning just on the sample mean of one of the clusters, and the data in the clusters that are not being combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Our method would then re-simulate the perturbations of each data point about its clustered mean, apply the existing post-selection inference method to each dataset, and calculate the weighted average as we do in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' We believe that this approach would have similar properties, of being a valid p-value regardless of the Monte Carlo sample size, and of having larger power as the Monte Carlo sample size increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Our approach for constructing valid p-values when using Monte Carlo to estimate selective p-values (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Saha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', 2022) may also be applicable more widely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' References Aston, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' and Kirch, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Evaluating stationarity via change-point alternatives with applications to fMRI data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' The Annals of Applied Statistics, 6(4):1906–1948.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Baranowski, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', and Fryzlewicz, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Narrowest-over-threshold detection of multi- ple change points and change-point-like features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Journal of the Royal Statistical Society: Series B (Statistical Methodology), 81(3):649–672.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Barrett, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', Bolz-Tereick, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', Killick, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', Mount, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', and Tratt, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Virtual ma- chine warmup blows hot and cold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Proceedings of the ACM on Programming Languages, 1(OOPSLA):1–27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Berk, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', Brown, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', Buja, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', Zhang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', and Zhao, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Valid post-selection inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' The Annals of Statistics, 41:802–837.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Braun, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', Braun, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', and M¨uller, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Multiple changepoint fitting via quasilikelihood, with application to DNA sequence segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Biometrika, 87(2):301–314.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' 20 Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', Jewell, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', and Witten, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Quantifying uncertainty in spikes estimated from calcium imaging data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' arXiv:2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='07818.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' and Witten, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Selective inference for k-means clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='15267.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Dubey, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' and M¨uller, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Fr´echet change-point detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' The Annals of Statistics, 48(6):3312–3335.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Duy, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' and Takeuchi, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' More powerful conditional selective inference for generalized lasso by parametric programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' arXiv:2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='04920.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Duy, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', Toda, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', Sugiyama, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', and Takeuchi, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Computing valid p-value for optimal changepoint by selective inference using dynamic programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 33:11356–11367.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Eichinger, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' and Kirch, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' A mosum procedure for the estimation of multiple random change points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Bernoulli, 24(1):526–564.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Fearnhead, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Exact and efficient Bayesian inference for multiple changepoint problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Statistics and Computing, 16(2):203–213.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Fearnhead, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' and Rigaill, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Changepoint detection in the presence of outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Journal of the American Statistical Association, 114(525):169–183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Fearnhead, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' and Rigaill, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Relating and comparing methods for detecting changes in mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Stat, 9(1):e291.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Fithian, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', Sun, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', and Taylor, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Optimal inference after model selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' arXiv:1410.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='2597.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Frick, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', Munk, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', and Sieling, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Multiscale change point inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Journal of the Royal Statistical Society: Series B (Statistical Methodology), 76(3):495–580.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Fryzlewicz, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Wild binary segmentation for multiple change-point detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' The Annals of Statistics, 42(6):2243–2281.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Fryzlewicz, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Narrowest significance pursuit: inference for multiple change-points in linear models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' arXiv:2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='05431.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Fryzlewicz, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Robust narrowest significance pursuit: inference for multiple change-points in the median.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' arXiv:2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='02487.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' 21 Gao, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', Bien, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', and Witten, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Selective inference for hierarchical clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Journal of the American Statistical Association, 00(0):1–27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Hao, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', Niu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', and Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Multiple change-point detection via a screening and ranking algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Statistica Sinica, 23(4):1553.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Heard, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' and Turcotte, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Monitoring a device in a communication network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' In Data Analysis for Network Cyber-security, pages 151–188.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' World Scientific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Hyun, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', G’Sell, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', and Tibshirani, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Exact post-selection inference for the generalized lasso path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Electronic Journal of Statistics, 12(1):1053–1097.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Hyun, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', Lin, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', G’Sell, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', and Tibshirani, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Post-selection inference for changepoint detection algorithms with application to copy number variation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Biometrics, 77(3):1037–1049.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Jewell, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', Fearnhead, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', and Witten, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Testing for a change in mean after changepoint detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' To appear in Journal of the Royal Statistical Society, Series B (Statistical Methodol- ogy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Jewell, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', Hocking, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', Fearnhead, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', and Witten, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Fast nonconvex deconvo- lution of calcium imaging data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Biostatistics, 21(4):709–726.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Killick, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', Fearnhead, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', and Eckley, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Optimal detection of changepoints with a linear computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Journal of the American Statistical Association, 107(500):1590–1598.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Kim, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', Koh, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', Boyd, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', and Gorinevsky, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' l1 trend filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' SIAM review, 51(2):339–360.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Kuchibhotla, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', Kolassa, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', and Kuffner, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Post-selection inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Annual Review of Statistics and Its Application, 9:505–527.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', Munk, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', and Sieling, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' FDR-control in multiscale change-point segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Electronic Journal of Statistics, 10(1):918–959.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Liu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', Markovic, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', and Tibshirani, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' More powerful post-selection inference, with application to the lasso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' arXiv:1801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='09037.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Maidstone, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', Hocking, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', Rigaill, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', and Fearnhead, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' On optimal multiple changepoint algorithms for large data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Statistics and Computing, 27(2):519–533.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' 22 Maleki, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', Bingham, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', and Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Development and realization of changepoint anal- ysis for the detection of emerging faults on industrial systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' IEEE Transactions on Industrial Informatics, 12(3):1180–1187.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Meier, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', Kirch, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', and Cho, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' mosum: A package for moving sums in change-point analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Journal of Statistical Software, 97:1–42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Neufeld, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', Gao, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', and Witten, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Tree-values: selective inference for regression trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Journal of Machine Learning Research, 23(305):1–43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Olshen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', Venkatraman, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', Lucito, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', and Wigler, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Circular binary segmentation for the analysis of array-based DNA copy number data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Biostatistics, 5(4):557–572.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Pein, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', Sieling, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', and Munk, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Heterogeneous change point inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Journal of the Royal Statistical Society: Series B (Statistical Methodology), 79(4):1207–1227.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Reeves, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', Chen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', Lund, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', and Lu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' A review and compari- son of changepoint detection techniques for climate data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Journal of Applied Meteorology and Climatology, 46(6):900–915.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Rinaldo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', Wasserman, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', and G’Sell, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Bootstrapping and sample splitting for high- dimensional, assumption-lean inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' The Annals of Statistics, 47(6):3438–3469.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Saha, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', Witten, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', and Bien, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Inferring independent sets of Gaussian variables after thresholding correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='01521.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Scott, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' and Knott, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (1974).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' A cluster analysis method for grouping means in the analysis of variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Biometrics, 30:507–512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Shi, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', Beaulieu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', Killick, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', and Lund, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Changepoint detection: An analysis of the central England temperature series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Journal of Climate, 35(19):2729–2742.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Shi, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', Gallagher, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', Lund, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', and Killick, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' A comparison of single and multi- ple changepoint techniques for time series data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Computational Statistics & Data Analysis, 170:107433.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Song, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' and Chen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Asymptotic distribution-free changepoint detection for data with repeated observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Biometrika, 109(3):783–798.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Tibshirani, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', Saunders, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', Rosset, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', Zhu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', and Knight, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Sparsity and smoothness via the fused lasso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Journal of the Royal Statistical Society: Series B (Statistical Methodology), 67(1):91–108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' 23 Tibshirani, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Adaptive piecewise polynomial estimation via trend filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' The Annals of Statistics, 42(1):285–323.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Truong, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', Oudre, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', and Vayatis, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Selective review of offline change point detection methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Signal Processing, 167:107299.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Wang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', Yu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', and Rinaldo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Optimal change point detection and localization in sparse dynamic networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' The Annals of Statistics, 49(1):203–232.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Wang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' and Samworth, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' High dimensional change point estimation via sparse projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Journal of the Royal Statistical Society: Series B (Statistical Methodology), 80(1):57– 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Zhao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', Witten, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=', and Shojaie, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' In defense of the indefensible: A very naive approach to high-dimensional inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Statistical Science, 36(4):562–577.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' 24 A Proofs A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='1 Proof of Theorem 1 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Fix α, and consider maximising Pr(P ∗ ≤ α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' This corresponds to choosing a rejection region Rα for (φ, ψ) that maximises � Rα ˜k(|φ|)f(φ) 2h−2 � i=1 g(ψi)dφdψ, subject to � Rα f(φ) 2h−2 � i=1 g(ψi)dφdψ ≤ α � S f(φ) 2h−2 � i=1 g(ψi)dφdψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' As ˜k is an increasing, it is straightforward that this is achieved for the region Rα = {(φ, ψ) : |φ| ≥ cα}, with cα defined by � Rα f(φ) 2h−2 � i=1 g(ψi)dφdψ = α � S f(φ) 2h−2 � i=1 g(ψi)dφdψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' This is precisely the form of PI, as PI ≤ α corresponds to |φ| ≥ cα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' The result follows directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content='2 Proof of Theorem 2 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' The p-value, ˆpN, is invariant to shuffling the labels of the ψ(j)s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Let ψ(1:N) denote the set of ψ(j) values after shuffling, and I the label of ψ(j) that corresponds to the observed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' The proof follows by calculating Pr(ˆpN > α|ψ(1:N)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' This requires calculating the distribution of φ given ψ(1:N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' As before, let f and g represent the pdfs under the null of φ and each component of ψ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Then f(φ, ψ(1:N), I|S) ∝ f(φ) � N � i=1 g(ψ(i)) � I{φ∈Sψ(I)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' If we condition on ψ(1:N), then we get f(φ, I|ψ(1:N), S) ∝ f(φ)I{φ∈Sψ(I)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' To normalize this, evaluate W = N � i=1 � φ f(φ)I{φ∈Sψ(I)}dφ = N � i=1 Pr(φ ∈ Sψ(i)) = N � i=1 wi, where wi is as defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' Thus, f(φ, I|ψ(1:N), S) = 1 W f(φ)I{φ∈Sψ(I)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' 25 We can now marginalise out I to get f(φ|ψ(1:N), S) = N � I=1 f(φ, I|ψ(1:N), S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' So, f(φ|ψ(1:N), S) = 1 W N � I=1 f(φ)I{φ∈Sψ(I)} = 1 W N � I=1 wI f(φ)I{φ∈Sψ(I)} wI = 1 W N � I=1 wIf(φ|φ ∈ Sψ(I)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' So Pr(|φ| > α|ψ(1:N), S) = 1 W N � I=1 wi Pr(|φ| > α|ψ(I), S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' This is the form of ˆpN, but with α replaced by φobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' So by the probability inverse transform, ˆpN will have a uniform distribution on [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} +page_content=' 26' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE5T4oBgHgl3EQfgw9z/content/2301.05636v1.pdf'} diff --git a/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf b/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf new file mode 100644 index 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Ex- +isting transfer learning approaches require much +information of the target PDEs such as its formu- +lation and/or data of its solution for pre-training. +In this work, we propose to construct transfer- +able neural feature spaces from purely function +approximation perspectives without using PDE +information. The construction of the feature space +involves re-parameterization of the hidden neu- +rons and uses auxiliary functions to tune the re- +sulting feature space. Theoretical analysis shows +the high quality of the produced feature space, +i.e., uniformly distributed neurons. Extensive nu- +merical experiments verify the outstanding perfor- +mance of our method, including significantly im- +proved transferability, e.g., using the same feature +space for various PDEs with different domains +and boundary conditions, and the superior accu- +racy, e.g., several orders of magnitude smaller +mean squared error than the state of the art meth- +ods. +1 +Introduction +Rapid advancement of deep learning has attracted signif- +icant attention of researchers to explore how to use deep +* Corresponding author +1Department of Mathematics, Florida State University, Tallahassee, +FL 32306, USA 2Department of Mathematics, University of South +Carolina, Columbia, SC 29208, USA 3Computer Science and +Mathematics Division, Oak Ridge National Laboratory, TN 37831, +USA. Correspondence to: Guannan Zhang . +Notice: This manuscript has been authored by Oak Ridge Na- +tional Laboratory, managed by UT-Battelle, LLC, under contract +DE-AC05-00OR22725 with the US Department of Energy (DOE). +The US government retains and the publisher, by accepting the ar- +ticle for publication, acknowledges that the US government retains +a nonexclusive, paid-up, irrevocable, worldwide license to pub- +lish or reproduce the published form of this manuscript, or allow +others to do so, for US government purposes. DOE will provide +public access to these results of federally sponsored research in +accordance with the DOE Public Access Plan. +learning to solve scientific and engineering problems. Since +numerical solutions of partial differential equations (PDEs) +sits at the heart of many scientific areas, there is a surge of +studies on how to use neural networks to leverage data and +physical knowledge to solve PDEs (Raissi et al., 2019; E +& Yu, 2018; Long et al., 2018; Zang et al., 2020; Li et al., +2021a; 2020; Lu et al., 2021a; Gin et al., 2021; Zhang et al., +2021; Teng et al., 2022; Clark Di Leoni et al., 2023). The +neural network-based methods have several advantages over +traditional numerical methods (e.g., finite element, finite +difference and finite volume), such as avoiding the need for +numerical integration, generating differentiable solutions, +exploiting advanced computing capabilities, e.g., GPUs. +Nevertheless, a major drawback of these deep learning meth- +ods for solving PDEs is high computational cost associated +with the neural network training/retraining using stochastic +gradient descent (SGD). One of the popular strategies to +alleviate this issue is transfer learning. +Transfer learning for PDEs is to develop a pre-trained neu- +ral network that can be effectively re-used to solve a PDE +with multiple coefficients or in various domains, or to solve +multiple types of PDEs. When transfer a pre-trained neural +network from one scenario to another, the feature space, +e.g., the hidden layers, are often frozen or slightly per- +turbed, which can dramatically reduce the training over- +head by orders of magnitude. However, existing transfer +learning approaches for PDEs, e.g., (Lu et al., 2021a; Li +et al., 2021a; Chakraborty, 2020; Desai et al., 2021), re- +quire information/knowledge of the target family of PDEs +to pre-train a neural network model. The needed information +could be the analytical definitions of the PDEs including +initial and boundary conditions, and/or measurement data +of the PDE’s solution. These requirement not only leads +to time-consuming simulation data generation using other +PDE solvers, but also limits the transferability of the pre- +trained neural network (i.e., the pre-trained network is only +transferable to the same or similar type of PDEs that are +used for pre-training). +To overcome the above challenges, in this paper we propose +a transferable neural network (TransNet) to improve the +transferability of neural networks for solving PDEs. The key +idea is construct a pre-trained neural feature space without +using any PDE information, so that the pre-trained feature +space could be transferred to a variety of PDEs with different +arXiv:2301.11701v1 [math.NA] 27 Jan 2023 + +TransNet: Transferable Neural Networks for PDEs +2 +domains and boundary conditions. We limit our attention to +single-hidden-layer fully-connected neural networks, which +have sufficient expressive power for low-dimensional PDEs +that are commonly used in science and engineering fields. +Specifically, we treat each hidden neuron as a basis function +and re-parameterize all the neurons to separate the parame- +ters that determine the neuron’s location and the ones that +control the shape (i.e., the slope) of the activation function. +Then, we develop a simple, yet very effective, approach to +generate uniformly distributed neurons in the unit ball, and +rigorously prove the uniform neuron distribution. Then, the +shape parameters of the neurons are tuned using auxiliary +functions, i.e., realizations of a Gaussian process. The entire +feature space construction (determining the hidden neurons’ +parameters) does not require the PDE’s formulation or data +of the PDE’s solution. When applying the constructed fea- +ture space to a PDE problem, we only need to solve for the +parameters of the output layer by minimizing the standard +PDE residual loss. This can be done by either solving a +simple least squares problem for linear PDE or combining +a least squares solver with a nonlinear iterative solver, e.g., +Pichard iteration, for nonlinear PDEs. +The major contributions of this work are summarized as +• We develop transferable neural feature spaces that are +independent of any PDE, and can be applied to effectively +solve various linear and nonlinear PDE problems. +• We theoretically and computationally prove the uniform +distribution of the hidden neurons, viewed as global non- +orthogonal basis, for the proposed TransNet in the unit +ball of any dimension. +• We demonstrate the superior accuracy and efficiency of +the proposed TransNet for solving PDEs, e.g., the mean +square errors of TransNet are several orders of magni- +tudes smaller than those by the state-of-the-art methods. +2 +Related work +Studies on using neural networks for solving PDEs can +be traced back to some early works, e.g., (Dissanayake & +Phan-Thien, 1994; Lagaris et al., 1998). Recent advances +mostly have been focused on physics-informed neural net- +work (PINN). The general idea of PINN is to represent +the PDE’s solution by a neural network, and then train the +network by minimizing certain measurement of the PDE’s +residual at a set of samples in the domain of computation. +Several improvements on the training and sampling were +proposed in (Lu et al., 2021b; Anitescu et al., 2019; Zhao +& Wright, 2021; Krishnapriyan et al., 2021). Besides direct +minimizing the PDE’s residual, there are studies on how +to combine traditional PDE solvers with neural networks. +For example, the deep Ritz method (E & Yu, 2018) uses the +variational form of PDEs and combines the stochastic gradi- +ent descent with numerical integration to train the network; +the deep Galerkin method (Sirignano & Spiliopoulos, 2018) +combines the Galerkin method with machine learning; the +PDE-Net (Long et al., 2018; 2019) uses a stack of neural +networks to approximate the PDE solutions over a multiple +of time steps. +Another type of deep learning method for PDEs is to use +neural networks to learn a family of PDE operators, in- +stead of a single equation. The Fourier neural operator +(FNO) (Li et al., 2021a) parameterizes the integral kernel in +Fourier space and is generalizable to different spatial/time +resolutions. The DeepONet (Lu et al., 2021a) extends the +universal approximation theorem (Chen & Chen, 1995) to +deep neural networks, and its variant (Wang et al., 2021) +further reduces the amount of data needed for training. The +physics-informed neural operator (PINO) (Li et al., 2021b) +combines operator learning with function approximation to +achieve higher accuracy. MIONet (Jin et al., 2022) was pro- +posed to learn multiple-input operators via tensor product +basd on low-rank approximation. +Random feature models have also been used to solve PDEs +(Sun et al., 2018; Liu et al., 2022b) or learn PDE operators +(Nelsen & Stuart, 2021). The theory of random feature +models for function approximation was developed due to its +natural connection with kernel methods (Liu et al., 2022a; +Bach, 2017). The proposed TransNet can be viewed as an +improved random feature model for PDEs from two per- +spectives: (1) the re-parameterization of the hidden neurons +to separate the parameters that determine locations of the +neurons and the ones that control the activation function +slope, (2) the usage of auxiliary functions to tune the neural +feature space, which makes a critical contribution to the +improvement of the accuracy of TransNet in solving PDEs. +3 +Transferable neural networks for PDEs +3.1 +Problem setting and background +We introduce the problem setup for using neural networks +to solve partial differential equations. The PDE of interest +can be presented in a general formulation, i.e., +� +L(u(y)) = f(y) +for y ∈ Ω, +B(u(y)) = g(y) +for y ∈ ∂Ω, +(1) +where Ω ⊂ Rd with the boundary ∂Ω is the spatial-temporal +bounded domain under consideration, y := (x, t) = +(x1, . . . , xd−1, t)⊤ is a column vector includes both spa- +tial and temporal variables, u denotes the unknown solution +of the PDE, L(·) is a differential operator, B(·) is the opera- +tor defining the initial and/or boundary conditions, f(y) and +g(y) are the right hand sides associated with the operators +L(·) and B(·), respectively. For notational simplicity, we +assume that the solution is a scalar function; the proposed +method can be extended to vector-valued functions without + +TransNet: Transferable Neural Networks for PDEs +3 +any essential difficulty. We limit our attention to the single- +hidden-layer fully-connected neural networks, denoted by +uNN(y) := +M +� +m=1 +αm σ(wmy + bm) + α0, +(2) +where M is the number of hidden neurons, the row vec- +tor wm = (wm,1, . . . , wm,d) and the scalar bm are the +weights and bias of the m-th hidden neuron, the row vec- +tor α = (α0, α1, . . . , αM) includes the weights and bias +of the output layer, and σ(·) is the activation function. As +demonstrated in Section 4, this type of neural networks have +sufficient expressive power for solving a variety of PDEs +with satisfactory accuracy. +A typical method (Karniadakis et al., 2021) for solving +the PDE in Eq. (1) is to directly parameterize the solution +u(y) as a neural network uNN(y) in Eq. (2) and optimize +the neural network’s parameters by minimizing the PDE +residual loss, e.g., L(y) = ∥L(u(y)) − L(uNN(y))∥2 + +∥B(u(y)) − B(uNN(y))∥2, at a set of spatial-temporal lo- +cations. Despite the good performance of these approaches +in solving PDE problems, its main drawback is the limited +transferability because of the high computational cost of +gradient-based re-training and hyperparameter re-tuning. +When there is any change to the operators L(·), B(·), the +right-hand-side functions f(y), g(y), or the shape of the +domain Ω, the neural network uNN(y) often needs to be +re-trained using gradient-based optimization (even though +the current parameter values could provide a good initial +guess for the re-training), or the hyperparameters associ- +ated with the network and the optimizer need to be re-tuned. +In comparison, the random feature models require much +lower re-training cost, which has been exploited in learning +operators (Nelsen & Stuart, 2021) and dynamical systems +(McDonald & ´Alvarez, 2021; Liu et al., 2022b). +3.2 +The neural feature space +We can treat each hidden neuron σ(wmy + bm) as a non- +linear feature map from the space of y ∈ Rd to the output +space R. From the perspective of approximation theory, the +set of hidden neurons {σ(wmy + bm)}M +m=1 can be viewed +as a globally supported basis in Rd. The neural feature +space, denoted by PNN, can be defined by the linear space +expanded by the basis {σ(wmy + bm)}M +m=1, i.e., +PNN = span +� +1, σ(w1y+b1), . . . , σ(wMy+bM) +� +, (3) +where the constant basis corresponds to the bias of the output +layer. Then, the neural network in Eq. (2) lives in the linear +space, i.e., uNN(y) ∈ PNN. In other words, the neural +network approximation can be viewed as a spectral method +with non-orthogonal basis, and the parameters α in Eq. (2) +of the output layer of uNN(y) contains the coefficients of +the expansion in the neural feature space PNN. +In the PINN methods, the neural feature space PNN and +the coefficient α are trained simultaneously using stochas- +tic gradient descent methods, which often leads to a non- +convex and ill-conditioned optimization problem. It has +been shown that the non-convexity and ill-conditioning in +the neural network training are major reasons of unsatis- +factory accuracy of the trained neural network. A natural +idea to reduce the complexity of the training is to decou- +ple the training of PNN from that of α. For example, in +random feature models, PNN is defined by randomly gen- +erating the parameters{(wm, bm)}M +m=1 from a user-defined +probability distribution; the coefficients α can then be ob- +tained by solving a linear system when the operators L, B +in Eq. (1) are linear. However, the numerical experiments +in Section 4 show that the random feature model based on +Eq. (2) converges very slowly with the increase of the num- +ber of features. This drawback motivates us to develop a +methodology to customize the neural feature space PNN to +improve the accuracy, efficiency and transferability of uNN +in solving PDEs. +3.3 +Constructing the transferable neural feature +space +This section contains the key ingredients of the proposed +TransNet. The goal is to construct a single neural feature +space PNN that can be used to solve various PDEs in differ- +ent domains. +3.3.1 +RE-PARAMETERIZATION OF PNN +The first step is to re-parameterize the hidden neuron +σ(wmy + bm), viewed as a basis function in Ω, to separate +the components that determine the location of the neuron +and the components that control the shape of the neuron. +The idea of handling the locations of the basis functions +is inspired by the studies on activation patterns of ReLU +networks. When σ is the ReLU function, there is a partition +hyperplane defined by +wm,1y1 + wm,2y2 + · · · + wm,dyd + bm = 0 +(4) +that separates the activated and inactivated regions for this +neuron. The intersections of multiple partition hyperplanes +associated with different neurons define a linear region of +ReLU network. Studies have shown that the expressive +power of a ReLU network is determined by the number of +linear regions and the distribution of those linear regions. +In principle, the more uniformly distributed linear regions +in the domain Ω, the more expressive power the ReLU +network has. For other activation functions, e.g., tanh(·) +that is widely used in solving PDEs due to its smoothness, +the partition hyperplane in Eq. (4) can be used to describe +the geometric property of the neuron. +Specifically, let us re-write Eq. (4) into the following point- + +TransNet: Transferable Neural Networks for PDEs +4 +slope form: +γm +� +am,1(y1 − rmam,1) + · · · + am,d(yd − rmam,d) +� += 0, +(5) +where am += +(am,1, . . . , am,d) is a unit vector, i.e., +∥am∥2 = 1, rm > 0 and γm ∈ R are two scalar parameters +for the m-th neuron. We can relate Eq. (5) to Eq. (4) by +� +� +� +� +� +� +� +wm,i = γmam,i, +i = 1, · · · , d, +bm = −γm +d +� +i=1 +a2 +m,irm, +(6) +which shows the desired geometric properties of the par- +tition hyperplane in Eq. (4). In terms of the location, the +unit vector am is the normal direction of the partition hy- +perplane in Rd, the vector (rmam,1, . . . , rmam,d) indicates +a point that the hyperplane passes, rm is the distance be- +tween the origin and the partition hyperplane. An illustra- +tion is shown in Figure 1(a). In terms of the shape, the +constant γm determines the steepness of the slope of the +activation function along the normal direction am. Thus, +the re-parameterization in Eq. (5) successfully separates the +parameters determining location from the ones determining +the shape. +3.3.2 +GENERATING UNIFORMLY DISTRIBUTED +NEURONS FOR PNN +The second step of constructing PNN is to determine the +parameters {(am, rm)}M +m=1 in Eq. (5), such that all the +neurons are uniformly distributed in Ω. We assume Ω is +a unit ball, i.e., B1(0) = {y : ∥y∥2 ≤ 1} ⊂ Rd in this +subsection. To proceed, we need to define a density function +that measures the neuron distribution. For a given y ∈ +Ω, the distance between y and the partition hyperplane in +Eq. (5) is given by +dist(y, m) = |am(y − rmam)|, +(7) +for m = 1, . . . , M. We use this distance to define how +close the point y to the m-th neuron. The density function, +denoted by DM(y), is defined using the above distance, i.e., +DM(y) = 1 +M +M +� +m=1 +1dist(y,m)<τ(y), +(8) +where 1dist(y,m)<τ(y) is the indicator function of the event +that the distance between y and the m-th neuron is smaller +than a prescribed tolerance τ > 0. Intuitively, DM(y) mea- +sures the percentage of neurons whose partition hyperplane +in Eq. (4) intersect the ball (with radius τ) around y. +Next we propose the following approach, illustrated in +Figure 1(b), to generate the parameters {(am, rm)}M +m=1. +Specifically, we first generate the normal directions +{am}M +m=1 uniformly distributed on the d − 1-dimensional +unit sphere. Note that when d > 2, sampling uniformly in +the angular space in the hyperspherical coordinate system +does not lead to uniformly distributed samples on the unit +sphere. This is known as the sphere point picking prob- +lem. To overcome this issue, we draw samples from the +d-dimensional Gaussian distribution in the Cartesian coor- +dinate system, and normalize the samples to unit vectors to +obtain {am}M +m=1. Then, we generate {rm}M +m=1 uniformly +from [0, 1] using the Monte Carlo method. The following +theorem shows that our approach provides a set of uniformly +distributed neurons in Ω, where the density is measured by +DM(y) in Eq. (8). +Theorem 1 (Uniform neuron distribution) Given the re- +parameterization in Eq. (5), if {am}M +m=1 are uniformly dis- +tributed random vectors on the d-dimensional unit sphere, +i.e., ∥am∥2 = 1, and {rm}M +m=1 are uniformly distributed +random variables in [0, 1], then, for a fixed τ ∈ (0, 1), +E[DM(y)] = τ for any ∥y∥2 ≤ 1 − τ, +where DM(y) is the density function defined in Eq. (8). +The proof is given in Appendix A; an illustration of the +density function is given in Figure 1(c). This result is a little +surprising that the distribution of {rmam}M +m=1, i.e., the red +dots in Figure 1(b)-middle, are not uniformly distributed +in the ball B1−τ(0), but the density function DM(y) is a +constant in the ball B1−τ(0). +Remark 1 (The dimentionality) Even though Theorem 1 +holds for any dimension d, the number of neurons required +to cover a high-dimensional unit ball still could be in- +tractable. On the other hand, the majority of PDEs com- +monly used in science and engineering are defined in low- +dimensional domains, e.g., 3D spatial domain + 1D time +domain. In this scenario, the proposed method is effective +and easy to implement, as demonstrated in Section 4. +3.3.3 +TUNING THE SHAPE OF THE NEURONS IN PNN +USING AUXILIARY FUNCTIONS +The third step is to tune the shape parameters {γm}M +m=1 +in Eq. (5) that controls the slope of the activation function. +The experimental tests in Section 4.1 show that the slope +parameters play a critical role in determining the accuracy +of the neural network approximator uNN. For simplicity, we +assume the same shape parameter value for all neurons, i.e., +γ = γm for m = 1, . . . , M. Because we intend to construct +a feature space PNN that can be used in multiple scenarios, +e.g., various PDEs with different domains and boundary +conditions, we do not want to tune the shape parameter γ +using any information about a specific PDE. +Our idea is to use auxiliary functions that have similar or +more complicated spatial-temporal variation frequency as + +TransNet: Transferable Neural Networks for PDEs +5 +Figure 1: (a) Illustrates how the re-parameterization in Eq. (5) characterizes the location of a neuron. The blue line is the +plane where tanh(·) = 0, am (the arrow) is the normal direction of the plane, the red dot is the location rmam that the +plane passes, rm is the distance between the origin and the plane. (b) illustrates how to generate uniformly distributed +neurons in the unit ball. The first step in (b)-left is to generate the normal directions {am}M +m=1 uniformly distributed on unit +sphere; the second step in (b)-middle is to generated {rm}M +m=1 uniformly from [0, 1] defining the locations the neurons’ +partition hyperplanes will pass; the blue lines in (b)-right show the distribution of the partition hyperplanes. (c) shows the +density function DM(y) with τ = 0.05 in Eq. (8) for a set of neurons generated using our approach. We can see that our +approach provides a uniformly distributed neurons in the ball B1−τ(0), which is consistent with Theorem 1. +the PDE solution to tune γ. Specifically, we propose to use +realizations of Gaussian processes to generate the auxiliary +functions. The advantage of Gaussian process is that one +can control the variation frequency of its realizations by +adjusting the correlation length. Additionally, the Guassian +process is independent of the coordinate system. Let us de- +note by G(y|ω, η) the Gaussian process, where ω represents +the abstract random variable and η is the correlation length. +Given a correlation length, we first generate a set of realiza- +tions of the Gaussian process, denoted by {G(y|ωk, η)}K +k=1. +For each realization, define the MSE loss as +MSE(uNN(y), G(y|ωk, η)) += 1 +J +J +� +j=1 +� M +� +m=1 +αmσ(wmyj + bm) + α0 − G(yj|wk, η) +�2 +, +(9) +where the parameters {wm}M +m=1 and {bm}M +m=1 are already +determined using the strategy in Section 3.3.2 and Eq. (6), +and J denotes the number of sample points. Unlike stan- +dard neural network training, the optimal coefficient α that +minimizing the MSE loss can be efficiently achieved by +solving the least squares problem. Hence, the shape parame- +ter γ can be tuned by solving the following one-dimensional +optimization problem +min +γ +� K +� +k=1 +min +α [MSE(uNN(y), G(y|ωk, η))] +� +, +(10) +where for each candidate γ, we solve K least squares prob- +lems to compute the total loss. +Remark 2 (The choice of the correlation length) There +are two strategies to choose the correlation length η. One is +to use the prior knowledge about the PDE. For example, for +the Naveier-Stokes equations with low Reynolds’ number, +we know the solution will not have very high-frequency +oscillation. The other is to use an over-killing correlation +length to ensure that the feature space has sufficient +expressive power to solve the target PDE. +3.4 +Applying TransNet to linear and nonlinear PDEs +Once the neural feature space PNN is constructed and tuned, +we can readily use it to solve PDE problems. Even though +PNN is defined on the unit ball, i.e., B1(0), we can always +place the (bounded) domain Ω for the target PDE in B1(0) +by simple translation and dilation. Thus, the feature space +can be used to handle PDEs defined in various domains, as +demonstrated in Section 4. +Linear PDEs. When L and B in Eq. (1) are linear opera- +tors, the unknown parameters α = (α0, . . . , αM) in Eq. (2) +can be easily determined by solving the following least +squares problem, i.e., +min +α +� +1 +J1 +J1 +� +j=1 +� M +� +m=1 +αm L(σ(wmyj + bm)) + α0 − f(yj) +�2 ++ 1 +J2 +J2 +� +j=1 +� M +� +m=1 +αm B(σ(wmyj + bm)) + α0 − g(yj) +�2 � +(11) +where the parameters {wm}M +m=1 and {bm}M +m=1 are first +computed using the strategy in Section 3.3.2 and Eq. (6). +Nonlinear PDEs. When one or both operators, L and B, +are nonlinear, there are two approaches to handle the situ- +ation. The first way is to wrap the least squares problem +with a well established nonlinear iterative solver, e.g., Pi- +card’s methods, to solve the PDE. Within each iteration, +the PDE is linearized such that we can update the coeffi- +cient α by solving the least squares problem as mentioned +above. When there is sufficient knowledge to choose a + +(a) +(b) +(c) +1.00 +1.00 +1.00 +1.00 +0.0489 +0.75 +0.75 +0.75 +0.75 +0.75 + 0.0462 +0.50 +0.5 0 + 0.50 +0.50 +0.0435 +0.25 +rm +0.25 + 0.25 +0.0408 +0.00 +0. 00 +0.00 +0.0381 +0.00 +am +0.0354 +0.25 +0.25 +0.25 +LZE0'0 +0.50 +amy +0.50 +-0.50 +0.0300 +-0.75 +0.75 +0.75 +0.0273 +-1.00 +0.0246 +1.00 +1.00 +-1.00-0.75-0.50-0.25 0.00 0.25 0.50 0.75 1.00 +-1.00-0.750.50-0.25 0.00 0.25 0.50 0.75 1.00 +.000.75 0.500.25 0.00 +0.50 +1.00 0.75 0.50 0.25 0.000.250.500.75 1.00TransNet: Transferable Neural Networks for PDEs +6 +proper nonlinear solver, we prefer this approach because +the well-established theory on nonlinear solvers can ensure +a good convergence rate. Thus, we in fcat adopt this ap- +proach for numerical experiments in this paper. The second +feasible approach is to wrap a gradient descent optimizer +around the total loss L(y) = ∥L(u(y)) − L(uNN(y))∥2 +2 + +∥B(u(y))−B(uNN(y))∥2 +2. Because the neural feature space +PNN is fixed, the optimization will be simpler than training +the entire neural network from scratch. This approach is +easier to implement and suitable for scenarios that standard +nonlinear solvers do not provide a satisfactory solution. +Remark 3 (Not using PDE’s solution data) In this work, +we do not rely on any measurement data of the solution u(y) +when using TransNet to solve PDEs, because the operators +L and B in Eq. (1) are sufficient to ensure the existence and +uniqueness of the PDE’s solution. On the other hand, if +any extra data of u(y) are available, TransNet can easily +incorporate it into the least squares problem in Eq. (11) as +a supervised learning loss. +3.5 +Complexity and accuracy of TransNet +The complexity of TransNet is greatly reduced compared to +the scenario of using SGD to train the entire network. The +construction of the neural feature space PNN only involves +random number generations and a simple one-dimensional +optimization in Eq. (10). Moreover, these cost are com- +pletely offline, and the constructed PNN is transferable to +various PDE problems. The online operation for solving +linear PDEs only requires solving one least squares problem, +where the assembling of the least squares matrix can be effi- +ciently done using the autograd function in Tensorflow or +Pytorch. The numerical experiments in Section 4 show that +that the accuracy and efficiency of TransNet is significantly +improved compared with several baseline methods, because +our method does not suffer from the slow convergence of +SGD in neural network training. +4 +Numerical experiments +We now demonstrate the performance of TransNet by testing +several classic steady-state or time-dependent PDEs in two +and three dimensional spaces. In Section 4.1, we illustrate +how to construct the transferable feature space PNN. To +test and demonstrate the transferability of our model, we +build and test two neural features spaces, one for the 2D +case and the other for the 3D case1. The constructed feature +spaces are then used in Section 4.2 to solve the model PDE +problems. +1Note that the dimension of the feature space is the sum of both +space and time dimensions since it doesn’t differ them. +4.1 +Uniform neuron distribution +This experiment is to use and test the algorithm proposed in +Section 3.3 to construct transferable neural feature spaces +PNN in the 2D and 3D unit balls. We tune the shape param- +eter γ = γm for m = 1, . . . , M in Eq. (5) with K = 50 +realizations of the Gaussian process. In addition, we also +test the effect of the correlation length and the number of +hidden neurons by setting different values for η and M. For +each setting of η and M, the shape parameter γ is tuned +separately. Additional information about the experiment +setup is given in Appendix B. +Figure 2: The loss landscapes of the optimizing problem +in Eq. (10) for tuning the shape parameter γ of the feature +space PNN in two and three dimensional cases. The blue star +is the optimal value for γ founded by our method. It shows +that the optimal value for γ varies with the number of hidden +neurons, meaning that tuning γ is a necessary operation to +achieve optimal accuracy of uNN when changing the number +of hidden neurons. +Figure 2 illustrates the landscapes of the loss function +�K +k=1 minα[MSE(uNN(y), G(y|ωk, η))] of the optimiza- +tion problem in Eq. (10) for 2D and 3D neural feature +spaces. We report the results for two correlation lengths +(η = 0.5 and η = 1.0) combined with three numbers of +hidden neurons (M = 100, 500, 1000 for 2D and M = +500, 1000, 5000 for 3D). We observe that the loss function +behaves roughly like a parabolic curve for a fixed number +of hidden neurons, so that the problem in Eq. (10) can be +solved by a simple solver for one-dimensional optimization. +More importantly, we observe that the optimal value for γ +varies with the number of hidden neurons. This provides +an important insight that tuning γ is a necessary operation +to achieve optimal accuracy of uNN when changing the +number of hidden neurons. + +Correlation length=0.5 (2D) +Correlation length=1 (2D) +10-2. +10-4. +10-5. +10-7. +MSE +GP fitting MSE +10-8 +GP fitting I +10-11 +10-13 +10-14 +10-16 +10-17. +10-19 +0 +2 +4 +0 +2 +4 +Shape parameter y +Shape parameter y +#(hidden neurons)=100 +#(hidden neurons)=500 +#(hidden neurons)=1000 +Correlation length=0.5 (3D) +Correlation length=1 (3D) +10-5. +10-7. +10-4 +MSE +10-9 +10-6. +GP fitting +10-11 +10-8. +10-13. +10-10 +10-15 +10-12 +10-17. +0.5 +1.0 +1.5 +2.0 +2.5 +0.5 +1.0 +1.5 +2.0 +2.5 +Shape parameter y +Shape parameter y +#(hidden neurons)=500 +#(hidden neurons)=1000 +#(hidden neurons)=5000TransNet: Transferable Neural Networks for PDEs +7 +Figure 3: Top row: three realizations of the auxiliary Gaus- +sian process with the correlation length η = 0.5. Bottom +row: the distribution of the MSE of TransNet’s approxima- +tion with 1000 hidden neurons. Thanks to the feature space +with the uniform density in the 2D unit ball (illustrated in +Figure 1(c)), we obtain a TransNet approximation with very +small MSE fluctuation. +Figure 3 illustrates the error distribution when using +TransNet to approximate three realizations of the Gaussian +process with correlation length η = 0.5 in the 2D unit ball. +Even though the purpose of TransNet is not to approximate +the Gaussian process, it is interesting to check whether the +uniform density DM(y) (proved in Theorem 1) leads to +uniform error distribution. We use 1000 hidden neurons +and the shape parameter γ is set to 2. The bottom row of +Figure 3 shows that the MSE error distributes uniformly in +the unit ball, which demonstrates the effectiveness of the +feature space generation method proposed in Section 3.3. +4.2 +PDE examples +We then use the constructed 2D and 3D neural feature spaces +from Section 4.1 to solve two steady-state PDEs (i.e., the +Poisson equation and the time-independent Navior-Stokes +equation) and two time-dependent PDEs (i.e., the Fokker- +Planck equation and the wave equation). The definitions of +the PDEs under consideration are given in Appendix C. We +perform the following testing cases: +(C1) Poisson equation (2D space) in a box domain; +(C2) Poisson equation (2D space) in a circular domain; +(C3) Poisson equation (2D space) in an L-shaped domain; +(C4) Poisson equation (2D space) in an annulus domain; +(C5) Poisson equation (3D space) in a box domain; +(C6) Steady-state Navier-Stokes equation (2D space); +(C7) Fokker-Planck equation (1D space + 1D time); +(C8) 2D Fokker-Planck equation (2D space + 1D time); +(C9) 1D wave equation (1D space + 1D time) +to demonstrate the transferability of TransNet in solving +various PDEs in different domains. Recall that for time- +dependent PDEs, the temporal variable is simply treated +as an extra dimension, so that we will use the 2D feature +space to solve problems (C7) and (C9) and the 3D feature +space to solve problem (C8). We compare our method with +two baseline methods, i.e., the random feature mode and the +PINN. All the methods use the same network architecture, +i.e., Eq. (2) with the tanh activation. Additional information +about the setup of the experiments are given in Appendix D. +Figure 4 shows the MSE decay with the increasing of the +number of the hidden neurons, where the number of hidden +neurons are chosen as M = 100, 200, 300, 400, 500, 600, +700, 800, 900, 1000, respectively, for the 2D feature space, +and M = 1000, 2000, 3000, 4000, 5000, respectively, for +the 3D feature space. We observe that our TransNet achieves +a superior performance for all the nine test cases, which +demonstrates the outstanding transferability of TransNet. +PINN with BFGS acceleration provides a good accuracy +gain compared with PINN with Adam, which means the +landscape of the PDE loss exhibits severe ill-conditioning as +the SGD method approaches the minimizer2. In comparison, +TransNet does not require SGD in solving the PDEs, so that +TransNet does not suffer from the slow convergence of SGD +used in PINN. +Figure 5 shows the density function DM(y) in Eq. (8) of the +feature spaces obtained by training PINN and the random +feature models in solving the Poisson equation in the 2D +space, i.e., case (C1) - (C4), where the constant τ in Eq. (8) +is set to 0.2. Compared with TransNet’s uniform density +shown in Figure 1(c), the feature spaces obtained by the +baseline methods have highly non-uniform densities in the +domain of computation. The random feature models tend +to have higher density, i.e., more hidden neurons, near the +center of the domain. The first row in Figure 5 can be +viewed as the initial densities of the feature space for PINN; +the second and the third rows are the final densities. We can +see that the training of PINN does not necessarily lead to a +more uniform density function DM(y), which is one of the +reasons why PINN cannot exploit the full expressive power +of the neural network uNN. +5 +Conclusion +We propose a transferable neural network model to advance +the state of the art of using neural networks to solve PDEs. +The key ingredient is to construct a neural feature space +independent of any PDE, which makes it easy to transfer the +neural feature space to various PDEs in different domains. +Moreover, because the feature space is in fact fixed when +using TransNet to solve a PDE, we only need to solve linear +least squares problems, which avoids the drawbacks of SGD- +2BFGS can alleviate ill-conditioning by exploiting the second- +order information, e.g., the approximate Hessian. + +3 +GP realizations +2 +1 +0 +-1 +2 +1e-15 +MSE of TransNet +8 +6 +4 +2TransNet: Transferable Neural Networks for PDEs +8 +Figure 4: The MSE decay along with the increasing of the number of hidden neurons for (C1) to (C9), where all the +methods use the same network architecture. Our TransNet significantly outperforms the baseline methods from two aspects: +(i) Transferability: for a fixed number of hidden neurons, TransNet only need use one 2D feature space and one 3D feature +space; (ii) Accuracy: TransNet achieves several orders of magnitude smaller MSE than PINN and the random feature models. +TransNet does not suffer from the slow convergence in SGD-based neural network training, and can exploit more expressive +power of a given neural network uNN to obtain more accurate PDE solutions. +(C1) +(C2) +(C3) +(C4) +(C5) +(C6) +(C7) +(C8) +(C9) +Random feature model +0.25s +0.22s +0.22s +0.19s +0.96s +12.85s +0.92s +1.21s +0.47s +PINN:Adam +29.69s +25.34s +24.57s +22.24s +110.59s +69.73s +61.45s +97.12s +49.25s +PINN:Adam+BFGS +125.78s +121.46s +120.93s +119.24s +264.62s +191.53s +172.86s +178.99s +152.71s +TransNet +0.27s +0.20s +0.20s +0.17s +1.03s +11.14s +0.97s +1.27s +0.51s +Table 1: The computing times of TransNet and the baselines in solving the nine PDE test cases with 1000 hidden neurons. +TransNet and the random feature model are significantly faster than PINN because SGD is not required in them. +Figure 5: The density function DM(y) with τ = 0.2 in +Eq. (8) of the neural feature spaces obtained by training +PINN and the random feature models in solving the Poisson +equation in the 2D space, i.e., problems (C1) - (C4). Com- +pared to the uniform density of TransNet in Figure 1(c), both +PINN and the random feature model cannot provide feature +spaces with uniform density, which is one explanation of +their under-performance shown in Figure 4. +based training algorithms, e.g., ill-conditioning. Numerical +experiments show that the proposed TransNet can exploit +more expressive power of a given neural network than the +compared baselines. This work is the first scratch in this +research direction, and there are multiple potential related +topics that will be studied in our future work, including (1) +theoretical analysis of the convergence rate of TransNet in +solving PDEs. We observe in Figure 4 that the MSE of +TransNet decays along with the increasing of the number of +hidden neurons. A natural question to study is that whether +TransNet can achieve the optimal convergence rate of the +single-hidden-layer fully-connected neural network. (2) Ex- +tension to multi-layer neural networks. Even though the +single-hidden-layer model has sufficient expressive power +for the PDEs tested in this work, there are more complicated +PDEs, e.g., turbulence models, that could require multi- +layer models with much higher expressive power. (3) The +properties of the least squares problem. In this work, we use +the standard least squares solver of Pytorch in the numerical +experiments. However, it is worth further investigation of +the properties of this specific least squares problem. For ex- +ample, since the set of neurons {σ(wmy + bm)}M +m=1 forms +a non-orthogonal basis, it is possible to have linearly corre- +lated neurons which will reduce the column rank of the least + +2D Poisson (box) +2D Poisson (circle) +2D Poisson (L-shape) +2D Poisson (annulus) +3D Poisson (box) +10° +10-2 +10-1+ +10-2. +10-3 +10-5 +10-4. +10-5 +10-2 +MSE +TestM +Test +10-9 +10-6 +10-12 +10-14. +10-13. +10-14 +10- +10-15 +10-17 +10-16 +10-17 +200400600 +8001000 +2004006008001000 +200400600 +8001000 +200400600800 1000 +1000 2000 3000 4000 5000 +The number of hidden neurons +The number of hidden neurons +The number of hidden neurons +The number of hidden neurons +2D Steady-state Navier Stokes +1D Fokker-Planck +2D Fokker-Planck +1D Wave equation +10-3 +10-4. +10-4. +Random feature model +10-6 +10-6. +10-4 +10-7. +PINN: Adam +Test MSE +MSE +10-8 +10-9 +PINN: Adam+BFGS + 10-10. +TransNet +10-12 +10-13 +10-12 +10-8. +10-15 +10-14. +10-16. +400600 +8001000 +200 +400600 +8001000 +100020003000 +40005000 +200　400　600　8001000 +The number of hidden neurons +The number of hidden neurons +The number of hidden neurons +The number of hidden neuronsRandom feature model(box) +Random feature model(annulus) +Random feature model(circle) +Random feature model(L-shape) +0.184 +0.185 +0.1825 +0.180 +0.172 +0.171 +0.175 +0.1725 +0.160 +0.162 +0.165 +0.1625 +0.148 +0.153 +0.155 +0.1525 +0.144 +0.136 +0.135 +0.145 +0.1425 +0.124 +0.126 +0.112 +0.135 +0.1325 +0.117 +0.100 +0.125 +0.1225 +0.108 +0.088 +0.115 +0.1125 +0.099 +PINN:Adam(box) +PINN:Adam(annulus) +PINN:Adam(circle) +PINN:Adam(L-shape) +0.216 +0.26 +0.288 +0.272 +0.198 +0.24 +0.270 +0.180 +0.240 +0.252 +0.22 +0.162 +0.234 +0.208 +0.20 +0.144 +0.216 +0.176 +0.18 +0.126 +0.198 +0.144 +0.16 +0.108 +0.180 +0.112 +0.090 +0.162 +0.14 +0.072 +0.144 +0.12 +0.080 +0.126 +0.048 +0.054 +0.10 +PINN:Adam+BFGS(box) +PINN:Adam+BFGS(annulus) +PINN:Adam+BFGS(circle) +PINN:Adam+BFGS(L-shape) +0.282 +0.210 +0.276 +0.33 +0.264 +0.192 +0.258 +0.30 +0.246 +0.174 +0.240 +0.27 +0.156 +0.228 +0.222 +0.24 +0.138 +0.210 +0.204 +0.21 +0.120 +0.186 +0.192 +0.18 +0.102 +0.168 +0.15 +0.174 +0.156 +0.12 +0.084 +0.150 +0.066 +0.132 +0.138 +0.09 +0.048 +0.114 +0.06 +0.120TransNet: Transferable Neural Networks for PDEs +9 +squares matrix, or even lead to an under-determined system. +This will require the use of some regularization techniques, +e.g., ridge regression, to stabilize the least squares system. +Additionally, compressed sensing, i.e., ℓ1 regularization, +could be added to remove redundant neurons from the fea- +ture space as needed and obtain a sparse neural network. +Acknowledgement +This work was supported by the U.S. Department of Energy, +Office of Science, Office of Advanced Scientific Computing +Research, Applied Mathematics Program, under the contract +number ERKJ387. This work was accomplished at Oak +Ridge National Laboratory (ORNL). ORNL is operated by +UT-Battelle, LLC., for the U.S. Department of Energy under +Contract DE-AC05-00OR22725. +References +Anitescu, C., Atroshchenko, E., Alajlan, N., and Rabczuk, +T. Artificial neural network methods for the solution +of second order boundary value problems. Computers, +Materials & Continua, 59(1):345–359, 2019. +Bach, F. On the equivalence between kernel quadrature +rules and random feature expansions. J. Mach. Learn. +Res., 18(1):714–751, jan 2017. ISSN 1532-4435. +Chakraborty, S. L. Transfer learning based multi-fidelity +physics informed deep neural network. J. Comput. Phys., +426:109942, 2020. +Chen, T. and Chen, H. Universal approximation to nonlinear +operators by neural networks with arbitrary activation +functions and its application to dynamical systems. IEEE +Transactions on Neural Networks, 6(4):911–917, 1995. +Clark Di Leoni, P., Lu, L., Meneveau, C., Karniadakis, +G. E., and Zaki, T. A. Neural operator prediction of linear +instability waves in high-speed boundary layers. Journal +of Computational Physics, 474:111793, 2023. +Desai, S., Mattheakis, M., Joy, H., Protopapas, P., and +Roberts, S. J. +One-shot transfer learning of physics- +informed neural networks. ArXiv, abs/2110.11286, 2021. +Dissanayake, M. and Phan-Thien, N. Neural-network-based +approximations for solving partial differential equations. +Communications in Numerical Methods in Engineering, +10(3):195–201, 1994. +E, W. and Yu, B. The deep Ritz method: a deep learning- +based numerical algorithm for solving variational prob- +lems. Communication in Mathematics and Statistics, 6 +(1):1–12, 2018. +Gin, C. R., Shea, D. E., Brunton, S. L., and Kutz, J. N. Deep- +green: deep learning of green’s functions for nonlinear +boundary value problems. Scientific Reports, 11(1):1–14, +2021. +Jin, P., Meng, S., and Lu, L. Mionet: Learning multiple- +input operators via tensor product. SIAM Journal on +Scientific Computing, 44(6):A3490–A3514, 2022. +Karniadakis, G. E., Kevrekidis, I. G., Lu, L., Perdikaris, +P., Wang, S., and Yang, L. Physics-informed machine +learning. Nature Reviews Physics, 3(6):422–440, January +2021. doi: 10.1038/s42254-021-00314-5. +Krishnapriyan, A., Gholami, A., Zhe, S., Kirby, R., and +Mahoney, M. W. Characterizing possible failure modes +in physics-informed neural networks. Advances in Neural +Information Processing Systems, 34, 2021. +Lagaris, I. E., Likas, A., and Fotiadis, D. I. Artificial neu- +ral networks for solving ordinary and partial differential +equations. IEEE transactions on neural networks, 9(5): +987–1000, 1998. +Li, Z., Kovachki, N., Azizzadenesheli, K., Liu, B., Stuart, +A., Bhattacharya, K., and Anandkumar, A. Multipole +graph neural operator for parametric partial differential +equations. Advances in Neural Information Processing +Systems, 33:6755–6766, 2020. +Li, Z., Kovachki, N. B., Azizzadenesheli, K., Bhattacharya, +K., Stuart, A., Anandkumar, A., et al. Fourier neural +operator for parametric partial differential equations. In +International Conference on Learning Representations, +2021a. +Li, Z., Zheng, H., Kovachki, N., Jin, D., Chen, H., Liu, +B., Azizzadenesheli, K., and Anandkumar, A. Physics- +informed neural operator for learning partial differential +equations. arXiv preprint arXiv:2111.03794, 2021b. +Liu, F., Huang, X., Chen, Y., and Suykens, J. A. K. Ran- +dom features for kernel approximation: A survey on algo- +rithms, theory, and beyond. IEEE Transactions on Pattern +Analysis and Machine Intelligence, 44(10):7128–7148, +2022a. +Liu, Y., McCalla, S. G., and Schaeffer, H. Random fea- +ture models for learning interacting dynamical systems, +2022b. +Long, Z., Lu, Y., Ma, X., and Dong, B. PDE-Net: Learning +PDEs from data. In International Conference on Machine +Learning, pp. 3214–3222, 2018. +Long, Z., Lu, Y., and Dong, B. PDE-Net 2.0: Learning +PDEs from data with a numeric-symbolic hybrid deep +network. Journal of Computational Physics, 399:108925, +2019. + +TransNet: Transferable Neural Networks for PDEs +10 +Lu, L., Jin, P., Pang, G., Zhang, Z., and Karniadakis, G. E. +Learning nonlinear operators via DeepONet based on the +universal approximation theorem of operators. Nature +Machine Intelligence, 3(3):218–229, 2021a. +Lu, L., Meng, X., Mao, Z., and Karniadakis, G. E. Deepxde: +A deep learning library for solving differential equations. +SIAM Review, 63(1):208–228, 2021b. +McDonald, T. and ´Alvarez, M. Compositional modeling +of nonlinear dynamical systems with ode-based random +features. In Ranzato, M., Beygelzimer, A., Dauphin, Y., +Liang, P., and Vaughan, J. W. (eds.), Advances in Neural +Information Processing Systems, volume 34, pp. 13809– +13819. Curran Associates, Inc., 2021. +Nelsen, N. H. and Stuart, A. M. The random feature model +for input-output maps between banach spaces. SIAM +Journal on Scientific Computing, 43(5):A3212–A3243, +2021. +Quarteroni, A., Sacco, R., and Saleri, F. Numerical Mathe- +matics, volume 332. Springer Science Business Media +&, 2007. +Raissi, M., Perdikaris, P., and Karniadakis, G. E. Physics- +informed neural networks: A deep learning framework for +solving forward and inverse problems involving nonlinear +partial differential equations. Journal of Computational +Physics, 378:686–707, 2019. +Sirignano, J. and Spiliopoulos, K. DGM: A deep learning al- +gorithm for solving partial differential equations. Journal +of Computational Physics, 375:1339–1354, 2018. +Sun, Y., Gilbert, A. C., and Tewari, A. On the approxima- +tion capabilities of relu neural networks and random relu +features. arXiv: Machine Learning, 2018. +Teng, Y., Zhang, X., Wang, Z., and Ju, L. Learning green’s +functions of linear reaction-diffusion equations with ap- +plication to fast numerical solver. In Mathematical and +Scientific Machine Learning Conference, 2022. +Wang, S., Wang, H., and Perdikaris, P. Learning the solution +operator of parametric partial differential equations with +physics-informed deeponets. Science Advances, 7(40): +eabi8605, 2021. +Zang, Y., Bao, G., Ye, X., and Zhou, H. Weak adversarial +networks for high dimensional partial differential equa- +tions. Journal of Computational Physics, 411:109409, +2020. +Zhang, X., Cheng, T., and Ju, L. Implicit form neural net- +work for learning scalar hyperbolic conservation laws. In +Mathematical and Scientific Machine Learning Confer- +ence, pp. 1082–1098, 2021. +Zhao, J. and Wright, C. L. Solving allen-cahn and cahn- +hilliard equations using the adaptive physics informed +neural networks. +Communications in Computational +Physics, 29:930–954, 2021. + +TransNet: Transferable Neural Networks for PDEs +11 +Appendix +A +The proof of Theorem 1 +For the re-parameterization in Eq. (5), we can treat {am}M +m=1 as M independent and identically distributed (i.i.d.) random +variables on the d-dimensional unit sphere, and {rm}M +m=1 as M i.i.d. random variables following the uniform distribution in +[0, 1]. For a fixed y ∈ Ω, the expectation of DM(y) is +E[DM(y)] = 1 +M +M +� +m=1 +E +� +1dist(y,m)<τ(y) +� +. +(12) +Because E[1dist(y,m)<τ(y)] = E[1dist(y,m′)<τ(y)], we only need to calculate one expectation E[1dist(y,m)<τ(y)]. There- +fore, we can drop the subscript of am and use a to denote am in the following derivation. +To proceed, we define the representations of the vectors a = (a1, . . . , ad) and y = (y1, . . . , yd) under different coordinate +systems. We denote by Coriginal the original Cartesian coordinate system and denote by a|Coriginal and y|Coriginal the +representations of a and y under Coriginal. Because a and y are defined in Coriginal, we have +a|Coriginal = (a1, . . . , ad) and y|Coriginal = (y1, . . . , yd). +We can also define a rotated Cartesian coordinate system, denoted by Crot, such that the first coordinate axis of Crot aligns +with the direction of y. We denote by c1, . . . , cd the directions of the coordinate axes of Crot, so the vector a can be +represented in Crot as +a|Crot = (˜a1, . . . , ˜ad) and a = ˜a1c1 + · · · + ˜adcd. +Because c1 = y/∥y∥2, we have +y|Crot = (∥y∥2, 0, · · · , 0). +Based on Crot, we define a d-dimensional hyperspherical coordinate system, denoted by Srot, with one radial variable r, +d − 2 polar angles (φ1, . . . , φd−2) ranging over [0, π] and one azimuthal angle φd−1 ranging over [0, 2π]. Then, the unit +vector a can be represented by the angular variables of Srot, i.e., +˜a1 = cos(φ1) +˜a2 = sin(φ1) cos(φ2) +... +˜ad−1 = sin(φ1) · · · sin(φd−2) cos(φd−1) +˜ad = sin(φ1) · · · sin(φd−2) sin(φd−1). +where (φ1, . . . , φd−1) are the representation of a under Srot. Since inner product is independent of coordinate system, the +inner product ay can be performed under Crot to obtain +ay = (a|Crot)(y|Crot) = ∥y∥2˜a1 + 0 ˜a2 + . . . + 0 ˜ad = ∥y∥2 cos(φ1), +which is independent of φ2, . . . , φd−1. +Now we derive the probability density function of the inner product ay for a fixed y. For any fixed φ2, . . . , φd−1, the set +Jφ1|φ2,...,φd−1 := {(1, φ1, φ2, . . . , φd−1) | φ1 ∈ [0, π] and φ2, . . . , φd−1 are fixed.}, +is a one-dimensional half circle on the d-dimensional unit sphere. When a is uniformly distributed on the d-dimensional unit +sphere, the conditional variable a|(φ2, . . . , φd−1) is uniformly distributed on the half circle Jφ1|φ2,...,φd−1 and φ1 follows a +uniform distribution over [0, π] (Quarteroni et al., 2007). Then, we have that the variable z = cos(φ1|φ2, . . . , φd−1) follows +the Chebyshev density +pZ(z) = 1 +π +1 +√ +1 − z2 z ∈ [−1, 1], +(13) +for any fixed (φ2, . . . , φd−1). Because the inner product ay = ∥y∥2 cos(φ1) is independent of (φ2, . . . , φd−1), the +conditional density in Eq. (13) is also the marginal density, i.e., pZ(z) in Eq. (13) is also the density of z = cos(φ1). + +TransNet: Transferable Neural Networks for PDEs +12 +Next we derive the analytical form of the expectation E[1dist(y,m)<τ(y)]. For the convenience of derivation, we temporarily +change the distribution of r to a uniform distribution in [−1, 0], which leads to an equivalent feature space to the one with +r ∈ [0, 1]. Since am is a unit vector, we have dist(y, m) = |ay + r|. Substituting ay = ∥y∥2z = ∥y∥2 cos(φ1) into +E[1dist(y,m)<τ(y)], we have +E[1dist(y,m)<τ(y)] = E[1|z∥y∥2+r|<τ(y)] += +� +{z∥y∥2+r<τ}∪{z∥y∥2+r>−τ} +pZ(z)pR(r)dzdr += +� +{z∥y∥2+r<τ}∪{z∥y∥2+r>−τ} +1 +π +1 +√ +1 − z2 dzdr. +The integral can be exactly calculated for the following two cases. +• Case 1: ∥y∥2 < τ meaning the integration range is below the line r = −z∥y∥2 + τ. In this case, we have +E[1dist(y,m)<τ(y)] = E[1|z∥y∥2+r|<τ(y)] += +� 1 +−1 +� −z∥y∥2+τ +0 +1 +π +1 +√ +1 − z2 drdz += +� 1 +−1 +−z∥y∥2 + τ +π +√ +1 − z2 dz += −∥y∥2 +π +� 1 +−1 +z +√ +1 − z2 dz + τ +π +� 1 +−1 +1 +√ +1 − z2 dz += 0 + τ +π +�π +2 − (−π +2 ) +� += τ. +• Case 2: τ ≤ ∥y∥2 ≤ 1 − τ meaning the integration range is between the line: r = −z∥y∥2 + τ and r = −z∥y∥2 − τ. +In this case, we have +E[1dist(y,m)<τ(y)] = E[1|z∥y∥2+r|<τ(y)] += +� 0 +−1 +� −z∥y∥2+τ +−z∥y∥2−τ +1 +π +1 +√ +1 − z2 drdz += +� 0 +−1 +2τ +π +√ +1 − z2 dz += τ. +Combining Case 1 and 2, we have +E[1dist(y,m)<τ(y)] = τ for any ∥y∥2 ≤ 1 − τ. +Substituting this into Eq. (12) concludes the proof. +B +Setup of the experiments in Section 4.1 +We use the python package gstools (https://github.com/GeoStat-Framework/GSTools/) to generate realizations of the +Gaussian process. For a fixed correlation length, we generate 10 realizations of the Gaussian process, i.e., K = 10 in +Eq. (10), to tune the shape parameter γ of the transferable feature space. For the feature space for the two-dimensional +PDEs, we sample each realization at 502 uniformly distributed locations in B1(0), i.e., J = 2500 in Eq. (9), to compute the +MSE in Eq. (9). For the feature space for the three-dimensional PDEs, we sample each realization at 503, i.e., J = 125, 000 +in Eq. (9), to compute the MSE in Eq. (9). A simple grid search is used to solve the one-dimensional optimization problem +in Eq. (10) to find the optimal shape parameter γ. + +TransNet: Transferable Neural Networks for PDEs +13 +C +Definitions of the PDEs in Section 4.2 +The definitions of the PDEs considered in Section 4.2 are given below. +The Poisson’s equation considered in case (C1)–(C5) is defined by +∆u(x) = f(x), +(14) +where the exact solution for the 2D settings, i.e., (C1)–(C4), is u(x) = sin(2πx1) sin(2πx2) sin(2πx3), and the exact +solution for the 3D setting, i.e., (C5), is u(x) = sin(2πx1) sin(2πx2). The forcing term f(x) can be obtained by applying +the Laplacian operator to the exact solution. The domains of computation for (C1)–(C5) are given below: +(C1) A 2D box domain: Ω = [−1, 1]2; +(C2) A 2D circular domain: Ω = B1(0); +(C3) A 2D L-shaped domain: Ω = [−1, 1]2\[0, 1]2; +(C4) A 2D annulus domain: Ω = B1(0)\B0.5(0); +(C5) A 3D box domain Ω = [−1, 1]3. +We consider the Dirichlet boundary condition in the experiments, where the boundary condition g(x) in Eq. (1) can be +obtained by restricting the exact solution on the boundary of Ω. Figure 6 illustrates how to place the domains of computation +into the unit ball for for the test cases (C1) – (C4) to use the transferable feature space. +Figure 6: Illustration of how to place the domains of computation for the test cases (C1) – (C4) in Section 4.2 into the unit +ball to use the transferable feature space to solve the Poisson’s equation in different domains. +The steady-state Navier-Stokes equation considered in case (C6) is defined by: +u · ∇u + ∇p − ν∆u = 0 +∇ · u = 0 +where u = (v1, v2) represents the velocity, p is the pressure, ν is the viscosity and Re = 1/ν is the Reynold’s number. The +domain of computation is Ω = [−0.5, 1] × [−0.5, 1.5] with Direchilet boundary condition. We consider the Kovasznay flow +problem that has the exact solution, i.e., +v1(x1, x2) = 1 − eλx1 cos(2πx2) +(15) +v2(x1, x2) = λ +2π eλx1 sin(2πx2) +(16) +p(x1, x2) = 1 +2(1 − e2πx1) +(17) +where λ = +1 +2ν − +� +1 +4ν2 + 4π2 and the Reynold’s number is set to 40. The Dirichlet boundary condition can be obtained by +restricting the exact solution on the boundary of Ω. +The Fokker-Planck equation considered in case (C7) and (C8) is defined by +∂u(t, x) +∂t ++ b(t, x) +d +� +i=1 +∂u +∂xi +(t, x) + σ2 +2 +d +� +i,j=1 +∂2u +∂xixj +(t, x) = 0, +u(0, x) = g(x), +(18) + +1.00 +1.00 +1.00 +1.00 +0.75 +0.75 +0.75 +0.75 +0.50 +0.50 +0.50 +0.50 +0.25 +0.25 +0.25 +0.25 +0.00 +0.00 +0.00 +0.00 +0.25 +0.25 +-0.25 +0.25 +0.50 +0.50 +0.50 +0.50 +0.75 +0.75 +0.75 +0.75 +1.00 +1.00 +1.00 +1.00 +1.000.750.500.250.000.250.500.751.00 +-1.00 0.75 0.50 0.25 0.000.250.500.751.00 +-1.00 0.75 0.50 0.25 0.000.250.500.751.00TransNet: Transferable Neural Networks for PDEs +14 +where the coefficients b(t, x), σ, g(x) and the exact solutions are +• (C7): b(x, t) = 2 cos (3t), σ = 0.3, u(x, 0) = p(x; 0, 0.42) and u(x, t) = p(x; 2 sin (3t) +3 +, 0.42 + t0.32), where +p(x; µ, Σ) denote the Gaussian density with mean µ and variance Σ. +• (C8): b(x1, x2, t) = [sin(2πt), cos(2πt)]T , σ = 0.3, u(x1, x2, 0) = p(x; [0, 0], 0.42I2), and u(x1, x2, t) = +p(x; [− cos(2πt)−1 +2π +, sin(2πt) +2π +], (0.42 + t0.32)I2), where p(x; µ, Σ) is the Gaussian density with mean µ and variance Σ. +The wave equation considered in case (C9) is defined by +∂2u +∂t2 = c∂2u +∂x2 , x ∈ [0, 1], t ∈ [0, 2] +u(x, 0) = sin(4πx) +u(0, t) = u(1, t) +where c = 1/(16π2). The domain of computation is Ω = [0, 1] × [0, 2]; the exact solution is +u(x, t) = 1 +2 (sin(4πx + t) + sin(4πx − t)) . +D +Setup of the experiments in Section 4.2 +We specify the setup for the test cases (C1) to (C9) as follows: +• (C1): We evaluate the loss function in Eq. (11) on a 50×50 uniform mesh in Ω = [−1, 1]2, i.e., J1 = 2500 in Eq. (11), +and on 200 uniformly distributed points on ∂Ω, i.e., J2 = 200. After solving the least squares problem, we compute +the error, i.e., the results shown in Figure 4 on a test set of 10,000 uniformly distributed random locations in Ω. +• (C2): We evaluate the loss function in Eq. (11) on a 50 × 50 uniform mesh in Ω = [−1, 1]2 and mask off the grid +points outside the domain Ω = B1(0), i.e., J1 = 1876, and evaluate the boundary loss on 200 uniformly distributed +points on ∂Ω, i.e., J2 = 200. After solving the least squares problem, we compute the error, i.e., the results shown in +Figure 4 on a test set of 10,000 uniformly distributed random locations in Ω. +• (C3): We evaluate the loss function in Eq. (11) on a 50 × 50 uniform mesh in Ω = [−1, 1]2 and mask off the grid +points outside the domain Ω = [−1, 1]2\[0, 1]2, i.e., J1 = 1875, and evaluate the boundary loss on 200 uniformly +distributed points on ∂Ω, i.e., J2 = 200. After solving the least squares problem, we compute the error, i.e., the results +shown in Figure 4 on a test set of 10,000 uniformly distributed random locations in Ω. +• (C4): We evaluate the loss function in Eq. (11) on a 50 × 50 uniform mesh in Ω = [−1, 1]2 and mask off the grid +points outside the domain Ω = B1(0)\B0.5(0), i.e., J1 = 1408, and evaluate the boundary loss on 200 uniformly +distributed points on ∂Ω, i.e., J2 = 200. After solving the least squares problem, we compute the error, i.e., the results +shown in Figure 4 on a test set of 10,000 uniformly distributed random locations in Ω. +• (C5): We evaluate the loss function in Eq. (11) on a 10,000 uniformly distributed random locations in Ω = [−1, 1]3, +i.e., J1 = 10000, and evaluate the boundary loss on 2400 uniformly distributed points on ∂Ω, i.e., J2 = 2400, 400 +points on each side of Ω. After solving the least squares problem, we compute the error, i.e., the results shown in Figure +4 on a test set of 10,000 uniformly distributed random locations in Ω. +• (C6): We evaluate the loss function in Eq. (11) on a 50 × 50 uniform mesh in Ω = [−0.5, 1] × [−0.5, 1.5], i.e., +J1 = 2500 in Eq. (11), and on 200 uniformly distributed points on ∂Ω (50 points on each side of the box), i.e., +J2 = 200. We use Pichard iteration to handle the nonlinearity. Specifically, the residual loss is defined by +loss = uk−1 +NN · ∇uk +NN + ∇pk +NN − ν∆uk +NN, +where k is the Picard iteration number. In the k-th iteration, the nonlinear term uk−1 +NN · ∇uk +NN becomes linear due to +the use of uk−1 +NN . After solving the least squares problem, we compute the error, i.e., the results shown in Figure 4 on a +test set of 10,000 uniformly distributed random locations in Ω. + +TransNet: Transferable Neural Networks for PDEs +15 +• (C7): The domain of computation is (t, x) ∈ [0, 1] × [−2, 2]. We evaluate the loss function on a 50 (time) × 200 +(space) = 10,000 grid points in the domain Ω. We use the absorbing boundary condition in the spatial domain. We +have a total of 3000 samples on the boundary of Ω, i.e., 1000 samples for each of u(x, 0), u(2, t) and u(−2, t). After +solving the least squares problem, we compute the error, i.e., the results shown in Figure 4 on a test set of 10,000 +uniformly distributed random locations in Ω. +• (C8): The domain of computation is t ∈ [0, 1] and (x1, x2) ∈ [−2, 2]2. We evaluate the loss function on 10000 +uniformly selected random points in the domain Ω. We use the absorbing boundary condition in the spatial domain. +In terms of samples on the boundary, we have 50 × 50 = 2500 grid points for the initial condition u(x1, x2, 0), +20(time) × 50(space) = 1000 grid points for each of u(±2, x2, t) and u(x1, ±2, t). After solving the least squares +problem, we compute the error, i.e., the results shown in Figure 4 on a test set of 10,000 uniformly distributed random +locations in Ω. +• (C9): We evaluate the loss function in Eq. (11) on 50(time) × 100(space) = 2500 grid points in domain, i.e., +J1 = 10000, and evaluate the boundary loss on 1000 uniformly distributed points on ∂Ω, i.e., J2 = 1500, 500 points +on each side of Ω. After solving the least squares problem, we compute the error, i.e., the results shown in Figure 4 on +a test set of 10,000 uniformly distributed random locations in Ω. +We use the standard least squares solver torch.linalg.lstsq in Pytorch to solve all the least squares problems. Our +code is implemented using Pytorch on a workstation with an NVIDIA Tesla V100 GPU. +Setup for PINN. For each test case, PINN uses exactly the same setting as TransNet, including network architecture, loss +function, data, to ensure fair comparison. In terms of training, we set learning rate to 0.001 with a decrease factor of 0.7 +every 1000 epochs. We first use Adam optimizer to train the neural networks for 5000 epochs, which gives us the results in +Figure 4 labeled by “PINN:Adam”. Then we continue training the network using LBFGS for another 200 iterations, which +gives us the results in Figure 4 labeled by “PINN:Adam+BFGS”. +Setup for for the random feature models. The random feature model use exactly the same setting as TransNet, including +network architecture, loss function, data, to ensure fair comparison. The parameters {wm, bm}M +m=1 are determined by the +default initialization methods in Pytorch, and the parameters in the output layer is obtained by the least squares solver +torch.linalg.lstsq in Pytorch. + diff --git a/fNFKT4oBgHgl3EQfAy3n/content/tmp_files/load_file.txt b/fNFKT4oBgHgl3EQfAy3n/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c39557affd8a25b87745d980e3a1c220a9dd740b --- /dev/null +++ b/fNFKT4oBgHgl3EQfAy3n/content/tmp_files/load_file.txt @@ -0,0 +1,1293 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFKT4oBgHgl3EQfAy3n/content/2301.11701v1.pdf,len=1292 +page_content='TransNet: Transferable Neural Networks for Partial Differential Equations Zezhong Zhang 1 Feng Bao 1 Lili Ju 2 Guannan Zhang 3 * Abstract Transfer learning for partial differential equations (PDEs) is to develop a pre-trained neural network that can be used to solve a wide class of PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFKT4oBgHgl3EQfAy3n/content/2301.11701v1.pdf'} +page_content=' Ex- isting transfer learning approaches require much information of the target PDEs such as its formu- lation and/or data of its solution for pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFKT4oBgHgl3EQfAy3n/content/2301.11701v1.pdf'} +page_content=' In this work, we propose to construct transfer- able neural feature spaces from purely function approximation perspectives without using PDE information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFKT4oBgHgl3EQfAy3n/content/2301.11701v1.pdf'} +page_content=' The construction of the feature space involves re-parameterization of the hidden neu- rons and uses auxiliary functions to tune the re- sulting feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFKT4oBgHgl3EQfAy3n/content/2301.11701v1.pdf'} +page_content=' Theoretical analysis shows the high quality of the produced feature space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFKT4oBgHgl3EQfAy3n/content/2301.11701v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFKT4oBgHgl3EQfAy3n/content/2301.11701v1.pdf'} +page_content=', uniformly distributed neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFKT4oBgHgl3EQfAy3n/content/2301.11701v1.pdf'} +page_content=' Extensive nu- merical experiments verify the outstanding perfor- mance of our method, including significantly im- proved transferability, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFKT4oBgHgl3EQfAy3n/content/2301.11701v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFKT4oBgHgl3EQfAy3n/content/2301.11701v1.pdf'} +page_content=', using the same feature space for various PDEs with different domains and boundary conditions, and the superior accu- racy, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFKT4oBgHgl3EQfAy3n/content/2301.11701v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFKT4oBgHgl3EQfAy3n/content/2301.11701v1.pdf'} +page_content=', several orders of magnitude smaller mean squared error than the state of the art meth- ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFKT4oBgHgl3EQfAy3n/content/2301.11701v1.pdf'} +page_content=' 1 Introduction Rapid advancement of deep learning has attracted signif- icant attention of researchers to explore how to use deep Corresponding author 1Department of Mathematics, Florida State University, Tallahassee, FL 32306, USA 2Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA 3Computer Science and Mathematics Division, Oak Ridge National Laboratory, TN 37831, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFKT4oBgHgl3EQfAy3n/content/2301.11701v1.pdf'} +page_content=' Correspondence to: Guannan Zhang –1.0 dex. For the latter stars, the Mg triplet at 6318-19 +Å should be preferred because these lines are still sensitive to the +Mg abundance. Therefore, for the stars for which the Mg line at +5711 Å turns out to be saturated, the Mg abundance has been +derived from the Mg triplet using spectral synthesis in order to +include the contribution of the close auto-ionization Ca line. +Finally, only for the Na lines used here (5682-88 Å and +6154-60 Å ) we corrected the derived abundances for departures +from the LTE assumption applying the corrections by Lind et al. +(2011). +The abundances are referred to the solar ones, taking as ref- +erence the values from Grevesse & Sauval (1998), apart from +oxygen for which the adopted value is from Caffau et al. (2011). +3.5. Abundance uncertainties +In the determination of the uncertainties in each derived abun- +dance ratio we take into account two main sources of error, +namely the errors arising from the measurement procedure (EW +or spectral synthesis) and those arising from atmospheric param- +eters. +(1) Uncertainties related to the measurement procedure are +computed as the dispersion of the mean normalised to the +root mean square of the number of used transitions. Properly, +this term includes both uncertainties from line fitting and from +adopted log gf values. For the elements measured from the EWs +and for which only one line is available, the DAOSPEC uncer- +tainty associated to the Gaussian fitting procedure (correspond- +ing to 1σ of the fit residuals) is assumed as internal error. For +the elements (O and La) for which only one transition has been +measured using spectral synthesis, the internal error has been +estimated by means of Monte Carlo simulations, creating a sam- +ple of 500 synthetic spectra with a Poissonian noise that repro- +duces the observed SNR and repeating the line-fitting procedure. +The dispersion of the abundance distribution obtained from these +noisy synthetic spectra is assumed as 1σ uncertainty. +(2) Uncertainties due to atmospheric parameters have been es- +timated by repeating the analysis by varying each time a given +parameter of the corresponding 1σ error and keeping fixed the +other parameters. +These two sources of uncertainties have been added in +quadrature. Since the abundance of the species X is expressed as +abundance ratios [X/Fe], also the uncertainties in the Fe abun- +dance have been taken into account. The final errors in [Fe/H] +and [X/Fe] abundance ratios are calculated as follows: +σ[Fe/H] = +� +σ2 +Fe +NFe ++ (δTeff +Fe )2 + (δlog g +Fe )2 + (δvt +Fe)2 +(1) +σ[X/Fe] = +� +σ2 +X +NX ++ σ2 +Fe +NFe ++ (δTeff +X +− δTeff +Fe )2 + (δlog g +X +− δlog g +Fe )2 + (δvt +X − δvt +Fe)2 +(2) +where σX,Fe is the dispersion around the mean of the chem- +ical abundances, NX,Fe is the number of lines used to derive +the abundances and δi +X,Fe are the abundance variations obtained +modifying the atmospheric parameter i. +3.6. Abundances of the MW control sample +Table 4 lists the average abundance ratios, together with the stan- +dard deviation and the average uncertainty in the abundance ra- +tio, for the 5 GCs of the MW control sample. We compared +the atmospheric parameters and [Fe/H] of the analysed stars +Article number, page 6 of 16 + +[0 1] +1.2 +Flux +Normalize +0.8 +0.6 +0.4 +0.2 +6298 +6300 +6302 +6304 +6306 +Wavelength (A)Mucciarelli et al.: SMC field stars +with those by Carretta et al. (2009) and Carretta et al. (2014) +that analysed the same spectroscopic dataset. The average differ- +ences between our analysis and the literature ones are +52±11 K +(σ= 50 K) for Teff , –0.01±0.01 (σ= 0.03) for log g, +0.07±0.04 +km s−1(σ= 0.19 km s−1) for vt and –0.03±0.02 dex (σ= 0.07 +dex) for [Fe/H]. +4. RV and [Fe/H] distributions +4.1. RV distribution +According to previous spectroscopic studies (Harris & Zaritsky +2006; Carrera et al. 2008; Dobbie et al. 2014a; De Leo et al. +2020; Hasselquist et al. 2021) we identified as members of the +SMC those stars with RV between +80 and +250 km s−1. The +membership is confirmed also by the proper motions measured +from Gaia EDR3 (Gaia Collaboration et al. 2021). We exclude +from the chemical analysis stars members of the GC associated +to each field (these stars will be discussed in a forthcoming paper +of the series), stars with spectra contaminated by prominent TiO +or C2 molecular bands or with too low SNR. The final sample +discussed in this work includes a total of 206 stars out of the 320 +observed stars. The RV and [Fe/H] for this sample are listed in +Table 2. Fig. 4 and 5 shows the RV and [Fe/H] discrete and ker- +nel density distributions of the three SMC fields. The advantage +of the latter representation is that the distribution is independent +of the choice of the bin width and of the starting bin, at variance +with the discrete distributions. +0 +2 +4 +6 +8 +100 +150 +200 +RV (km/s) +count +0 +2 +4 +6 +8 +−2.5 +−2.0 +−1.5 +−1.0 +−0.5 +[Fe/H] +count +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +100 +150 +200 +RV (km/s) +count +0 +5 +10 +15 +−2.5 +−2.0 +−1.5 +−1.0 +−0.5 +[Fe/H] +count +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +100 +150 +200 +RV (km/s) +count +0 +5 +10 +15 +−2.5 +−2.0 +−1.5 +−1.0 +−0.5 +[Fe/H] +count +Fig. 4. RV and [Fe/H] distributions (left and right panel, respectively) +of the three fields. Colours indicate the different fields: FLD-121 (red), +FLD-339 (green) and FLD-419 (blue). +The RV distributions of the three fields appear significantly +different with each other, both in terms of the main peak and +shape. The RV distribution of FLD-121 peaks at RV≈+125 +km s−1 , that of FLD-339 displays a peak at RV≈+160 km s−1 +, while that of FLD-419 exhibits two distinct peaks, the main +one at ≈+150 km s−1 and the second one at ≈+180 km s−1 . A +Kolmogorov-Smirnov test performed on these distributions con- +firms that the RV distributions of FLD-339 and FLD-419 are sig- +nificantly different with respect to that of FLD-121 (with statistic +significance larger than 99.9%), while we cannot reject the hy- +pothesis that FLD-339 and FLD-419 may derive from the same +population. +The differences in the peak of these three RV distributions +are compatible with the rotation pattern of the SMC as inferred +from low-resolution spectroscopic surveys of giant stars (Dob- +bie et al. 2014a; De Leo et al. 2020), from the HI column den- +sity map (Di Teodoro et al. 2019) and from the APOGEE re- +sults from 17th Data Release of the Sloan Digital Sky Survey +(Abdurro’uf et al. 2022). All these studies show that the western +side of the SMC, where FLD-121 is located, has a lower velocity +with respect to the eastern side. However, the presence of multi- +ple peaks, clearly visible in the distribution of FLD-419, seems +to suggest a more complex kinematic pattern (as discussed be- +low). +4.2. [Fe/H] distribution +The [Fe/H] distribution of the entire sample is peaked at +[Fe/H]∼–1.0 dex, with about 95% of the stars having [Fe/H] be- +tween –1.5 and –0.5 dex and with a weak but extended metal- +poor tail reaching [Fe/H]∼–2.2 dex. This distribution is qualita- +tively similar to those obtained from low-resolution spectra us- +ing Ca II triplet (Carrera et al. 2008; Dobbie et al. 2014a; Parisi +et al. 2016) and that from APOGEE data (Nidever et al. 2020). +However, similar to what we see with the RV distributions, when +the individual fields are considered, the metallicity distributions +appear different with each other. The distributions of FLD-339 +and FLD-419 are confined between –1.5 and –0.5 dex, with only +one star per field (∼1%) with [Fe/H]<–1.5 dex . On the other +hand the distribution of FLD-121 ranges from –0.8 dex down to +–2.2 dex, with ∼20% of the stars more metal-poor than –1.5 dex. +Note that the APOGEE field 47Tuc, superimposed to our field +FLD-121, exhibits a lower fraction of metal-poor stars, ∼2%, +probably reflecting some selection bias against metal-poor stars +in the APOGEE observations. +The peaks of the distributions of FLD-339 and FLD-419 are sep- +arated by ∼0.2 dex and located at [Fe/H]∼–0.9 and ∼–1.1 dex, +respectively. Also, the two distributions seem to be not symmet- +ric, with the presence a secondary peak at [Fe/H]∼–1.1 dex in +FLD-339 and a heavily-populated metal-rich tail or a secondary +peak in FLD-419 (see Sec. 4.5). +4.3. [Fe/H] distribution and the age-metallicity relation +We try to interpret the derived [Fe/H] distributions in terms of +ages, using as a guidance the SF histories recovered from Hub- +ble Space Telescope (Noel et al. 2007; Sabbi et al. 2009; Cignoni +et al. 2012, 2013) and ground-based (Massana et al. 2022) pho- +tometry, and the theoretical age-metallicity relations available +for the SMC (Pagel & Tautvaisiene 1998; Tsujimoto & Bekki +2009; Cignoni et al. 2013). +All these works agree that the early epochs of the SMC have +been characterised by a significant SF activity followed by a long +quiescent period, interrupted between ∼3 and ∼4 Gyr ago by sig- +nificant SF episodes, likely due to some merger events. The old- +est SMC GC, NGC 121, has an age of ∼10.5±0.5 Gyr (Glatt et +al. 2008) and a metallicity of [Fe/H]∼–1.2/–1.3 dex (Dalessan- +dro et al. 2016, A. Minelli et al. in prep.). This suggests that the +SF activity in the first Gyrs was able to increase the metallicity +to values as high as [Fe/H]∼–1.2/–1.3 dex. We can consider the +Article number, page 7 of 16 + +A&A proofs: manuscript no. smc_fld_v7 +Table 4. Average abundance ratios, the corresponding standard deviation and the average uncertainty for the five GCs of the MW control sample. +Ratio +NGC 104 +NGC 1851 +NGC 5904 +NGC 1904 +NGC 4833 +<> +σ +<> +σ +<> +σ +<> +σ +<> +σ +< σ[X/Fe] > +[Fe/H] +–0.84 +0.02 +–1.15 +0.03 +–1.29 +0.01 +–1.57 +0.04 +–2.11 +0.03 +0.07 +[O/Fe] ++0.42 +0.04 ++0.40 +0.03 ++0.47 +0.04 ++0.54 +0.04 ++0.59 +0.03 +0.08 +[Na/Fe] ++0.00 +0.03 +–0.15 +0.04 +–0.35 +0.04 +–0.40 +0.03 +–0.50 +0.05 +0.08 +[Mg/Fe] ++0.31 +0.04 ++0.34 +0.04 ++0.33 +0.03 ++0.35 +0.04 ++0.38 +0.02 +0.10 +[Si/Fe] ++0.28 +0.03 ++0.26 +0.05 ++0.28 +0.02 ++0.30 +0.06 ++0.46 +0.06 +0.11 +[Ca/Fe] ++0.29 +0.04 ++0.25 +0.03 ++0.26 +0.03 ++0.25 +0.01 ++0.26 +0.06 +0.09 +[Sc/Fe] ++0.35 +0.03 ++0.17 +0.04 ++0.26 +0.05 ++0.13 +0.03 ++0.28 +0.04 +0.08 +[Ti/Fe] ++0.24 +0.02 ++0.06 +0.01 ++0.16 +0.03 ++0.15 +0.01 ++0.20 +0.06 +0.09 +[V/Fe] ++0.19 +0.04 +–0.15 +0.03 +–0.06 +0.05 +–0.05 +0.03 +–0.09 +0.03 +0.10 +[Ni/Fe] +–0.04 +0.02 +–0.10 +0.04 +–0.11 +0.03 +–0.08 +0.02 +–0.11 +0.04 +0.06 +[Cu/Fe] +–0.02 +0.04 +–0.41 +0.02 +–0.37 +0.05 +–0.52 +0.03 +–0.60 +0.04 +0.08 +[Zr/Fe] ++0.30 +0.04 ++0.11 +0.03 ++0.10 +0.03 ++0.14 +0.06 ++0.06 +0.06 +0.12 +[Ba/Fe] ++0.04 +0.05 ++0.13 +0.04 ++0.08 +0.06 ++0.10 +0.04 ++0.31 +0.05 +0.12 +[La/Fe] ++0.29 +0.03 ++0.35 +0.05 ++0.24 +0.04 ++0.14 +0.03 ++0.27 +0.06 +0.08 +SMC field stars in our sample with [Fe/H]<–1.3 dex (which are +almost all located in FLD-121) as formed in the first 1-2 Gyr of +the life of the galaxy. +The subsequent evolution of the SMC and the corresponding +metallicity distribution can be interpreted in the light of the the- +oretical age-metallicity relations: Fig. 6 shows that by Pagel & +Tautvaisiene (1998) assuming a burst of SF at an age of ∼4 Gyr. +After a long period characterised by a low SF efficiency (and +where the metallicity remains almost constant), the SF in the +SMC re-ignites with a prominent burst, likely triggered by the +first close encounter between SMC and LMC (Bekki et al. 2004; +Bekki & Chiba 2005). The most recent SF history for the SMC +provided by Massana et al. (2022) using the SMASH photom- +etry identified the re-ignition of the SF at ∼3.5 Gyr ago, simul- +taneously in both the Clouds. The stars with [Fe/H]>–1.3 dex +analysed here should be a mixture of stars with different ages +(from ∼1 to ∼10-11 Gyr). It is not easy to separate the different +populations in terms of age due to the almost constant [Fe/H] +over a large age range. Massana et al. (2022) identified in the +SF history of the SMC five peaks (at ∼3, 2, 1, 0.45 Gyr ago and +one still ongoing) occurring simultaneously also in the LMC. A +fascinating possibility is that the different peaks in the metallic- +ity distributions of FLD-339 and FLD-419 could be associated +to some of these different bursts of SF. Finally, we can suppose +that the stars with [Fe/H] around –0.6/–0.5 dex are likely formed +with the burst at 1 Gyr. This is confirmed also by the metallici- +ties of the stellar clusters with ages around 1 Gyr (see e.g. Parisi +et al. 2022) +4.4. Run of [Fe/H] with the distance +Previous spectroscopic studies (Carrera et al. 2008; Dobbie et +al. 2014a; Parisi et al. 2016; Choudhury et al. 2020; Grady et al. +2021) found evidence of a shallow (from –0.03 to –0.07 dex/deg) +metallicity gradient, within 3◦–5◦. Fig. 7 shows the run of [Fe/H] +of the spectroscopic targets with their projected distance from +the SMC centre (Ripepi et al. 2017). The mean metallicity in +three fields is consistent with the shallow gradient previously +proposed by Choudhury et al. (2020). However two main dif- +ferences between the external field FLD-121 and the two inter- +nal ones are evident. First, in FLD-121 the fraction of metal-poor +stars ([Fe/H]<–1.5 dex) is about ∼20%, against ∼1% in the other +two fields. The fraction of metal-poor stars increases outward re- +flecting a larger fraction of old stars with respect to those formed +subsequently during the long quiescent period and the recent SF +bursts and that are preferentially confined in the innermost re- +gion of the SMC (see e.g. Rubele et al. 2018). +Second, in the metallicity distribution of FLD-121 there is a +clear lack of stars with [Fe/H] between –0.8 and –0.5 dex, instead +detected in FLD-339 and FLD-419. Following the discussion +above, these stars should have ∼1 Gyr (the youngest stars among +the intermediate-age SMC populations). Again, this is consistent +with a scenario where the younger, metal-richer populations are +progressively more concentrated toward the innermost regions. +Age-metallicity gradients of this kind are quite common in dwarf +galaxies (see e.g. Taibi et al. 2022, and references therein). +4.5. Possible kinematic/chemically distinct sub-structures? +The distribution of the SMC stars in the RV-[Fe/H] plane seems +to suggest the presence of sub-structures, in particular the two +different peaks of the [Fe/H] distribution of FLD-339, the large +and asymmetric [Fe/H] distribution of FLD-419 and the double- +peak of the RV distribution of FLD-419. +We used the gaussian mixture package Mclust (Scrucca et al. +2016), within the R environment, to analyse the distribution of +FLD-339 and FLD-419 stars in the [Fe/H] - RV space. Mclust +choose the best model, both in terms of number and form (e.g., +equal or variable variance, orientation etc., see Scrucca et al. +2016) of the gaussian components, by means of the Bayesian +Information Criterion. Since we are interested in substructures +within the bulk of the metallicity distribution we exclude from +the analysis the two metal-poor outliers, one per field. While for +FLD-339 a single elliptical gaussian model is the preferred solu- +tion, the [Fe/H] - RV distribution of FLD-419 is best described +with two elliptical gaussian components with the same variance +both in [Fe/H] and RV. The gain of this model with respect to a +single elliptical gaussian is only marginal, in practice they pro- +vide an equally good representation of the data. Still, the solu- +tion synthesise the properties of the hypothesised two compo- +nents. The first component has (µ[Fe/H], µRV)= ( -0.85 dex, 171.8 +km s−1), and it accounts for 33% of the sample, the second com- +ponent has (µ[Fe/H], µRV)= ( -1.13 dex, 154.5 km s−1), accounting +for the remaining 67% of the sample. The standard deviations are +σ[Fe/H]= 0.10 dex, and σRV= 22.7 km s−1. It seems that the most +metal rich component has a larger systemic RV than its metal- +poor counterpart. +Article number, page 8 of 16 + +Mucciarelli et al.: SMC field stars +100 +150 +200 +−2.5 +−2.0 +−1.5 +−1.0 +−0.5 +[Fe/H] +RV (Km/s) +Fig. 5. RVs are plotted against [Fe/H] for target stars in the central panel of the figure. Colour-shaded areas denote the contours of the three clusters +RV vs [Fe/H] distributions. Side plots show the kernel distributions of the RV (right-hand panel) and the [Fe/H] values (top panel) for each cluster. +Same colours of Fig. 4 +. +As additional check, we performed a Kolmogorov-Smirnov +test on the RV sub-populations of FLD-419, separated according +to the metallicity of their member stars (and assuming [Fe/H]=– +1.05 dex as a boundary between the two groups of stars). We +obtained that the two RV distributions cannot be extracted from +the same population with a significance of 98%. +The size of the FLD-419 sample is not sufficient to put this +odd result on sound statistical bases, still it may be suggestive +of the presence of some chemo-kinematic substructures in the +SMC along this line of sight. In this respect, it is worth recalling +that the SMC has a substantial line-of-sight depth, depending +on the used tracers and ranging from a few kpc up to about 20 +kpc (see e.g. de Grijs & Bono 2015; Subramanian et al. 2017). +Therefore, when we observe stars in an individual SMC field we +are likely crossing different depths and we are sampling different +populations in terms of kinematics and metallicity. +5. Chemical abundance ratios +We derived abundances of Na, O, Mg, Si, Ca, Sc, Ti, V, Fe, +Ni, Cu, Zr, Ba and La for 206 SMC RGB stars. All the abun- +dances, with the corresponding uncertainties, are available in the +electronic form (Table 5). With respect to the APOGEE sam- +ple by Hasselquist et al. (2021) we measured a larger number +of species, in particular Na, Sc, Ti, V, Cu, Zr, Ba and La, not +included in that study. Fig. 8-11 show the behaviour of derived +abundance ratios as a function of [Fe/H] for the analysed SMC +stars, highlighting stars belonging to the different fields. These +abundance ratios are compared with those obtained for the con- +trol sample of 5 Galactic GCs, adopting the same assumptions +in the chemical analysis and therefore removing most of the sys- +tematics of the analyses. This comparison allows us to high- +light the real difference between SMC and MW stars of simi- +lar [Fe/H]. Additionally, we show abundance ratios for Galactic +field stars from the literature as reference. The comparison with +the literature is affected by the systematics among the different +analyses (in terms of model atmospheres, solar abundance val- +Article number, page 9 of 16 + +A&A proofs: manuscript no. smc_fld_v7 +Fig. 6. Main panel: age-metallicity relation by Pagel & Tautvaisiene +(1998). Side panel: the kernel [Fe/H] distributions for the individual +SMC field stars discussed in this work. +Fig. 7. Behaviour of [Fe/H] as a function of the projected distance from +the SMC centre (Ripepi et al. 2017), same colours of Fig. 1. The thick +grey line is the linear fit for the metallicity gradient estimated by Choud- +hury et al. (2020). +ues, NLTE corrections, linelists, use of dwarf and giant stars). +However, it is useful to display the overall trends in the MW +based on a large number of stars. In the following, we refer to the +MW control sample to quantify the main differences and simi- +larities between MW and SMC stars. +5.1. Na +Sodium is mainly produced in massive stars during the hydro- +static C and Ne burning, with a strong dependence of its yields +on the metallicity. Also, a smaller contribution is provided by +asymptotic giant branch (AGB) stars. In Galactic stars (both in +the control sample and in literature data), [Na/Fe] increases by +increasing [Fe/H] until it reaches solar values around [Fe/H]>–1 +dex. An offset is evident between the values in the control sam- +ple and in the literature, especially in the metal-poor regime and +likely due to the different NLTE corrections. Top-left panel of +Fig. 8 shows the distribution of [Na/Fe] of the observed targets. +The bulk of the SMC stars exhibits sub-solar [Na/Fe] abundance +ratios at any metallicities, with an average value of about –0.4/– +0.5 dex, similar to the typical [Na/Fe] measured in the LMC stars +(Van der Swaelmen et al. 2013; Minelli et al. 2021) but at higher +[Fe/H]. The low [Na/Fe] values measured in the SMC stars may +point to a lower contribution by massive stars, besides the larger +impact of Type Ia supernovae (SNe Ia) at low metallicities in +dwarf galaxies (Tolstoy et al. 2009). We observe a large scatter +of [Na/Fe], not fully explainable within the typical uncertainties, +and already detected in spectroscopic samples of LMC and SMC +metal-rich stars (Pompéia et al. 2008; Van der Swaelmen et al. +2013; Minelli et al. 2021; Hasselquist et al. 2021). This scat- +ter could reflect that multiple sites of Na production are taking +place. Finally, we note a systematic difference between the me- +dian [Na/Fe] values in FLD-339 and FLD-419, where the latter +displays [Na/Fe] higher by 0.1-0.15 dex. A systematic difference +in [Na/Fe] of different regions of the parent galaxy has been also +observed in the LMC (Van der Swaelmen et al. 2013) with the +stars in the LMC bar more enriched in [Na/Fe] by 0.2 dex with +respect to the LMC disc stars. +5.2. α-elements +The α-elements are produced mainly in short-lived massive stars +exploding as core-collapse supernovae (CC-SNe), while a mi- +nor fraction (depending on the element) is synthesised in SNe +Ia. Due to the time delay between the enrichment of the two +classes of SNe, the [α/Fe] abundance ratios are the classical trac- +ers of the relative timescales of the different SNe. In particular, +the metallicity of the knee (marking the onset of a significant +chemical contribution by SNe Ia) can be used as a proxy of the +SF efficiency of the galaxy (Tinsley 1979; Matteucci & Greggio +1986). +O and Mg (the so-called hydrostatic α-elements) are pro- +duced mainly in stars with masses larger than ∼30-35 M⊙ and +without contribution by SNe Ia. On the other hand, Si, Ca and +Ti (explosive α-elements) are produced in less massive stars +(∼15-25 M⊙) and with a smaller (but not negligible) contribu- +tion by SNe Ia (see e.g. Kobayashi et al. 2020b). Fig. 8 shows +the behaviour with [Fe/H] of individual [α/Fe] abundance ratios, +while Fig. 9 shows the run of the average values of hydrostatic +and explosive [α/Fe]. These abundance ratios in the SMC stars +clearly display a decrease by increasing the metallicity, moving +from enhanced values for the most metal-poor stars ([Fe/H]<– +1.5 dex) down to solar-scaled values in the dominant population. +This trend is in contrast with that obtained by the APOGEE sur- +vey, where "there is a slight increase in [Mg/Fe] beginning at +[Fe/H]∼–1.3 dex, with a peak at [Fe/H]∼–1.0 dex, followed by a +slight decrease. The [O/Fe], [Si/Fe] and [Ca/Fe] abundance pat- +terns are flat over this range" (Hasselquist et al. 2021). +The most metal-poor stars in our sample exhibit enhanced +values of [α/Fe] and in agreement with the results by Nidever +et al. (2020) and Reggiani et al. (2021) for SMC stars of sim- +ilar metallicity. Oxygen and magnesium, that are mainly pro- +duced by stars with masses larger than ∼30 M⊙, are, however, +slightly underabundant at low [Fe/H] with respect to the MW +sample, which points to a lower contribution from the most mas- +sive stars to the overall chemical enrichment of the SMC. The +subsequent decrease of [α/Fe] at higher [Fe/H] indicates that +these stars formed from a gas enriched by SNe Ia. For stars with +Article number, page 10 of 16 + +0.5 +1.0 +[Fe/H] +1.5 +-2.0 +2.5 +0 +5 +10 +Age (Gyr)0 +-0.5 +[Fe/H] +-1.5 +-2.5 +1 +1.5 +2 +2.5 +Distance (degrees)Mucciarelli et al.: SMC field stars +Fig. 8. Behaviour of the light element [Na/Fe] and α-elements [O/Fe], [Mg/Fe], [Si/Fe], [Ca/Fe] and [Ti/Fe] abundance ratios as a function of +[Fe/H] for SMC stars located in the fields FLD-419, FLD-339 and FLD-121 (blue, green and red circles, respectively). Arrows indicate upper +limits. The errorbars in the bottom-right corner indicate the typical uncertainties. Grey squares are the average values for the five Galactic GCs of +the control sample. Abundances of Galactic stars from the literature are also plotted as a reference: Edvardsson et al. (1993); Gratton et al. (2003); +Reddy et al. (2003, 2006); Bensby et al. (2005, 2014) for all the elements, Fulbright (2000); Stephens & Boesgaard (2002); Roederer et al. (2014) +for Na, Mg, Si, Ca and Ti, Adibekyan et al. (2012) for Na, Mg, Si and Ca, Barklem et al. (2005) for Mg. +Article number, page 11 of 16 + +0.5 +0 +Mg/Fe] +-0.5 +-2.5 +-2 +-1.5 +-0.5 +1 +[Fe/H]0.5 +0 +[Si/Fe] +-0.5 +-2.5 +-2 +-1.5 +-1 +-0.5 +[Fe/H]0.5 +0 +[Ca/Fe] +0.5 +-2.5 +-2 +-1.5 +-0.5 +[Fe/H]0.5 +0 +[Ti/Fe] +-0.5 +-2 +-2.5 +1.5 +-0.5 +[Fe/H]0.5 +0 +-0.5 +Na/Fe] +-2.5 +-1.5 +-0.5 +[Fe/H]0.5 +0 +[O/Fe] +-0.5 +2.5 +-2 +-1.5 +-0.5 +[Fe/H]A&A proofs: manuscript no. smc_fld_v7 +[Fe/H]>–1.5 dex the difference in [α/Fe] between SMC and MW +stars becomes more significant. In particular, the SMC-MW dif- +ference is more pronounced for hydrostatic α-elements, again +suggesting a lower contribution by stars with masses larger than +30-35 M⊙ to the chemical enrichment of the SMC. +We note, as for Na, that the metal-rich stars in FLD-419 are +slightly enhanced in [Ti/Fe], by ∼0.1 dex, with respect to the +stars of the other two fields with similar [Fe/H]. +Fig. 9. Behaviour of the hydrostatic and explosive average [α/Fe] abun- +dance ratios as a function of [Fe/H]. Same symbols of Fig. 8. +5.3. Iron-peak elements +Iron-peak elements are produced mainly in massive stars, +through different nucleosynthesis paths (Limongi & Chieffi +2003; Romano et al. 2010; Kobayashi et al. 2020b), and ejected +in the interstellar medium both from normal CC-SNe and hyper- +novae. These elements are also partly produced by SNe Ia on +longer time scales (Leung & Nomoto 2018; Lach et al. 2020). +Sc and V are produced mainly in massive stars, with a small +contribution by SN Ia only for V (Kobayashi et al. 2020b). +The SMC stars with [Fe/H]<–1.5 dex have Sc and V abun- +dances compatible with those measured in the control sample. +We note some offsets between these abundance ratios in the con- +trol sample and in the literature data, likely attributable to differ- +ent linelists (in terms of log gf and/or hyperfine structures. On the +other hand, the metal-rich SMC stars have abundances of Sc and +V significantly lower than the MW stars (see Fig. 10). For both +elements, we observed a decrease of the abundance ratio with in- +creasing [Fe/H] because of the overwhelming delayed contribu- +tion to Fe by SNe Ia. This behaviour resembles those observed in +metal-rich stars of dwarf galaxies, like LMC and Sagittarius (see +e.g. Sbordone et al. 2007; Minelli et al. 2021). We note that also +[V/Fe] in metal-rich stars of FLD-419 is systematically higher +by ∼0.15 dex with respect to the stars of FLD-339. A compara- +ble shift has been detected also for [V/Fe] in the LMC disc and +bar stars (Van der Swaelmen et al. 2013). +Ni is largely produced by SN Ia, with production also by +CC-SNe, similar to the production of Fe. The SMC stars have +[Ni/Fe] values compatible with those measured in the GCs of the +control sample until [Fe/H]∼–1.0 dex, while for higher metal- +licities this abundance ratio slightly decreases, reaching values +around [Ni/Fe]∼–0.2 dex (see Fig. 10). A similar behaviour in +the SMC stars has been observed by Hasselquist et al. (2021). +This mild trend resembles that observed for [Ni/Fe] in the +LMC and in Sagittarius at higher [Fe/H] (Minelli et al. 2021). +The decrease of [Ni/Fe] at higher metallicities is not observed +in MW stars, where [Ni/Fe] remains constant. In this respect, +Kobayashi et al. (2020a) suggested a lower contribution by sub- +Chandrasekhar mass SN Ia to reproduce the [Ni/Fe] measured in +dwarf spheroidal galaxies. +Cu is produced mainly in massive stars through the weak s- +process (Romano & Matteucci 2007), with a small contribution +by AGB stars (Travaglio et al. 2004) and a negligible contri- +bution by SN Ia (Iwamoto et al. 1999; Romano & Matteucci +2007). The [Cu/Fe] abundance ratio in the SMC stars exhibits a +large star-to-star dispersion and it is difficult to establish its real +trend. However, it is clear that the most metal-rich SMC stars +have [Cu/Fe] lower than that measured in MW stars, indicating +again a lower contribution to the chemical enrichment by mas- +sive stars. Values of [Cu/Fe] lower than those measured in MW +stars have been observed also in the LMC (Van der Swaelmen et +al. 2013), Sagittarius (Sbordone et al. 2007) and Omega Centauri +(Cunha et al. 2002). +5.4. Neutron capture elements +Elements heavier than the iron-peak group are produced through +neutron capture processes on seed nuclei, followed by β decays +(Burbidge et al. 1957). The neutron capture elements measured +here (namely Zr, Ba and La) are produced mainly by the slow +process occurring in low-mass (1-3 M⊙) AGB stars and in a mi- +nor amount in more massive stars (Busso et al. 1999; Cristallo +et al. 2015). At low metallicities these elements are produced +also through rapid processes (Truran 1981), occurring in rare and +energetic events like neutron star mergers or collapsars. In this +spectroscopic dataset, there are no transition of pure r-process +elements (i.e. Eu) and we cannot discuss the relative contribu- +tion of these two production channels. However, Reggiani et al. +(2021) analysed 4 metal-poor SMC giant stars finding [Eu/Fe] +values higher than those of the MW stars, supporting a strong +contribution at these metallicities by r-process. +The SMC stars show [Zr/Fe] and [La/Fe] abundance ratios +similar, within the star-to-star scatter, to those observed in MW +stars, and slightly higher [Ba/Fe]. Generally, these results sug- +gest that the enrichment by AGB stars in the SMC has been +Article number, page 12 of 16 + +0.5 +0 +[(O+Mg)/Fe] +-0.5 +-2.5 +1.5 +-0.5 +1 +[Fe/H]0.5 +[(Si+Ca+Ti)/Fe +-0.5 +-2.5 +-2 +-1.5 +-0.5 +[Fe/H]Mucciarelli et al.: SMC field stars +Fig. 10. Behaviour of the iron-elements [Sc/Fe], [V/Fe], [Ni/Fe] and [Cu/Fe] abundance ratios as a function of [Fe/H]. Abundances of Galactic +field stars are from Reddy et al. (2003, 2006); Roederer et al. (2014) for all the elements, Gratton et al. (2003) for Sc, V and Ni, Fulbright (2000) +for V and Ni, Adibekyan et al. (2012) for Sc and Ni, Edvardsson et al. (1993); Stephens & Boesgaard (2002); Bensby et al. (2005) for Ni, Bihain +et al. (2004); Yan et al. (2015) for Cu. Same symbols of Fig. 8. +comparable to that in the MW. [Ba/Fe] in the SMC stars is en- +hanced (∼+0.3/+0.4 dex) and higher than the values measured +in the MW stars. [Ba/Fe] displays a large scatter at all the metal- +licities and not explainable in light of the typical uncertainties +in the abundance ratios (∼0.15 dex). At [Fe/H]∼–1.0 dex, the +SMC stars have values of [Ba/Fe] higher than those observed +in the MW stars, suggesting a galaxy-wide initial mass func- +tion (IMF) biased in favour of the low-mass stars in the SMC. A +similar behaviour is observed for [La/Fe], while [Zr/Fe] presents +a trend in agreement with that observed for the MW. Similar +to what we observe for Na and V, also for [Zr/Fe] we found a +shift (∼0.2 dex) between FLD-339 and FLD-419 that resemble +those observed for the same ratio between LMC disc and bar +stars (Van der Swaelmen et al. 2013). The high values of [Ba/Fa] +and [La/Fe], together with the large star-to-star scatter, suggest +that the production of s-process elements has been very efficient +in the SMC, while the large star-to-star scatter could arise from +enrichment from AGB stars of different metallicities, being the +yields of AGB stars for these elements extremely metallicity- +dependent. +Finally, we identified a few stars with high [Ba/Fe] and +[La/Fe] values (>0.5-0.7 dex, reaching also +1.3 dex). A similar +enhancement of s-process elements could be due to mass transfer +processes from an AGB companion star in a binary system. +6. Conclusions +The analysis of optical spectra of 206 SMC RGB stars located +in three different positions of the parent galaxy has allowed us +to highlight some finer details of the complex and still poorly +known nature of this galaxy. The main results are summarised as +follows: +– The RV and [Fe/H] distributions of the three fields are dif- +ferent with each other. The fields FLD-339 and FLD-419, +despite the same distance from the SMC centre, have [Fe/H] +distributions peaked at different values, separated by 0.2 dex. +Article number, page 13 of 16 + +0.5 +0 +-0.5 +[ScI/Fe] +-2.5 +-2 +1.5 +1 +-0.5 +[Fe/H]0.5 +0 +-0.5 +[V/Fe] +2.5 +-2 +-1.5 +-0.5 +[Fe/H]0.5 +0 +-0.5 +[Ni/Fe] +-2.5 +-2 +1.5 +1 +-0.5 +[Fe/H]0.5 +0 +-0.5 +-1.5 +[Cu/Fe] +-2.5 +-2 +-1.5 +1 +-0.5 +[Fe/H]A&A proofs: manuscript no. smc_fld_v7 +These two populations could be connected to different bursts +of SF occurring in the recent life of the SMC (Massana et al. +2022) or the result of a different chemical enrichment path in +these regions (despite their similar projected distance from +the SMC centre). +– The fraction of metal-poor ([Fe/H]<–1.5 dex) stars increases +outward, being ∼1% in the two internal fields and ∼20% +in FLD-121. This run likely reflects an age gradient in the +SMC, with the internal regions dominated by intermediate- +age, metal-rich stars and the outskirts by the old, metal-poor +spheroid (see e.g. Rubele et al. 2018). +– The RV-[Fe/H] distribution of the observed fields seems +to suggest the possible existence of chemically/kinematic +distinct substructures. In particular, we potentially identi- +fied two groups of stars, one around [Fe/H]∼–1.1 dex and +RV∼+154 km s−1 and the other around [Fe/H]∼–0.9 dex and +RV∼+172 km s−1 . More data are needed to confirm the +statistical significance of these chemo-kinematical substruc- +tures. +– The SMC displays, especially for the dominant, metal-rich +component, distinct abundance patterns with respect to the +MW stars. In particular, those elements mainly produced by +massive stars (Na, α, Sc, V and Cu) have abundance ratios +lower than those measured in the MW stars. This suggests +that the gas from which these stars formed has been poorly +enriched by the most massive stars. This can be explained +in light of the low SF rate expected for a galaxy as small as +the SMC, leading to a lower contribution by massive stars +to the overall chemical enrichment of the galaxy (Jeˇrábková +et al. 2018; Yan et al. 2020). This is confirmed also by the +most metal-poor stars of the sample that exhibit [O/Fe] and +[Mg/Fe] ratios slightly lower than those in MW stars of sim- +ilar [Fe/H]. +– The [s/Fe] abundance ratios are enriched with respect to +the MW stars, with a large star-to-star scatter, suggesting +that these elements are produced by AGB stars of different +masses and metallicities. Also, the enhancement of the [s/Fe] +abundance ratios in the SMC seems to suggest a galaxy-wide +IMF biased in favour of the low-mass stars in the SMC. +– The possibility that the IMF is not universal, but varies with +the environment is the subject of lively debate (Bastian et +al. 2010; Hopkins 2018; Smith 2020). Theoretically, if stars +form in clusters according to IMFs that depend on the metal- +licity and density of the parent gaseous clumps, it is possi- +ble to calculate the integrated galaxy-wide IMF that in turn +depends on the metallicity and star formation rate of the +host galaxy (Jeˇrábková et al. 2018; Yan et al. 2020). More- +over, the abundance ratios of chemical elements produced in +stars with initial masses falling in narrow and well-detached +ranges can be used as powerful, indirect probes of the shape +of the galaxy-wide IMF (e.g. Romano et al. 2017). +Observationally, the possibility that the Sagittarius dwarf +spheroidal galaxy had a stronger contribution from AGB +stars to its chemical enrichment than the MW and the LMC is +discussed in Hasselquist et al. (2021). Similarly, Hallakoun +& Maoz (2021), resting on Gaia DR2 data, point to a bottom- +heavy IMF for the Gaia-Enceladus progenitor. Finally, Muc- +ciarelli et al. (2021) claim that the LMC GC NGC 2005 must +have formed in an accreted system that experienced an ex- +tremely low star formation rate and, hence, an extremely low +number of hypernova explosions, in order to explain the pe- +culiarly low Zn abundance of the cluster. +On the other hand, Hill et al. (2019) do not find any clear +cut evidence in favour of a non-standard IMF in the Sculp- +tor dwarf spheroidal galaxy. In a forthcoming paper, we will +quantitatively deal with the issue of IMF variations in the +SMC by computing chemical evolution models specifically +tailored to this galaxy (Romano et al., in preparation). +– The three fields exhibit similar chemical patterns for all the +elements but Na, V, Zr and Ti showing subtle differences +among the fields. Differences in the same abundance ratios +have been observed also in the LMC between bar and disk +stars (Van der Swaelmen et al. 2013). These differences con- +firm that the chemical enrichment history in the SMC has +been not uniform but depends on the position within the +galaxy. +These promising results enforces the need to study the prop- +erties of the SMC stars locally rather than globally, with an effort +to enlarge the samples of high-resolution spectra located in dif- +ferent regions of the galaxy. In this respect, the advent of the +multi-object spectrographs like MOONS at the Very Large Tele- +scope (Cirasuolo et al. 2020) and 4MOST at the VISTA Tele- +scope (de Jong et al. 2019) will allow us a significant improve- +ment in the investigation of possible chemically-distinct sub- +structures in the Magellanic Clouds (Gonzalez et al. 2020). +Acknowledgements. We thanks the referee, Mathieu Van der Swaelmen, for the +useful comments and suggestions. This research is funded by the project "Light- +on-Dark" , granted by the Italian MIUR through contract PRIN-2017K7REXT. +C. Lardo acknowledges funding from Ministero dell’Università e della Ricerca +through the Programme Rita Levi Montalcini (grant PGR18YRML1). This work +has made use of data from the European Space Agency (ESA) mission Gaia +(https://www.cosmos.esa.int/gaia), processed by the Gaia Data Process- +ing and Analysis Consortium (DPAC,https://www.cosmos.esa.int/web/ +gaia/dpac/consortium). Funding for the DPAC has been provided by na- +tional institutions, in particular the institutions participating in the Gaia Mul- +tilateral Agreement. +Article number, page 14 of 16 + +Mucciarelli et al.: SMC field stars +Fig. 11. Behaviour of the neutron capture-elements [Zr/Fe], [Ba/Fe] and +[La/Fe] abundance ratios as a function of [Fe/H]. Abundances of Galac- +tic field stars are from Mishenina et al. (2013); Roederer et al. (2014) +for all the elements, Edvardsson et al. (1993); Fulbright (2000); Reddy +et al. (2003) for Zr and Ba, Burris et al. (2000); Battistini & Bensby +(2016) for Zr and La, Stephens & Boesgaard (2002); Barklem et al. +(2005); Bensby et al. (2005) for Ba. +References +Abdurro’uf, Accetta, K., Aerts, C., et al. 2022, ApJS, 259, 35. doi:10.3847/1538- +4365/ac4414 +Adibekyan, V. Z., Sousa, S. G., Santos, N. C., et al. 2012, A&A, 545, A32. +doi:10.1051/0004-6361/201219401 +Andrae, R., Fouesneau, M., Creevey, O., et al. 2018, A&A, 616, A8 +Barklem, P. S., Christlieb, N., Beers, T. C., et al. 2005, A&A, 439, 129. +doi:10.1051/0004-6361:20052967 +Bastian, N., Covey, K. R., & Meyer, M. R. 2010, ARA&A, 48, 339. +doi:10.1146/annurev-astro-082708-101642 +Battistini, C. & Bensby, T. 2016, A&A, 586, A49. doi:10.1051/0004- +6361/201527385 +Bekki, K., Couch, W. J., Beasley, M. A., et al. 2004, ApJ, 610, L93 +Bekki, K., & Chiba, M., 2005, MNRAS, 356, 680 +Besla, G., Kallivayalil, N., Hernquist, L., et al. 2007, ApJ, 668, 949. +doi:10.1086/521385 +Besla, G., Kallivayalil, N., Hernquist, L., et al. 2010, ApJ, 721, L97. +doi:10.1088/2041-8205/721/2/L97 +Besla, G. 2015, arXiv:1511.03346 +Bensby, T., Feltzing, S., Lundström, I., et al. 2005, A&A, 433, 185. +doi:10.1051/0004-6361:20040332 +Bensby, T., Feltzing, S., & Oey, M. S. 2014, A&A, 562, A71. doi:10.1051/0004- +6361/201322631 +Bertaux, J. L., Lallement, R., Ferron, S., Boonne, C., & Bodichon, R., 2014, +A&A, 564, 46 +Bihain, G., Israelian, G., Rebolo, R., et al. 2004, A&A, 423, 777. +doi:10.1051/0004-6361:20035913 +Burbidge, E. M., Burbidge, G. R., Fowler, W. A., et al. 1957, Reviews of Modern +Physics, 29, 547. doi:10.1103/RevModPhys.29.547 +Burris, D. L., Pilachowski, C. A., Armandroff, T. E., et al. 2000, ApJ, 544, 302. +doi:10.1086/317172 +Busso, M., Gallino, R., & Wasserburg, G. J. 1999, ARA&A, 37, 239. +doi:10.1146/annurev.astro.37.1.239 +Caffau, E., Ludwig, H.-G., Steffen, M., et al. 2011, Sol. Phys., 268, 255 +Carrera, R., Gallart, C., Aparicio, A., Costa, E., Mendez, R. A., & Noel, N. E. +D., 2008, AJ, 136, 1039 +Carretta, E., Bragaglia, A., Gratton, R. G., et al. 2009, A&A, 505, 117. +doi:10.1051/0004-6361/200912096 +Carretta, E., Bragaglia, A., Gratton, R. G., et al. 2014, A&A, 564, A60. +doi:10.1051/0004-6361/201323321 +Castelli, F., & Kurucz, R. L., 2004, astro.ph-5087 +Choudhury, S., de Grijs, R., Rubele, S., et al. 2020, MNRAS, 497, 3746. +doi:10.1093/mnras/staa2140 +Cignoni, M., Cole, A. A., Tosi, M., Gallagher, J. S., Sabbi, E. Anderson, J., +Grebel, E. K. & Nota, A., 2012, ApJ, 754, 130 +Cignoni, M., Cole, A. A., Tosi, M., Gallagher, J. S., Sabbi, E. Anderson, J., +Grebel, E. K. & Nota, A., 2013, ApJ, 775, 83 +Cioni, M.-R.-L., van der Marel, R. P., Loup, C., & Habing, H. J., 2000, A&A, +359, 601 +Cirasuolo, M., Fairley, A., Rees, P., et al. 2020, The Messenger, 180, 10. +doi:10.18727/0722-6691/5195 +Cristallo, S., Straniero, O., Piersanti, L., et al. 2015, ApJS, 219, 40. +doi:10.1088/0067-0049/219/2/40 +Cunha, K., Smith, V. V., Suntzeff, N. B., et al. 2002, AJ, 124, 379. +doi:10.1086/340967 +Dalessandro, E., Lapenna, E., Mucciarelli, A., Origlia, L., Ferraro, F. R., & Lan- +zoni, B., 2016, ApJ, 829, 77 +de Grijs, R. & Bono, G. 2015, AJ, 149, 179. doi:10.1088/0004-6256/149/6/179 +de Jong, R. S., Agertz, O., Berbel, A. A., et al. 2019, The Messenger, 175, 3. +doi:10.18727/0722-6691/5117 +De Leo, M., Carrera, R., Noël, N. E. D., et al. 2020, MNRAS, 495, 98. +doi:10.1093/mnras/staa1122 +Di Teodoro, E. M., McClure-Griffiths, N. M., Jameson, K. E., et al. 2019, MN- +RAS, 483, 392. doi:10.1093/mnras/sty3095 +Dobbie, P. D., Cole, A. A., Subramaniam, A., & Keller, S., 2014, MNRAS, 442, +1663 +Dobbie, P. D., Cole, A. A., Subramaniam, A., & Keller, S., 2014, MNRAS, 442, +1680 +Edvardsson, B., Andersen, J., Gustafsson, B., et al. 1993, A&A, 500, 391 +Fuhr, J. R., Martin, G. A., & Wiese, W. L. 1988, Journal of Physical and Chemi- +cal Reference Data, 17 +Fulbright, J. P. 2000, AJ, 120, 1841. doi:10.1086/301548 +Fuhr, J. R. & Wiese, W. L. 2006, Journal of Physical and Chemical Reference +Data, 35, 1669. doi:10.1063/1.2218876 +Gaia Collaboration, Prusti, T., de Bruijne, J. H. J., et al. 2016, A&A, 595, A1 +Gaia Collaboration, Babusiaux, C., van Leeuwen, F., et al. 2018, A&A, 616, A10 +Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2021, A&A, 649, A1. +doi:10.1051/0004-6361/202039657 +Glatt, K. et al., 2008, AJ, 135, 1106 +Article number, page 15 of 16 + +0.5 +0 +Zr/Fe +-0.5 +-2.5 +-2 +-1.5 +-0.5 +[Fe/H]0.5 +0 +-0.5 +[Ba/Fe] +-2.5 +-2 +1.5 +-0.5 +[Fe/H]1.5 +0.5 +0 +[La/Fe +-0.5 +-2.5 +-2 +-1.5 +-0.5 +[Fe/H]A&A proofs: manuscript no. smc_fld_v7 +onzalez, O. A., Mucciarelli, A., Origlia, L., et al. 2020, The Messenger, 180, 18. +doi:10.18727/0722-6691/5196 +Graczyk, D. etal., 2014, ApJ, 780, 59 +Grady, J., Belokurov, V., & Evans, N. W. 2021, ApJ, 909, 150. doi:10.3847/1538- +4357/abd4e4 +Gratton, R. G., Carretta, E., Claudi, R., et al. 2003, A&A, 404, 187. +doi:10.1051/0004-6361:20030439 +Grevesse, N., & Sauval, A. J. 1998, Space Sci. Rev., 85, 161 +Jeˇrábková, T., Hasani Zonoozi, A., Kroupa, P., et al. 2018, A&A, 620, A39. +doi:10.1051/0004-6361/201833055 +Hallakoun, +N. +& +Maoz, +D. +2021, +MNRAS, +507, +398. +doi:10.1093/mnras/stab2145 +Harris, J., & Zaritsky, D, 2004, AJ, 127, 1531 +Harris, J. & Zaritsky, D., 2006, AJ, 131, 2514 +Hasselquist, S., Hayes, C. R., Lian, J., et al. 2021, ApJ, 923, 172. +doi:10.3847/1538-4357/ac25f9 +Helmi, A., Babusiaux, C., Koppelman, H. H., et al. 2018, Nature, 563, 85. +doi:10.1038/s41586-018-0625-x +Hill, V., Barbuy, B., & Spite, M., 1997, A&A, 323, 461 +Hill, V., François, P., Spite, M., et al. 2000, A&A, 364, L19 +Hill, +V., +Skúladóttir, +Á., +Tolstoy, +E., +et +al. +2019, +A&A, +626, +A15. +doi:10.1051/0004-6361/201833950 +Hopkins, A. M. 2018, PASA, 35, e039. doi:10.1017/pasa.2018.29 +Iwamoto, K., Brachwitz, F., Nomoto, K., et al. 1999, ApJS, 125, 439. +doi:10.1086/313278 +Kobayashi, +C., +Leung, +S.-C., +& +Nomoto, +K. +2020, +ApJ, +895, +138. +doi:10.3847/1538-4357/ab8e44 +Kobayashi, C., Karakas, A. I., & Lugaro, M. 2020, ApJ, 900, 179. +doi:10.3847/1538-4357/abae65 +Kallivayalil, N., van der Marel, R. P., Besla, G., et al. 2013, ApJ, 764, 161. +doi:10.1088/0004-637X/764/2/161 +Kurucz, R. L., 2005, MSAIS, 8, 14 +Lach, F., Röpke, F. K., Seitenzahl, I. R., et al. 2020, A&A, 644, A118. +doi:10.1051/0004-6361/202038721 +Lapenna, E., Mucciarelli, A., Origlia, L., et al. 2012, ApJ, 761, 33. +doi:10.1088/0004-637X/761/1/33 +Leung, S.-C. & Nomoto, K. 2018, ApJ, 861, 143. doi:10.3847/1538-4357/aac2df +Limongi, M. & Chieffi, A. 2003, ApJ, 592, 404. doi:10.1086/375703 +Lind, K., Asplund, M., Barklem, P. S., & Belyaev, A. K., 2011, A&A, 528, 103 +Martin, G. A., Fuhr, J. R., & Wiese, W. L. 1988, New York: American Institute +of Physics (AIP) and American Chemical Society, 1988 +Massana, P., Ruiz-Lara, T., Noël, N. E. D., et al. 2022, MNRAS, 513, L40. +doi:10.1093/mnrasl/slac030 +Matteucci, F. & Greggio, L. 1986, A&A, 154, 279 +McCall, M. L. 2004, AJ, 128, 2144. doi:10.1086/424933 +Minelli, A., Mucciarelli, A., Romano, D., et al. 2021, ApJ, 910, 114. +doi:10.3847/1538-4357/abe3f9 +Mishenina, T. V., Pignatari, M., Korotin, S. A., et al. 2013, A&A, 552, A128. +doi:10.1051/0004-6361/201220687 +Mucciarelli, A., Origlia, L., Maraston, C., & Ferraro, F. R., 2009, ApJ, 690, 288 +Mucciarelli, A., Origlia, L., & Ferraro, F. R. 2010, ApJ, 717, 277. +doi:10.1088/0004-637X/717/1/277 +Mucciarelli, A., Pancino, E., Lovisi, L., Ferraro, F. R., & Lapenna, E., 2013, ApJ, +766, 78 +Mucciarelli, A., 2013, arXiv1311.1403 +Mucciarelli, A., Monaco, L., Bonifacio, P., & Saviane, I., 2017, A&A, 603L, 7 +Mucciarelli, A. & Bonifacio, P. 2020, A&A, 640, A87. doi:10.1051/0004- +6361/202037703 +Mucciarelli, A., Bellazzini, M., & Massari, D. 2021, A&A, 653, A90. +doi:10.1051/0004-6361/202140979 +Mucciarelli, A., Massari, D., Minelli, A., et al. 2021, Nature Astronomy, 5, 1247. +doi:10.1038/s41550-021-01493-y +Nidever, D. L., Hasselquist, S., Hayes, C. R., et al. 2020, ApJ, 895, 88. +doi:10.3847/1538-4357/ab7305 +Noel, N. E D., Gallart, C., Costa, E., & Mendez, R. A., 2007, AJ, 133, 2037 +Pagel, B. E. J. & Tautvaisiene, G. 1998, MNRAS, 299, 535. doi:10.1046/j.1365- +8711.1998.01792.x +Parisi, M. C., Geisler, D., Carraro, G., et al. 2016, AJ, 152, 58. doi:10.3847/0004- +6256/152/3/58 +Parisi, M. C., Gramajo, L. V., Geisler, D., et al. 2022, A&A, 662, A75. +doi:10.1051/0004-6361/202142597 +Pasquini, L., et al., 2000, SPIE, 4008, 129 +Pompéia, L., Hill, V., Spite, M., et al. 2008, A&A, 480, 379. doi:10.1051/0004- +6361:20064854 +Reddy, B. E., Tomkin, J., Lambert, D. L., et al. 2003, MNRAS, 340, 304. +doi:10.1046/j.1365-8711.2003.06305.x +Reddy, B. E., Lambert, D. L., & Allende Prieto, C. 2006, MNRAS, 367, 1329. +doi:10.1111/j.1365-2966.2006.10148.x +Reggiani, H., Schlaufman, K. C., Casey, A. R., et al. 2021, AJ, 162, 229. +doi:10.3847/1538-3881/ac1f9a +Ripepi, V., Cioni, M.-R. L., Moretti, M. I., et al. 2017, MNRAS, 472, 808. +doi:10.1093/mnras/stx2096 +Roederer, I. U., Preston, G. W., Thompson, I. B., et al. 2014, AJ, 147, 136. +doi:10.1088/0004-6256/147/6/136 +Romaniello, M., Primas, F., Mottini, M., Pedicelli, S., Lemasle, B., Bono, G., +Francois, P., Groenewegen, M. A. T., & Laney, C. D., 2008, A&A, 488, 731 +Romano, D. & Matteucci, F. 2007, MNRAS, 378, L59. doi:10.1111/j.1745- +3933.2007.00320.x +Romano, D., Karakas, A. I., Tosi, M., et al. 2010, A&A, 522, A32. +doi:10.1051/0004-6361/201014483 +Romano, D., Matteucci, F., Zhang, Z.-Y., et al. 2017, MNRAS, 470, 401. +doi:10.1093/mnras/stx1197 +Rubele, S., Pastorelli, G., Girardi, L., et al. 2018, MNRAS, 478, 5017. +doi:10.1093/mnras/sty1279 +Sabbi, E., et al., 2009, ApJ, 703, 721 +Sbordone, L., Bonifacio, P., Castelli, F., & Kurucz, R. L., 2004, MSAIS, 5, 93 +Sbordone, L., Bonifacio, P., Buonanno, R., et al. 2007, A&A, 465, 815. +doi:10.1051/0004-6361:20066385 +Schlafly, E. F., & Finkbeiner, D. P. 2011, ApJ, 737, 103 +Scrucca, L., Fop, M., Murphy, T. B., & Raftery, A. E. 2016, The R Journal, 8, +289, https://doi.org/10.32614/RJ-2016-021 +Skrutskie, M. F., et al., 2006, AJ, 131, 1163 +Smith, G. & Raggett, D. S. J. 1981, Journal of Physics B Atomic Molecular +Physics, 14, 4015. doi:10.1088/0022-3700/14/21/016 +Smith, R. J. 2020, ARA&A, 58, 577. doi:10.1146/annurev-astro-032620-020217 +Spite, F., Spite M., & Francois, P., 1989a, A&A, 210, 25 +Spite, F., Spite M., & Barbuy, B., 1989b, A&A, 222, 35 +Stanimirovi´c, S., Staveley-Smith, L., & Jones, P. A. 2004, ApJ, 604, 176. +doi:10.1086/381869 +Stephens, A. & Boesgaard, A. M. 2002, AJ, 123, 1647. doi:10.1086/338898 +Stetson, P. B., & Pancino, E., PASP, 120, 1332 K. 2005, PASJ, 57, 27 +Subramanian, S. & Subramaniam, A. 2009, A&A, 496, 399. doi:10.1051/0004- +6361/200811029 +Subramanian, S., Rubele, S., Sun, N.-C., et al. 2017, MNRAS, 467, 2980. +doi:10.1093/mnras/stx205 +Taibi, +S., +Battaglia, +G., +Leaman, +R., +et +al. +2022, +A&A, +665, +A92. +doi:10.1051/0004-6361/202243508 +Tinsley, B. M. 1979, ApJ, 229, 1046. doi:10.1086/157039 +Tolstoy, E., Hill, V., & Tosi, M. 2009, ARA&A, 47, 371. doi:10.1146/annurev- +astro-082708-101650 +Travaglio, C., Gallino, R., Arnone, E., et al. 2004, ApJ, 601, 864. +doi:10.1086/380507 +Truran, J. W. 1981, A&A, 97, 391 +Tsujimoto, +T. +& +Bekki, +K. +2009, +ApJ, +700, +L69. +doi:10.1088/0004- +637X/700/2/L69 +van der Marel, R. P., Kallivayalil, N., & Besla, G. 2009, The Magellanic System: +Stars, Gas, and Galaxies, 256, 81. doi:10.1017/S1743921308028299 +Van der Swaelmen, M., Hill, V., Primas, F., et al. 2013, A&A, 560, A44. +doi:10.1051/0004-6361/201321109 +Yan, H. L., Shi, J. R., & Zhao, G. 2015, ApJ, 802, 36. doi:10.1088/0004- +637X/802/1/36 +Yan, Z., Jerabkova, T., & Kroupa, P. 2020, A&A, 637, A68. doi:10.1051/0004- +6361/202037567 +Article number, page 16 of 16 + diff --git a/iNFAT4oBgHgl3EQf9x57/content/tmp_files/load_file.txt b/iNFAT4oBgHgl3EQf9x57/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ef8f4f252f6eab2d06f8137bf553bb732fcbcbe9 --- /dev/null +++ b/iNFAT4oBgHgl3EQf9x57/content/tmp_files/load_file.txt @@ -0,0 +1,2157 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf,len=2156 +page_content='Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' smc_fld_v7 ©ESO 2023 January 24, 2023 The chemical DNA of the Magellanic Clouds I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The chemical composition of 206 Small Magellanic Cloud red giant stars ⋆ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Mucciarelli1, 2, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Minelli1, 2, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Bellazzini2, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Lardo1, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Romano2, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Origlia2, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Ferraro1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2 1 Dipartimento di Fisica e Astronomia “Augusto Righi”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Alma Mater Studiorum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Università di Bologna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Via Gobetti 93/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' I-40129 Bologna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Italy 2 INAF - Osservatorio di Astrofisica e Scienza dello Spazio di Bologna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Via Gobetti 93/3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' I-40129 Bologna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Italy January 24,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2023 ABSTRACT We present the chemical composition of 206 red giant branch stars members of the Small Magellanic Cloud (SMC) using optical,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' high- resolution spectra collected with the multi-object spectrograph FLAMES-GIRAFFE at the ESO Very Large Telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' This sample includes stars in three fields located in different positions within the parent galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' We analysed the main groups of elements, namely light- (Na), α- (O, Mg, Si, Ca, Ti), iron-peak (Sc, V, Fe, Ni, Cu) and s-process elements (Zr, Ba, La).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The metallicity distribution of the sample displays a main peak around [Fe/H]∼–1 dex and a weak metal-poor tail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' However, the three fields display [Fe/H] distributions different with each other, in particular a difference of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='2 dex is found between the mean metallicities of the two most internal fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The fraction of metal-poor stars increases significantly (from ∼1 to ∼20%) from the innermost fields to the most external one, likely reflecting an age gradient in the SMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Also, we found a hint of possible chemically/kinematic distinct substructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The SMC stars have abundance ratios clearly distinct with respect to the Milky Way stars, in particular for the elements produced by massive stars (like Na, α and most iron-peak elements) that have abundance ratios systematically lower than those measured in our Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' This points out that the massive stars contributed less to the chemical enrichment of the SMC with respect to the Milky Way, according to the low star formation rate expected for this galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Finally, we identified small systematic differences in the abundances of some elements (Na, Ti, V and Zr) in the two innermost fields, suggesting that the chemical enrichment history in the SMC has been not uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Galaxies: Magellanic Clouds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Techniques: spectroscopic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Stars: abundances 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Introduction The Local Universe provides an unique window into the process of hierarchical mass assembly on all scales,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' allowing us to inves- tigate a plethora of systems,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' all of them satellites of the major assemblies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' like the Milky Way (MW) and M33: for instance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' galaxies in relative isolation (like most of the nearby dwarf galaxies),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' in close interaction with other systems (the Large and Small Magellanic Cloud,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' LMC and SMC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' respectively) or con- sumed by large galaxies (like the Sagittarius dwarf remnant and the satellites engulfed by the MW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Among them, the Magellanic Clouds, thanks to their proximity and the possibility to resolve individual stars, provide an unique close-up of a pair of interact- ing dwarf galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' They are gas-rich irregular galaxies, gravitationally bound each other and likely at the first peri-Galactic passage with the MW (Besla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2007, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Kallivayalil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Besla 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The galaxy discussed in this paper, the SMC, is the sec- ond most massive MW satellite after the LMC, with a total mass of ∼ 2 · 109M⊙ (Stanimirovi´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2004), about one order of magnitude lower than that of the LMC, and a stellar mass of ∼ 5 − 6 · 108M⊙ (van der Marel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Rubele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2018), comparable with that of the main merger of the Milky Way, the former galaxy Gaia-Enceladus (Helmi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' There are several signatures of their mutual interaction and of the interac- ⋆ Based on observations collected at the ESO-VLT under the pro- grams 072.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='D-0507, 083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='D-0208 and 086.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='D-0665.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' tion between the Clouds and the MW, like the Magellanic Bridge connecting SMC and LMC and the Magellanic Stream embrac- ing these two galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The history of the stellar populations of the SMC is intimately linked to the interplay of these three galax- ies (Massana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2022): the multiple episodes of star formation (SF) occurring in their history are likely the result of the periodic close encounters between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The colour-magnitude diagrams (CMD) of different SMC fields (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Harris & Zaritsky 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Noel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Cignoni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2012, 2013) reveal a mixture of stellar populations, with a prominent red giant branch (RGB), signature of stellar populations older than 1-2 Gyr, and the pres- ence of an extended blue main sequence, hinting at the presence of younger stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Our current picture of the SMC SF history (Cignoni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2012, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Rubele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Massana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2022) is that this galaxy formed in isolation, with a SF activity starting ∼13 Gyr ago and a prolonged period of low-level SF ac- tivity until ∼3-4 Gyr ago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' At this epoch, the SMC has been likely tidally captured by the LMC, becoming gravitationally bound to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' This capture should have triggered new, vigorous and syn- chronised SF bursts in both the galaxies (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Bekki & Chiba 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Massana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2022), likely forming most of the stars that we observe today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' According to Rubele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (2018), the SMC formed (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='31±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='05)×108M⊙ of stars over an Hubble time, 2/3 of which are now found in stellar remnants or living stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' At variance with the LMC stars whose chemical composition has been widely studied using high resolution spectroscopy (Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Pompéia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Mucciarelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Lapenna Article number, page 1 of 16 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='08758v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='GA] 20 Jan 2023 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' smc_fld_v7 et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Van der Swaelmen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Nidever et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Mucciarelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2021), the chemical composition of the SMC stars has received less attention, despite the proximity of this galaxy (∼62 kpc, Graczyk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' For decades, the only high-resolution spectroscopic studies of SMC stars were mainly focused on bright supergiant stars and cepheids, hence sampling stellar populations younger than ∼200 Myr (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Spite et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 1989a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Hill, Barbuy & Spte 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Ro- maniello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Most of the information about the metallic- ity distribution of the SMC RGB stars came from low-resolution spectroscopy in I-band, using the calibrated strength of the Ca II triplet as a proxy of [Fe/H] (Carrera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Dobbie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2014a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Parisi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' De Leo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The metallicity distribution of the SMC stars as derived from these studies dis- plays a main peak around [Fe/H]∼–1 dex and a weak metal-poor tail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' A clear decrease of the mean metallicity has been observed at distance larger than ∼ 3◦ from the galaxy centre (Carrera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Also, evidence of a shallow metallicity gradient within the SMC’s inner ∼3◦ have been found, between –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='07 dex/deg (Dobbie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2014a) and –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='03 dex/deg (Choudhury et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Only recently, chemical analyses of high-resolution spectra of SMC RGB stars have been presented (Nidever et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Reggiani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Hasselquist et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2021), allowing us to investigate in details the chemical composition of these stellar populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Nidever et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (2020) discussed [Mg/Fe], [Si/Fe] and [Ca/Fe] abundance ratios for about 1000 RGB SMC stars, find- ing a quite flat behaviour of these abundance ratios in the range of [Fe/H] between –1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='2 and –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='2 dex, and a knee (the metallic- ity corresponding to the decrease of the [α/Fe] abundance ratios) located at [Fe/H] lower than –2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='2 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The same sample of SMC stars is discussed by Hasselquist et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (2021) that includes also the abundances of Al, O, Ni and Ce, and compare them with those of other Milky Way satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Reggiani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (2021) discussed the chemical composition of four metal-poor ([Fe/H]<–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='0 dex) SMC stars, finding that these stars have abundances comparable to those of the MW halo stars for all the main groups of elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' On the other hand, these stars are more enriched in [Eu/Fe] (a pure r-process element) with respect to the MW stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' This paper is the first of a series dedicated to the investi- gation of the chemical properties of the LMC/SMC (field and stellar clusters) stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' In this work, we present the chemical anal- ysis of 206 RGB stars members of the SMC observed with the high-resolution spectrograph FLAMES mounted at the ESO Very Large Telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Observations and data reduction 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' SMC sample A total of 320 stars in the direction of the SMC has been observed (ID program 086.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='D-0665, PI: Mucciarelli) with the multi-object spectrograph FLAMES (Pasquini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2000) in the GIRAFFE-MEDUSA mode that allows us the simultaneous al- location of 132 high-resolution (R∼20,000) fibres over a patrol field of about 25 arcmin diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Three different fields have been observed, centred around three globular clusters (GCs), namely NGC 121, NGC 339 and NGC 419 (hereafter these fields will be indicated as FLD-121, FLD-339 and FLD-419, respec- tively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 1 shows the spatial location of the three FLAMES fields superimposed to the map of the SMC stars ob- tained with the early third data release (EDR3) of the Gaia/ESA mission (Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2016, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' ‘ The fields are located in different positions of the SMC, with FLD-121, FLD- 339 and FLD-419 at ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='4◦ northern-western, ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='4◦ southern- eastern and ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5◦ eastern from the SMC centre (Ripepi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2017), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The field FLD-121 partially overlaps with the APOGEE field 47Tuc (two only stars in common), the field FLD-419 is adjacent to the APOGEE field SMC2 (one only star in common), while the field FLD-339 samples a region not ob- served by APOGEE (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 1 by Nidever et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The adopted GIRAFFE-MEDUSA setups are HR11, with a spectral resolution of 24200 and ranging from 5597 to 5840 Å and HR13, with a spectral resolution of 22500 and a spectral coverage between 6120 and 6405 Å .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' These two setups allow us to measure lines of the main groups of elements, like odd-Z (Na), α (O, Mg, Si, Ca and Ti), iron-peak (Sc, V, Fe, Ni, Cu) and s-process elements (Zr, Ba, La).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The UVES fibers have been allocated to targets belonging to the three GCs and discussed in separated papers (Dalessandro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2016, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Minelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Table 1 lists the exposure times and the number of individual exposures for each setup and field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The spectroscopic targets for each field have been origi- nally selected from near-infrared (Ks,J-Ks) CMDs, using the SofI@NTT catalogues for the region within 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 arcmin from the cluster centres (Mucciarelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2009, for NGC 339 and NGC 419, and unpublished proprietary photometry for NGC 121), and the 2MASS database (Skrutskie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2006) for the external regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The targets have been selected accord- ing to the following criteria: (1) stars fainter than the RGB Tip (Ks=12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='62, Cioni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2000);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (2) stars brighter than Ks= 14 for FLD-339 and FLD-419, and brighter than Ks= 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='4 for FLD- 121, in order to guarantee a signal-to-noise ratio (SNR) per pixel larger than ∼30 in both setups and in all the observed fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Due to the paucity of RGB stars in the SMC outskirts, a fainter (by ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='4 mag) threshold has been adopted for FLD-121 in order to enlarge the number of observed SMC stars;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (3) stars without close stars brighter than < Kstar s +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='0 within 2” ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (4) for the tar- gets from the 2MASS catalogue (the majority of the observed targets) only stars with J and Ks magnitudes flagged as A (pho- tometric uncertainties smaller than 10%) have been selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' All the targets have been recovered in the Gaia EDR3 cata- log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 1 shows the position in the (G, BP-RP) CMDs of the observed targets considered as SMC stars accord- ing to their radial velocity (RV), see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Table 2 lists for all the SMC targets coordinates and the Gaia EDR3 identifica- tion number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The spectra have been reduced with the dedicated ESO GIRAFFE pipeline1, including bias-subtraction, flat-fielding, wavelength calibration with a standard Th-Ar lamp and spectral extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The contribution of the sky has been subtracted from each spectrum by using a median sky spectrum, as obtained by combining ∼15-20 spectra from fibres allocated to sky positions within each exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The final SNR per pixel of the spectra is of ∼30-50 for HR11 spectra and ∼40-60 for HR13 spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2 shows, as an example of the spectral quality, the spectra of two SMC giant stars with very similar atmospheric parameters and a large (∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 dex) difference in [Fe/H].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' MW control sample As discussed in Minelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (2021), the comparison between chemical abundances obtained from different works can be ham- pered by various systematics characterising the chemical anal- yses, for instance the method used to infer the stellar parame- ters, the adopted atomic data for the analysed transitions, model 1 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='eso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='org/sci/software/pipelines/ Article number, page 2 of 16 Mucciarelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' : SMC field stars Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Left panel: spatial distribution of the three fields observed with FLAMES (marked as red, green and blue circles for FLD-121, FLD-339 and FLD-419, respectively) superimposed to the map of the SMC RGB stars with G between 16 and 19 from Gaia EDR3 (Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2021), revealing the SMC old spheroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The white plus symbol marks the position of the SMC centre derived by Ripepi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Right panel: Gaia EDR3 CMDs of the three SMC fields (grey points, Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2021) with superimposed the spectroscopic GIRAFFE targets (same colours of left panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' In the CMD of FLD-121 is clearly visible the main sequence of the MW GC 47 Tucanae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Coordinates of the FLAMES pointing, number of exposures and exposure times for the two FLAMES setups, adopted colour excess (Schlafly & Finkbeiner 2011) and the number of SMC stars analysed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Field RA Dec HR11 HR13 E(B-V) NS MC (J2000) (J2000) (mag) FLD-121 00:26:49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='0 –71:32:09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='9 7x2700sec 5x2700sec 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='028 37 1x2200sec FLD-339 00:57:48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='9 –74:28:00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1 9x2700sec 5x2700sec 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='042 78 FLD-419 01:08:17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='7 –72:53:02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='7 6x2700sec 4x2700sec 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='089 91 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Information about the SMC spectroscopic targets: ID for our internal catalogues, ID and coordinates from Gaia EDR3 (Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2021), measured RV, derived atmospheric parameters and [Fe/H] abundance ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The entire table is available in electronic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' ID ID Gaia EDR3 RA Dec RV Teff log g vt [Fe/H] (degree) (degree) (km s−1) (K) (cgs) (km s−1) (dex) FLD-121_23 4689857932203222528 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='6427941 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5293047 144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='3± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='2 4115 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='79 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='8 –1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='58± 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='25± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='17 Article number, page 3 of 16 70 FLD-121 FLD-33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='9 16 法 71 FLD - 121 O 18 72 (degrees) 20 FLD-419O 73 G 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 2 Dec FLD-419 16 74 FLD-339O N 18 75 20 76 E 20 18 16 14 12 10 8 6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 2 BP-RP RA (degrees)A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' smc_fld_v7 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Comparison between the HR13 spectra of the stars FLD- 419_102664 (upper panel) and FLD-121_100683 (lower panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The stars have very similar atmospheric parameters but different iron con- tent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Arrows mark the position of some metallic lines of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' atmospheres and solar reference abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' For this reason, when chemical analyses of extra-galactic stars are performed (with the aim to compare their abundances with those of MW stars), it is crucial to consider also a control sample of MW stars analysed in an homogeneous way, in order to erase the main sys- tematics quoted above and highlight and quantify possible dif- ferences and similarities between the abundance ratios of stars from different galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' We defined a control sample of MW stars analysed with the same assumptions used for the SMC stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' We analysed five MW GCs covering the same metallicity range of the SMC stars ([Fe/H] between ∼–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='2 and ∼–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 dex) and for which FLAMES spectra obtained with the GIRAFFE HR11 and HR13 setups are available in the ESO archive (ID programs: 072.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='D-0507 and 083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='D-0208, PI: Carretta).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The selected GCs are NGC 104, NGC 1851, NGC 1904, NGC 4833 and NGC 5904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The use of the same GIRAFFE setups allows us to derive chemical abun- dances in these MW GCs from the same transitions used for the SMC stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' We restrict the analysis only to the stars with effective temperatures and surface gravities comparable with those of the SMC stars studied here six stars for NGC 104, 2 for NGC 1851, 6 for NGC 5904, 3 for NGC 1904 and 4 for NGC 4833.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Only for O and Na that exhibit large star-to-star variations in each of these GCs, we analysed stars belonging to the so-called first popula- tion and selected according to Carretta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The O and Na abundances of these first population stars can be considered as a good proxy of the chemical composition of the MW field at those metallicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Spectral analysis 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Line selection We selected for each star an appropriate set of unblended metal- lic lines, selected by visual inspection of suitable synthetic spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The latter have been calculated with the code SYNTHE (Sbordone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Kurucz 2005), using the typical atmo- spheric parameters of the observed stars (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='2), adopt- ing ATLAS9 model atmospheres (Castelli & Kurucz 2004)2 and including all the atomic and molecular transitions in the Kurucz/Castelli linelist3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The synthetic spectra have been con- voluted with Gaussian profiles in order to reproduce the ob- served line broadening, mainly dominated by the instrumental resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' We privileged transitions with laboratory oscillator strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Only for the Sc II line at 6245.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='6 Å , for the Si I lines at 6155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1 and 6237.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='3 Å , and for the Cu I line at 5782 Å we adopted solar oscillator strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Because the level of blending of a given transition depends on the metallicity, in this case not known a priori, we adopted an iterative process to define the linelist of each target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' A preliminary linelist has been defined by adopting a metal- licity [M/H]=–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='0 dex for all the used synthetic spectra, accord- ing to the mean metallicity of the SMC derived from previous studies (Carrera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Dobbie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2014a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Parisi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Nidever et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' After a first chemical analysis, a new set of unblended lines has been defined for each star using a syn- thetic spectrum calculated with the appropriate chemical com- position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' This procedure has been specifically necessary for the most metal-poor stars of our sample, with [Fe/H] significantly lower than the mean value of [Fe/H]=–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='0 dex, and for a few stars with enhancement of s-process elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The average num- ber of selected metallic lines is of about 80-90 for most of the stars (with [Fe/H]∼–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='0 dex), decreasing down to 40-50 for the most metal-poor ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Most of the lines used in the metal-poor stars are still available for metal-rich stars, while some features are excluded because saturated or blended with other lines at higher metallicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' However, we checked that the use of differ- ent samples of lines depending on the stellar metallicity does not introduce biases in the abundances at different metallicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' All the used lines are listed in Table 3 together with the cor- responding log gf and excitation potential χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Atmospheric parameters The derived atmospheric parameters are listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Effec- tive temperatures (Teff) and surface gravities (log g) have been estimated from the photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' In particular, Teff have been obtained from the broad-band colour (G − Ks)0 adopting the (G−Ks)0-Teff transformation provided by Mucciarelli, Bellazzini & Massari (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' We adopted G magnitudes from Gaia EDR3 and Ks from 2MASS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' G magnitudes have been corrected for ex- tinction following the prescriptions by Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (2018), while Ks magnitudes adopting the extinction coefficient by McCall (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The colour excess values E(B-V) are from the infrared dust maps by Schlafly & Finkbeiner (2011) and listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Uncertainties in Teff have been estimated by propagating for any individual star the errors in the adopted colour and in the colour excess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The typical error in the (G − Ks)0colours is of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='03-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='05 mag, dominated by the uncertainty of the Ks magnitude, and translating in 20-40 K of uncertainty in Teff .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' For 2 http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='oact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='inaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='it/castelli/castelli/sources/atlas9codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='html 3 http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='oact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='inaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='it/castelli/castelli/linelists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='html Article number, page 4 of 16 Fe Fe Na Ca FeCa C2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='2 Flux 1 Normalized 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='4 FLD-419_1026641Fe/H|=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='2 lux 1 lized 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='8 Jormal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='4 FLD-121_100683[Fe/Hl=-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='2 6155 6160 6165 6170 Wavelength (A)Mucciarelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' : SMC field stars Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' List of the used transitions together with oscillator strengths, the excitation potential and the reference of the atomic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Wavelengths without some decimal digits indicate transitions affected by hyperfine/isotopic splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The entire table is available in electronic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Wavelength Ion loggf χ Reference (Å ) (eV) 5590.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='720 Co I 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='870 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='042 Fuhr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (1988) 5598.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='480 Fe I 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='087 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='521 Fuhr & Wiese (2006) 5601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='277 Ca I 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='523 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='526 Smith & Raggett (1981) 5611.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='356 Fe I 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='990 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='635 Fuhr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (1988) 5615.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='644 Fe I 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='050 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='332 Fuhr & Wiese (2006) 5618.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='632 Fe I 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='276 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='209 Fuhr & Wiese (2006) 5624.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='542 Fe I 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='755 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='417 Fuhr & Wiese (2006) 5633.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='946 Fe I 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='320 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='991 Fuhr & Wiese (2006) 5638.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='262 Fe I 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='840 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='220 Fuhr & Wiese (2006) 5647.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='234 Co I 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='560 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='280 Fuhr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (1988) 5648.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='565 Ti I 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='260 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='495 Martin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (1988) 5650.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='689 Fe I 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='960 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='085 Fuhr & Wiese (2006) 5651.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='469 Fe I 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='000 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='473 Fuhr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (1988) 5652.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='318 Fe I 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='920 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='260 Fuhr & Wiese (2006) 5653.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='867 Fe I 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='610 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='386 Fuhr & Wiese (2006) 5661.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='345 Fe I 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='756 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='284 Fuhr & Wiese (2006) 5662.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='516 Fe I 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='573 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='178 Fuhr & Wiese (2006) 5670.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='8** V I 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='420 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='081 Martin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (1988) 5679.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='023 Fe I 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='900 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='652 Fuhr & Wiese (2006) 5682.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='633 Na I 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='706 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='102 NIST the colour excess, we adopted a conservative error of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='01 mag for all the three fields, despite the lower errors quoted by Schlafly & Finkbeiner (2011), leading to a negligible (a few K) uncer- tainty in Teff .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' These uncertainties have been added in quadrature to the typical error associated to the (G−Ks)0-Teff transformation (46 K), estimated as 1σ dispersion of the fit residuals (Muccia- relli, Bellazzini & Massari 2021), and that dominates the total Teff error (typically ∼50-60 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The log g values have been calculated through the Stefan- Boltzmann relation adopting the photometric Teff, a true dis- tance modulus (m − M)0=18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='965±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='025 (Graczyk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2014), the bolometric corrections by Andrae et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (2018) and a stellar mass of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='0M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Concerning the distance modulus, it is worth to recall that the SMC has a substantial line-of-sight depth, not easy to properly take into account for each individual target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Accord- ing to the depth maps provided by Subramanian & Subramaniam (2009), the three fields studied in this study should cover a depth range between 2 and 6 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Considering a conservative varia- tion of the distance by 3 kpc increases the quoted uncertainties in log g only by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='02 dex, translating in variations of less of less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='02 in the abundances of single ionised (but without im- pact on the abundances of neutral lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Uncertainties in log g are of the order of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1, including the uncertainties in Teff dis- tance modulus and stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The final error budget in log g is dominated by the uncertainty in the stellar mass, assumed to be ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='2 M⊙, reflecting the possible spread in ages of our targets (older than ∼1-2 Gyr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Microturbulent velocities (vt) are usu- ally derived spectroscopically by erasing any trend between iron abundance and the reduced equivalent widths (defined as the log- arithm of the EW normalised to the wavelength).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Because of the relatively small number (∼30-40 or less) of available Fe I lines in the adopted spectral ranges, vt obtained spectroscopically risk to be uncertain or unreliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' In order to avoid significant fluc- tuations in vt (with an impact on the derived abundances), we adopted the log g - vt relations provided by Mucciarelli & Boni- facio (2020) and based on the spectroscopic vt obtained from high-resolution, high-SNR spectra of giant stars in 16 Galac- tic GCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The uncertainty in vt has been estimated by adding in quadrature the error arising from the uncertainty in log g and that of the adopted log g - vt relation and is of the order of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='2 km s−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Radial velocities RVs have been measured by using the code DAOSPEC (Stetson & Pancino 2008) that performs a line fitting assuming a Gaussian profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The code is automatically launched by using the software 4DAO (Mucciarelli 2013) that allows us a visual inspection of all the fitted lines in order to directly evaluate the quality of the fitting procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' RVs have been measured by the position of about 100 metallic lines for each star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The internal uncertainty of the RV for each spectrum is estimated as the standard error of the mean, of the order of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='3 km/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The final RV for each target is obtained as the weighted mean of the values obtained from the two setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The accuracy of the wavelength calibration has been checked by measuring the position of the strong emission sky line at 6300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='3 Å in the HR13 setup, finding no significant offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' No sky emission lines are available in the HR11 setup and we cannot directly check the accuracy of the wavelength calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' However, the RVs obtained from the two setups agree each other with an average difference between RV from HR11 and HR13 of +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='12±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='06 km s−1(σ= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='8 km s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' This excludes any offset for the two setups and confirming the accuracy also of the RVs from HR11 spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Chemical abundances The chemical abundances of Na, Mg (from the line at 5711 Å), Si, Ca, Ti, Fe, Ni and Zr have been derived from the measure of the equivalent widths (EWs) of unblended lines by using the code GALA (Mucciarelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' EWs have been measured by using the code DAOSPEC (Stetson & Pancino 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Article number, page 5 of 16 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' smc_fld_v7 For species whose lines are affected by blending (O and the Mg lines at 6318-19 Å ) or by hyperfine/isotopic splitting (Sc, V, Cu, Ba and La), abundances have been derived using our own code SALVADOR that performs a χ2-minimisation between the observed lines and a grid of synthetic spectra calculated on-the- fly with the code SYNTHE (Sbordone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2004) and including all the atomic and molecular lines available in the Kurucz/Castelli linelists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' For all the species investigated, when the lines are not clearly detectable, we provide upper limits based on the compar- ison between observed and synthetic spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The [OI] line at 6300 Å can be also contaminated by telluric lines, depending on the stellar RV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' This possible contamination has been checked with suitable synthetic spectra of the Earth atmosphere calculated with the TAPAS tool (Bertaux et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' These synthetic spectra are calculated assuming the appropriate date of observation and airmass of our targets, in order to account for the proper weather conditions of the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' In case of contamination, the line profile has been cleaned by dividing the observed spectrum by the telluric one and visually checking that no discontinuities were introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 3 shows an example of a stellar spectrum around the [OI] line before and after the telluric correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Example of the telluric correction around the [O I] line in the star FLD-339_466: black curve is the original (not corrected for RV) spec- trum, the blue curve is the spectrum corrected for the telluric lines and the red one is the synthetic spectrum of the Earth atmosphere (shifted for sake of clarity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' In the calculation of the synthetic spectra used to measure the oxygen abundance, the Ni abundance of each star has been included to account for the blending of the O feature with a Ni line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Mg abundances have been obtained for most of the stars from the EW of the line at 5711 Å .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' As discussed in Minelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (2021), this transition is heavily saturated for giant stars with [Fe/H] > –1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='0 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' For the latter stars, the Mg triplet at 6318-19 Å should be preferred because these lines are still sensitive to the Mg abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Therefore, for the stars for which the Mg line at 5711 Å turns out to be saturated, the Mg abundance has been derived from the Mg triplet using spectral synthesis in order to include the contribution of the close auto-ionization Ca line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Finally, only for the Na lines used here (5682-88 Å and 6154-60 Å ) we corrected the derived abundances for departures from the LTE assumption applying the corrections by Lind et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The abundances are referred to the solar ones, taking as ref- erence the values from Grevesse & Sauval (1998), apart from oxygen for which the adopted value is from Caffau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Abundance uncertainties In the determination of the uncertainties in each derived abun- dance ratio we take into account two main sources of error, namely the errors arising from the measurement procedure (EW or spectral synthesis) and those arising from atmospheric param- eters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (1) Uncertainties related to the measurement procedure are computed as the dispersion of the mean normalised to the root mean square of the number of used transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Properly, this term includes both uncertainties from line fitting and from adopted log gf values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' For the elements measured from the EWs and for which only one line is available, the DAOSPEC uncer- tainty associated to the Gaussian fitting procedure (correspond- ing to 1σ of the fit residuals) is assumed as internal error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' For the elements (O and La) for which only one transition has been measured using spectral synthesis, the internal error has been estimated by means of Monte Carlo simulations, creating a sam- ple of 500 synthetic spectra with a Poissonian noise that repro- duces the observed SNR and repeating the line-fitting procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The dispersion of the abundance distribution obtained from these noisy synthetic spectra is assumed as 1σ uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (2) Uncertainties due to atmospheric parameters have been es- timated by repeating the analysis by varying each time a given parameter of the corresponding 1σ error and keeping fixed the other parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' These two sources of uncertainties have been added in quadrature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Since the abundance of the species X is expressed as abundance ratios [X/Fe], also the uncertainties in the Fe abun- dance have been taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The final errors in [Fe/H] and [X/Fe] abundance ratios are calculated as follows: σ[Fe/H] = � σ2 Fe NFe + (δTeff Fe )2 + (δlog g Fe )2 + (δvt Fe)2 (1) σ[X/Fe] = � σ2 X NX + σ2 Fe NFe + (δTeff X − δTeff Fe )2 + (δlog g X − δlog g Fe )2 + (δvt X − δvt Fe)2 (2) where σX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='Fe is the dispersion around the mean of the chem- ical abundances,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' NX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='Fe is the number of lines used to derive the abundances and δi X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='Fe are the abundance variations obtained modifying the atmospheric parameter i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Abundances of the MW control sample Table 4 lists the average abundance ratios, together with the stan- dard deviation and the average uncertainty in the abundance ra- tio, for the 5 GCs of the MW control sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' We compared the atmospheric parameters and [Fe/H] of the analysed stars Article number, page 6 of 16 [0 1] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='2 Flux Normalize 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='2 6298 6300 6302 6304 6306 Wavelength (A)Mucciarelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' : SMC field stars with those by Carretta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (2009) and Carretta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (2014) that analysed the same spectroscopic dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The average differ- ences between our analysis and the literature ones are +52±11 K (σ= 50 K) for Teff , –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='01±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='01 (σ= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='03) for log g, +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='07±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='04 km s−1(σ= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='19 km s−1) for vt and –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='03±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='02 dex (σ= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='07 dex) for [Fe/H].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' RV and [Fe/H] distributions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' RV distribution According to previous spectroscopic studies (Harris & Zaritsky 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Carrera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Dobbie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2014a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' De Leo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Hasselquist et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2021) we identified as members of the SMC those stars with RV between +80 and +250 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The membership is confirmed also by the proper motions measured from Gaia EDR3 (Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' We exclude from the chemical analysis stars members of the GC associated to each field (these stars will be discussed in a forthcoming paper of the series), stars with spectra contaminated by prominent TiO or C2 molecular bands or with too low SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The final sample discussed in this work includes a total of 206 stars out of the 320 observed stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The RV and [Fe/H] for this sample are listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 4 and 5 shows the RV and [Fe/H] discrete and ker- nel density distributions of the three SMC fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The advantage of the latter representation is that the distribution is independent of the choice of the bin width and of the starting bin, at variance with the discrete distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 0 2 4 6 8 100 150 200 RV (km/s) count 0 2 4 6 8 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='0 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 [Fe/H] count 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 100 150 200 RV (km/s) count 0 5 10 15 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='0 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 [Fe/H] count 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 100 150 200 RV (km/s) count 0 5 10 15 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='0 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 [Fe/H] count Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' RV and [Fe/H] distributions (left and right panel, respectively) of the three fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Colours indicate the different fields: FLD-121 (red), FLD-339 (green) and FLD-419 (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The RV distributions of the three fields appear significantly different with each other, both in terms of the main peak and shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The RV distribution of FLD-121 peaks at RV≈+125 km s−1 , that of FLD-339 displays a peak at RV≈+160 km s−1 , while that of FLD-419 exhibits two distinct peaks, the main one at ≈+150 km s−1 and the second one at ≈+180 km s−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' A Kolmogorov-Smirnov test performed on these distributions con- firms that the RV distributions of FLD-339 and FLD-419 are sig- nificantly different with respect to that of FLD-121 (with statistic significance larger than 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='9%), while we cannot reject the hy- pothesis that FLD-339 and FLD-419 may derive from the same population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The differences in the peak of these three RV distributions are compatible with the rotation pattern of the SMC as inferred from low-resolution spectroscopic surveys of giant stars (Dob- bie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2014a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' De Leo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2020), from the HI column den- sity map (Di Teodoro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2019) and from the APOGEE re- sults from 17th Data Release of the Sloan Digital Sky Survey (Abdurro’uf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' All these studies show that the western side of the SMC, where FLD-121 is located, has a lower velocity with respect to the eastern side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' However, the presence of multi- ple peaks, clearly visible in the distribution of FLD-419, seems to suggest a more complex kinematic pattern (as discussed be- low).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' [Fe/H] distribution The [Fe/H] distribution of the entire sample is peaked at [Fe/H]∼–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='0 dex, with about 95% of the stars having [Fe/H] be- tween –1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 and –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 dex and with a weak but extended metal- poor tail reaching [Fe/H]∼–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='2 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' This distribution is qualita- tively similar to those obtained from low-resolution spectra us- ing Ca II triplet (Carrera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Dobbie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2014a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Parisi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2016) and that from APOGEE data (Nidever et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' However, similar to what we see with the RV distributions, when the individual fields are considered, the metallicity distributions appear different with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The distributions of FLD-339 and FLD-419 are confined between –1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 and –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 dex, with only one star per field (∼1%) with [Fe/H]<–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 dex .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' On the other hand the distribution of FLD-121 ranges from –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='8 dex down to –2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='2 dex, with ∼20% of the stars more metal-poor than –1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Note that the APOGEE field 47Tuc, superimposed to our field FLD-121, exhibits a lower fraction of metal-poor stars, ∼2%, probably reflecting some selection bias against metal-poor stars in the APOGEE observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The peaks of the distributions of FLD-339 and FLD-419 are sep- arated by ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='2 dex and located at [Fe/H]∼–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='9 and ∼–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1 dex, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Also, the two distributions seem to be not symmet- ric, with the presence a secondary peak at [Fe/H]∼–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1 dex in FLD-339 and a heavily-populated metal-rich tail or a secondary peak in FLD-419 (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' [Fe/H] distribution and the age-metallicity relation We try to interpret the derived [Fe/H] distributions in terms of ages, using as a guidance the SF histories recovered from Hub- ble Space Telescope (Noel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Sabbi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Cignoni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2012, 2013) and ground-based (Massana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2022) pho- tometry, and the theoretical age-metallicity relations available for the SMC (Pagel & Tautvaisiene 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Tsujimoto & Bekki 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Cignoni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' All these works agree that the early epochs of the SMC have been characterised by a significant SF activity followed by a long quiescent period, interrupted between ∼3 and ∼4 Gyr ago by sig- nificant SF episodes, likely due to some merger events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The old- est SMC GC, NGC 121, has an age of ∼10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 Gyr (Glatt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2008) and a metallicity of [Fe/H]∼–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='2/–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='3 dex (Dalessan- dro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2016, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Minelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' This suggests that the SF activity in the first Gyrs was able to increase the metallicity to values as high as [Fe/H]∼–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='2/–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='3 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' We can consider the Article number, page 7 of 16 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' smc_fld_v7 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Average abundance ratios, the corresponding standard deviation and the average uncertainty for the five GCs of the MW control sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Ratio NGC 104 NGC 1851 NGC 5904 NGC 1904 NGC 4833 <> σ <> σ <> σ <> σ <> σ < σ[X/Fe] > [Fe/H] –0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='03 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='06 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='12 [Ba/Fe] +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='05 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='04 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='06 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='04 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='12 [La/Fe] +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='03 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='05 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='04 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='03 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='08 SMC field stars in our sample with [Fe/H]<–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='3 dex (which are almost all located in FLD-121) as formed in the first 1-2 Gyr of the life of the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The subsequent evolution of the SMC and the corresponding metallicity distribution can be interpreted in the light of the the- oretical age-metallicity relations: Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 6 shows that by Pagel & Tautvaisiene (1998) assuming a burst of SF at an age of ∼4 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' After a long period characterised by a low SF efficiency (and where the metallicity remains almost constant), the SF in the SMC re-ignites with a prominent burst, likely triggered by the first close encounter between SMC and LMC (Bekki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Bekki & Chiba 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The most recent SF history for the SMC provided by Massana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (2022) using the SMASH photom- etry identified the re-ignition of the SF at ∼3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 Gyr ago, simul- taneously in both the Clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The stars with [Fe/H]>–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='3 dex analysed here should be a mixture of stars with different ages (from ∼1 to ∼10-11 Gyr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' It is not easy to separate the different populations in terms of age due to the almost constant [Fe/H] over a large age range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Massana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (2022) identified in the SF history of the SMC five peaks (at ∼3, 2, 1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='45 Gyr ago and one still ongoing) occurring simultaneously also in the LMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' A fascinating possibility is that the different peaks in the metallic- ity distributions of FLD-339 and FLD-419 could be associated to some of these different bursts of SF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Finally, we can suppose that the stars with [Fe/H] around –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='6/–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 dex are likely formed with the burst at 1 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' This is confirmed also by the metallici- ties of the stellar clusters with ages around 1 Gyr (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Parisi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2022) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Run of [Fe/H] with the distance Previous spectroscopic studies (Carrera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Dobbie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2014a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Parisi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Choudhury et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Grady et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2021) found evidence of a shallow (from –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='03 to –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='07 dex/deg) metallicity gradient, within 3◦–5◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 7 shows the run of [Fe/H] of the spectroscopic targets with their projected distance from the SMC centre (Ripepi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The mean metallicity in three fields is consistent with the shallow gradient previously proposed by Choudhury et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' However two main dif- ferences between the external field FLD-121 and the two inter- nal ones are evident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' First, in FLD-121 the fraction of metal-poor stars ([Fe/H]<–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 dex) is about ∼20%, against ∼1% in the other two fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The fraction of metal-poor stars increases outward re- flecting a larger fraction of old stars with respect to those formed subsequently during the long quiescent period and the recent SF bursts and that are preferentially confined in the innermost re- gion of the SMC (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Rubele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Second, in the metallicity distribution of FLD-121 there is a clear lack of stars with [Fe/H] between –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='8 and –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 dex, instead detected in FLD-339 and FLD-419.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Following the discussion above, these stars should have ∼1 Gyr (the youngest stars among the intermediate-age SMC populations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Again, this is consistent with a scenario where the younger, metal-richer populations are progressively more concentrated toward the innermost regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Age-metallicity gradients of this kind are quite common in dwarf galaxies (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Taibi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2022, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Possible kinematic/chemically distinct sub-structures?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The distribution of the SMC stars in the RV-[Fe/H] plane seems to suggest the presence of sub-structures, in particular the two different peaks of the [Fe/H] distribution of FLD-339, the large and asymmetric [Fe/H] distribution of FLD-419 and the double- peak of the RV distribution of FLD-419.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' We used the gaussian mixture package Mclust (Scrucca et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2016), within the R environment, to analyse the distribution of FLD-339 and FLD-419 stars in the [Fe/H] - RV space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Mclust choose the best model, both in terms of number and form (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', equal or variable variance, orientation etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', see Scrucca et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2016) of the gaussian components, by means of the Bayesian Information Criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Since we are interested in substructures within the bulk of the metallicity distribution we exclude from the analysis the two metal-poor outliers, one per field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' While for FLD-339 a single elliptical gaussian model is the preferred solu- tion, the [Fe/H] - RV distribution of FLD-419 is best described with two elliptical gaussian components with the same variance both in [Fe/H] and RV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The gain of this model with respect to a single elliptical gaussian is only marginal, in practice they pro- vide an equally good representation of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Still, the solu- tion synthesise the properties of the hypothesised two compo- nents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The first component has (µ[Fe/H], µRV)= ( -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='85 dex, 171.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='8 km s−1), and it accounts for 33% of the sample, the second com- ponent has (µ[Fe/H], µRV)= ( -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='13 dex, 154.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 km s−1), accounting for the remaining 67% of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The standard deviations are σ[Fe/H]= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='10 dex, and σRV= 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='7 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' It seems that the most metal rich component has a larger systemic RV than its metal- poor counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Article number, page 8 of 16 Mucciarelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' : SMC field stars 100 150 200 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='0 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 [Fe/H] RV (Km/s) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' RVs are plotted against [Fe/H] for target stars in the central panel of the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Colour-shaded areas denote the contours of the three clusters RV vs [Fe/H] distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Side plots show the kernel distributions of the RV (right-hand panel) and the [Fe/H] values (top panel) for each cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Same colours of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' As additional check, we performed a Kolmogorov-Smirnov test on the RV sub-populations of FLD-419, separated according to the metallicity of their member stars (and assuming [Fe/H]=– 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='05 dex as a boundary between the two groups of stars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' We obtained that the two RV distributions cannot be extracted from the same population with a significance of 98%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The size of the FLD-419 sample is not sufficient to put this odd result on sound statistical bases, still it may be suggestive of the presence of some chemo-kinematic substructures in the SMC along this line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' In this respect, it is worth recalling that the SMC has a substantial line-of-sight depth, depending on the used tracers and ranging from a few kpc up to about 20 kpc (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' de Grijs & Bono 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Subramanian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Therefore, when we observe stars in an individual SMC field we are likely crossing different depths and we are sampling different populations in terms of kinematics and metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Chemical abundance ratios We derived abundances of Na, O, Mg, Si, Ca, Sc, Ti, V, Fe, Ni, Cu, Zr, Ba and La for 206 SMC RGB stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' All the abun- dances, with the corresponding uncertainties, are available in the electronic form (Table 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' With respect to the APOGEE sam- ple by Hasselquist et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (2021) we measured a larger number of species, in particular Na, Sc, Ti, V, Cu, Zr, Ba and La, not included in that study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 8-11 show the behaviour of derived abundance ratios as a function of [Fe/H] for the analysed SMC stars, highlighting stars belonging to the different fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' These abundance ratios are compared with those obtained for the con- trol sample of 5 Galactic GCs, adopting the same assumptions in the chemical analysis and therefore removing most of the sys- tematics of the analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' This comparison allows us to high- light the real difference between SMC and MW stars of simi- lar [Fe/H].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Additionally, we show abundance ratios for Galactic field stars from the literature as reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The comparison with the literature is affected by the systematics among the different analyses (in terms of model atmospheres, solar abundance val- Article number, page 9 of 16 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' smc_fld_v7 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Main panel: age-metallicity relation by Pagel & Tautvaisiene (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Side panel: the kernel [Fe/H] distributions for the individual SMC field stars discussed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Behaviour of [Fe/H] as a function of the projected distance from the SMC centre (Ripepi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2017), same colours of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The thick grey line is the linear fit for the metallicity gradient estimated by Choud- hury et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' ues, NLTE corrections, linelists, use of dwarf and giant stars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' However, it is useful to display the overall trends in the MW based on a large number of stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' In the following, we refer to the MW control sample to quantify the main differences and simi- larities between MW and SMC stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Na Sodium is mainly produced in massive stars during the hydro- static C and Ne burning, with a strong dependence of its yields on the metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Also, a smaller contribution is provided by asymptotic giant branch (AGB) stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' In Galactic stars (both in the control sample and in literature data), [Na/Fe] increases by increasing [Fe/H] until it reaches solar values around [Fe/H]>–1 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' An offset is evident between the values in the control sam- ple and in the literature, especially in the metal-poor regime and likely due to the different NLTE corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Top-left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 8 shows the distribution of [Na/Fe] of the observed targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The bulk of the SMC stars exhibits sub-solar [Na/Fe] abundance ratios at any metallicities, with an average value of about –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='4/– 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 dex, similar to the typical [Na/Fe] measured in the LMC stars (Van der Swaelmen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Minelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2021) but at higher [Fe/H].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The low [Na/Fe] values measured in the SMC stars may point to a lower contribution by massive stars, besides the larger impact of Type Ia supernovae (SNe Ia) at low metallicities in dwarf galaxies (Tolstoy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' We observe a large scatter of [Na/Fe], not fully explainable within the typical uncertainties, and already detected in spectroscopic samples of LMC and SMC metal-rich stars (Pompéia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Van der Swaelmen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Minelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Hasselquist et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' This scat- ter could reflect that multiple sites of Na production are taking place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Finally, we note a systematic difference between the me- dian [Na/Fe] values in FLD-339 and FLD-419, where the latter displays [Na/Fe] higher by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='15 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' A systematic difference in [Na/Fe] of different regions of the parent galaxy has been also observed in the LMC (Van der Swaelmen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2013) with the stars in the LMC bar more enriched in [Na/Fe] by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='2 dex with respect to the LMC disc stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' α-elements The α-elements are produced mainly in short-lived massive stars exploding as core-collapse supernovae (CC-SNe), while a mi- nor fraction (depending on the element) is synthesised in SNe Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Due to the time delay between the enrichment of the two classes of SNe, the [α/Fe] abundance ratios are the classical trac- ers of the relative timescales of the different SNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' In particular, the metallicity of the knee (marking the onset of a significant chemical contribution by SNe Ia) can be used as a proxy of the SF efficiency of the galaxy (Tinsley 1979;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Matteucci & Greggio 1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' O and Mg (the so-called hydrostatic α-elements) are pro- duced mainly in stars with masses larger than ∼30-35 M⊙ and without contribution by SNe Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' On the other hand, Si, Ca and Ti (explosive α-elements) are produced in less massive stars (∼15-25 M⊙) and with a smaller (but not negligible) contribu- tion by SNe Ia (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Kobayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 8 shows the behaviour with [Fe/H] of individual [α/Fe] abundance ratios, while Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 9 shows the run of the average values of hydrostatic and explosive [α/Fe].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' These abundance ratios in the SMC stars clearly display a decrease by increasing the metallicity, moving from enhanced values for the most metal-poor stars ([Fe/H]<– 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 dex) down to solar-scaled values in the dominant population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' This trend is in contrast with that obtained by the APOGEE sur- vey, where "there is a slight increase in [Mg/Fe] beginning at [Fe/H]∼–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='3 dex, with a peak at [Fe/H]∼–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='0 dex, followed by a slight decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The [O/Fe], [Si/Fe] and [Ca/Fe] abundance pat- terns are flat over this range" (Hasselquist et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The most metal-poor stars in our sample exhibit enhanced values of [α/Fe] and in agreement with the results by Nidever et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (2020) and Reggiani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (2021) for SMC stars of sim- ilar metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Oxygen and magnesium, that are mainly pro- duced by stars with masses larger than ∼30 M⊙, are, however, slightly underabundant at low [Fe/H] with respect to the MW sample, which points to a lower contribution from the most mas- sive stars to the overall chemical enrichment of the SMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The subsequent decrease of [α/Fe] at higher [Fe/H] indicates that these stars formed from a gas enriched by SNe Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' For stars with Article number, page 10 of 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='0 [Fe/H] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 0 5 10 Age (Gyr)0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 [Fe/H] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 Distance (degrees)Mucciarelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' : SMC field stars Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Behaviour of the light element [Na/Fe] and α-elements [O/Fe], [Mg/Fe], [Si/Fe], [Ca/Fe] and [Ti/Fe] abundance ratios as a function of [Fe/H] for SMC stars located in the fields FLD-419, FLD-339 and FLD-121 (blue, green and red circles, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Arrows indicate upper limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The errorbars in the bottom-right corner indicate the typical uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Grey squares are the average values for the five Galactic GCs of the control sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Abundances of Galactic stars from the literature are also plotted as a reference: Edvardsson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (1993);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Gratton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (2003);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Reddy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (2003, 2006);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Bensby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (2005, 2014) for all the elements, Fulbright (2000);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Stephens & Boesgaard (2002);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Roederer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (2014) for Na, Mg, Si, Ca and Ti, Adibekyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (2012) for Na, Mg, Si and Ca, Barklem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (2005) for Mg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Article number, page 11 of 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 0 Mg/Fe] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 1 [Fe/H]0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 0 [Si/Fe] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 [Fe/H]0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 0 [Ca/Fe] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 [Fe/H]0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 0 [Ti/Fe] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 [Fe/H]0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 Na/Fe] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 [Fe/H]0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 0 [O/Fe] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 [Fe/H]A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' smc_fld_v7 [Fe/H]>–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 dex the difference in [α/Fe] between SMC and MW stars becomes more significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' In particular, the SMC-MW dif- ference is more pronounced for hydrostatic α-elements, again suggesting a lower contribution by stars with masses larger than 30-35 M⊙ to the chemical enrichment of the SMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' We note, as for Na, that the metal-rich stars in FLD-419 are slightly enhanced in [Ti/Fe], by ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1 dex, with respect to the stars of the other two fields with similar [Fe/H].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Behaviour of the hydrostatic and explosive average [α/Fe] abun- dance ratios as a function of [Fe/H].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Same symbols of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Iron-peak elements Iron-peak elements are produced mainly in massive stars, through different nucleosynthesis paths (Limongi & Chieffi 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Romano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Kobayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2020b), and ejected in the interstellar medium both from normal CC-SNe and hyper- novae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' These elements are also partly produced by SNe Ia on longer time scales (Leung & Nomoto 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Lach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Sc and V are produced mainly in massive stars, with a small contribution by SN Ia only for V (Kobayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The SMC stars with [Fe/H]<–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 dex have Sc and V abun- dances compatible with those measured in the control sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' We note some offsets between these abundance ratios in the con- trol sample and in the literature data, likely attributable to differ- ent linelists (in terms of log gf and/or hyperfine structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' On the other hand, the metal-rich SMC stars have abundances of Sc and V significantly lower than the MW stars (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' For both elements, we observed a decrease of the abundance ratio with in- creasing [Fe/H] because of the overwhelming delayed contribu- tion to Fe by SNe Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' This behaviour resembles those observed in metal-rich stars of dwarf galaxies, like LMC and Sagittarius (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Sbordone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Minelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' We note that also [V/Fe] in metal-rich stars of FLD-419 is systematically higher by ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='15 dex with respect to the stars of FLD-339.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' A compara- ble shift has been detected also for [V/Fe] in the LMC disc and bar stars (Van der Swaelmen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Ni is largely produced by SN Ia, with production also by CC-SNe, similar to the production of Fe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The SMC stars have [Ni/Fe] values compatible with those measured in the GCs of the control sample until [Fe/H]∼–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='0 dex, while for higher metal- licities this abundance ratio slightly decreases, reaching values around [Ni/Fe]∼–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='2 dex (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' A similar behaviour in the SMC stars has been observed by Hasselquist et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' This mild trend resembles that observed for [Ni/Fe] in the LMC and in Sagittarius at higher [Fe/H] (Minelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The decrease of [Ni/Fe] at higher metallicities is not observed in MW stars, where [Ni/Fe] remains constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' In this respect, Kobayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (2020a) suggested a lower contribution by sub- Chandrasekhar mass SN Ia to reproduce the [Ni/Fe] measured in dwarf spheroidal galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Cu is produced mainly in massive stars through the weak s- process (Romano & Matteucci 2007), with a small contribution by AGB stars (Travaglio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2004) and a negligible contri- bution by SN Ia (Iwamoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Romano & Matteucci 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The [Cu/Fe] abundance ratio in the SMC stars exhibits a large star-to-star dispersion and it is difficult to establish its real trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' However, it is clear that the most metal-rich SMC stars have [Cu/Fe] lower than that measured in MW stars, indicating again a lower contribution to the chemical enrichment by mas- sive stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Values of [Cu/Fe] lower than those measured in MW stars have been observed also in the LMC (Van der Swaelmen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2013), Sagittarius (Sbordone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2007) and Omega Centauri (Cunha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Neutron capture elements Elements heavier than the iron-peak group are produced through neutron capture processes on seed nuclei, followed by β decays (Burbidge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 1957).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The neutron capture elements measured here (namely Zr, Ba and La) are produced mainly by the slow process occurring in low-mass (1-3 M⊙) AGB stars and in a mi- nor amount in more massive stars (Busso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Cristallo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' At low metallicities these elements are produced also through rapid processes (Truran 1981), occurring in rare and energetic events like neutron star mergers or collapsars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' In this spectroscopic dataset, there are no transition of pure r-process elements (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Eu) and we cannot discuss the relative contribu- tion of these two production channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' However, Reggiani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (2021) analysed 4 metal-poor SMC giant stars finding [Eu/Fe] values higher than those of the MW stars, supporting a strong contribution at these metallicities by r-process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The SMC stars show [Zr/Fe] and [La/Fe] abundance ratios similar, within the star-to-star scatter, to those observed in MW stars, and slightly higher [Ba/Fe].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Generally, these results sug- gest that the enrichment by AGB stars in the SMC has been Article number, page 12 of 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 0 [(O+Mg)/Fe] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 1 [Fe/H]0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 [(Si+Ca+Ti)/Fe 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 [Fe/H]Mucciarelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' : SMC field stars Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Behaviour of the iron-elements [Sc/Fe], [V/Fe], [Ni/Fe] and [Cu/Fe] abundance ratios as a function of [Fe/H].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Abundances of Galactic field stars are from Reddy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (2003, 2006);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Roederer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (2014) for all the elements, Gratton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (2003) for Sc, V and Ni, Fulbright (2000) for V and Ni, Adibekyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (2012) for Sc and Ni, Edvardsson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (1993);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Stephens & Boesgaard (2002);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Bensby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (2005) for Ni, Bihain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (2004);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (2015) for Cu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Same symbols of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' comparable to that in the MW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' [Ba/Fe] in the SMC stars is en- hanced (∼+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='3/+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='4 dex) and higher than the values measured in the MW stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' [Ba/Fe] displays a large scatter at all the metal- licities and not explainable in light of the typical uncertainties in the abundance ratios (∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='15 dex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' At [Fe/H]∼–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='0 dex, the SMC stars have values of [Ba/Fe] higher than those observed in the MW stars, suggesting a galaxy-wide initial mass func- tion (IMF) biased in favour of the low-mass stars in the SMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' A similar behaviour is observed for [La/Fe], while [Zr/Fe] presents a trend in agreement with that observed for the MW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Similar to what we observe for Na and V, also for [Zr/Fe] we found a shift (∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='2 dex) between FLD-339 and FLD-419 that resemble those observed for the same ratio between LMC disc and bar stars (Van der Swaelmen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The high values of [Ba/Fa] and [La/Fe], together with the large star-to-star scatter, suggest that the production of s-process elements has been very efficient in the SMC, while the large star-to-star scatter could arise from enrichment from AGB stars of different metallicities, being the yields of AGB stars for these elements extremely metallicity- dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Finally, we identified a few stars with high [Ba/Fe] and [La/Fe] values (>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='7 dex, reaching also +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='3 dex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' A similar enhancement of s-process elements could be due to mass transfer processes from an AGB companion star in a binary system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Conclusions The analysis of optical spectra of 206 SMC RGB stars located in three different positions of the parent galaxy has allowed us to highlight some finer details of the complex and still poorly known nature of this galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The main results are summarised as follows: – The RV and [Fe/H] distributions of the three fields are dif- ferent with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' The fields FLD-339 and FLD-419, despite the same distance from the SMC centre, have [Fe/H] distributions peaked at different values, separated by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='2 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Article number, page 13 of 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 [ScI/Fe] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 [Fe/H]0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 [V/Fe] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 [Fe/H]0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 [Ni/Fe] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 [Fe/H]0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 [Cu/Fe] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 [Fe/H]A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' smc_fld_v7 These two populations could be connected to different bursts of SF occurring in the recent life of the SMC (Massana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2022) or the result of a different chemical enrichment path in these regions (despite their similar projected distance from the SMC centre).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' – The fraction of metal-poor ([Fe/H]<–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 dex) stars increases outward, being ∼1% in the two internal fields and ∼20% in FLD-121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' This run likely reflects an age gradient in the SMC, with the internal regions dominated by intermediate- age, metal-rich stars and the outskirts by the old, metal-poor spheroid (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Rubele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' – The RV-[Fe/H] distribution of the observed fields seems to suggest the possible existence of chemically/kinematic distinct substructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' In particular, we potentially identi- fied two groups of stars, one around [Fe/H]∼–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1 dex and RV∼+154 km s−1 and the other around [Fe/H]∼–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='9 dex and RV∼+172 km s−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' More data are needed to confirm the statistical significance of these chemo-kinematical substruc- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' – The SMC displays, especially for the dominant, metal-rich component, distinct abundance patterns with respect to the MW stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' In particular, those elements mainly produced by massive stars (Na, α, Sc, V and Cu) have abundance ratios lower than those measured in the MW stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' This suggests that the gas from which these stars formed has been poorly enriched by the most massive stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' This can be explained in light of the low SF rate expected for a galaxy as small as the SMC, leading to a lower contribution by massive stars to the overall chemical enrichment of the galaxy (Jeˇrábková et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' This is confirmed also by the most metal-poor stars of the sample that exhibit [O/Fe] and [Mg/Fe] ratios slightly lower than those in MW stars of sim- ilar [Fe/H].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' – The [s/Fe] abundance ratios are enriched with respect to the MW stars, with a large star-to-star scatter, suggesting that these elements are produced by AGB stars of different masses and metallicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Also, the enhancement of the [s/Fe] abundance ratios in the SMC seems to suggest a galaxy-wide IMF biased in favour of the low-mass stars in the SMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' – The possibility that the IMF is not universal, but varies with the environment is the subject of lively debate (Bastian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Hopkins 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Smith 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Theoretically, if stars form in clusters according to IMFs that depend on the metal- licity and density of the parent gaseous clumps, it is possi- ble to calculate the integrated galaxy-wide IMF that in turn depends on the metallicity and star formation rate of the host galaxy (Jeˇrábková et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' More- over, the abundance ratios of chemical elements produced in stars with initial masses falling in narrow and well-detached ranges can be used as powerful, indirect probes of the shape of the galaxy-wide IMF (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Romano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Observationally, the possibility that the Sagittarius dwarf spheroidal galaxy had a stronger contribution from AGB stars to its chemical enrichment than the MW and the LMC is discussed in Hasselquist et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Similarly, Hallakoun & Maoz (2021), resting on Gaia DR2 data, point to a bottom- heavy IMF for the Gaia-Enceladus progenitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Finally, Muc- ciarelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (2021) claim that the LMC GC NGC 2005 must have formed in an accreted system that experienced an ex- tremely low star formation rate and, hence, an extremely low number of hypernova explosions, in order to explain the pe- culiarly low Zn abundance of the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' On the other hand, Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (2019) do not find any clear cut evidence in favour of a non-standard IMF in the Sculp- tor dwarf spheroidal galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' In a forthcoming paper, we will quantitatively deal with the issue of IMF variations in the SMC by computing chemical evolution models specifically tailored to this galaxy (Romano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', in preparation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' – The three fields exhibit similar chemical patterns for all the elements but Na, V, Zr and Ti showing subtle differences among the fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Differences in the same abundance ratios have been observed also in the LMC between bar and disk stars (Van der Swaelmen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' These differences con- firm that the chemical enrichment history in the SMC has been not uniform but depends on the position within the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' These promising results enforces the need to study the prop- erties of the SMC stars locally rather than globally, with an effort to enlarge the samples of high-resolution spectra located in dif- ferent regions of the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' In this respect, the advent of the multi-object spectrographs like MOONS at the Very Large Tele- scope (Cirasuolo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2020) and 4MOST at the VISTA Tele- scope (de Jong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2019) will allow us a significant improve- ment in the investigation of possible chemically-distinct sub- structures in the Magellanic Clouds (Gonzalez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' We thanks the referee, Mathieu Van der Swaelmen, for the useful comments and suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' This research is funded by the project "Light- on-Dark" , granted by the Italian MIUR through contract PRIN-2017K7REXT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Lardo acknowledges funding from Ministero dell’Università e della Ricerca through the Programme Rita Levi Montalcini (grant PGR18YRML1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='int/gaia), processed by the Gaia Data Process- ing and Analysis Consortium (DPAC,https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='int/web/ gaia/dpac/consortium).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Funding for the DPAC has been provided by na- tional institutions, in particular the institutions participating in the Gaia Mul- tilateral Agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Article number, page 14 of 16 Mucciarelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' : SMC field stars Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Behaviour of the neutron capture-elements [Zr/Fe], [Ba/Fe] and [La/Fe] abundance ratios as a function of [Fe/H].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Abundances of Galac- tic field stars are from Mishenina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Roederer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (2014) for all the elements, Edvardsson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (1993);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Fulbright (2000);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Reddy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (2003) for Zr and Ba, Burris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (2000);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Battistini & Bensby (2016) for Zr and La, Stephens & Boesgaard (2002);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Barklem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (2005);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Bensby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' (2005) for Ba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' References Abdurro’uf, Accetta, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Aerts, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2022, ApJS, 259, 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='3847/1538- 4365/ac4414 Adibekyan, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Sousa, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Santos, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2012, A&A, 545, A32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1051/0004-6361/201219401 Andrae, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Fouesneau, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Creevey, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2018, A&A, 616, A8 Barklem, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Christlieb, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Beers, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2005, A&A, 439, 129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1051/0004-6361:20052967 Bastian, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Covey, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', & Meyer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2010, ARA&A, 48, 339.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1146/annurev-astro-082708-101642 Battistini, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' & Bensby, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2016, A&A, 586, A49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1051/0004- 6361/201527385 Bekki, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Couch, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Beasley, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2004, ApJ, 610, L93 Bekki, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', & Chiba, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', 2005, MNRAS, 356, 680 Besla, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Kallivayalil, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Hernquist, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2007, ApJ, 668, 949.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1086/521385 Besla, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Kallivayalil, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Hernquist, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2010, ApJ, 721, L97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1088/2041-8205/721/2/L97 Besla, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2015, arXiv:1511.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='03346 Bensby, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Feltzing, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Lundström, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2005, A&A, 433, 185.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1051/0004-6361:20040332 Bensby, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Feltzing, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', & Oey, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2014, A&A, 562, A71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1051/0004- 6361/201322631 Bertaux, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Lallement, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Ferron, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Boonne, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', & Bodichon, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', 2014, A&A, 564, 46 Bihain, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Israelian, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Rebolo, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2004, A&A, 423, 777.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1051/0004-6361:20035913 Burbidge, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Burbidge, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Fowler, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 1957, Reviews of Modern Physics, 29, 547.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1103/RevModPhys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='547 Burris, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Pilachowski, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Armandroff, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2000, ApJ, 544, 302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1086/317172 Busso, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Gallino, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', & Wasserburg, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 1999, ARA&A, 37, 239.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1146/annurev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='239 Caffau, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Ludwig, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Steffen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2011, Sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', 268, 255 Carrera, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Gallart, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Aparicio, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Costa, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Mendez, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', & Noel, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', 2008, AJ, 136, 1039 Carretta, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Bragaglia, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Gratton, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2009, A&A, 505, 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1051/0004-6361/200912096 Carretta, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Bragaglia, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Gratton, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2014, A&A, 564, A60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1051/0004-6361/201323321 Castelli, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', & Kurucz, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', 2004, astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='ph-5087 Choudhury, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', de Grijs, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Rubele, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2020, MNRAS, 497, 3746.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1093/mnras/staa2140 Cignoni, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Cole, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Tosi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Gallagher, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Sabbi, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Anderson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Grebel, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' & Nota, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', 2012, ApJ, 754, 130 Cignoni, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Cole, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Tosi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Gallagher, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Sabbi, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Anderson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Grebel, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' & Nota, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', 2013, ApJ, 775, 83 Cioni, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', van der Marel, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Loup, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', & Habing, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', 2000, A&A, 359, 601 Cirasuolo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Fairley, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Rees, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2020, The Messenger, 180, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='18727/0722-6691/5195 Cristallo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Straniero, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Piersanti, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2015, ApJS, 219, 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1088/0067-0049/219/2/40 Cunha, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Smith, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Suntzeff, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2002, AJ, 124, 379.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1086/340967 Dalessandro, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Lapenna, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Mucciarelli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Origlia, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Ferraro, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', & Lan- zoni, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', 2016, ApJ, 829, 77 de Grijs, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' & Bono, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2015, AJ, 149, 179.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1088/0004-6256/149/6/179 de Jong, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Agertz, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Berbel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2019, The Messenger, 175, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='18727/0722-6691/5117 De Leo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Carrera, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Noël, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2020, MNRAS, 495, 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1093/mnras/staa1122 Di Teodoro, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', McClure-Griffiths, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Jameson, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2019, MN- RAS, 483, 392.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1093/mnras/sty3095 Dobbie, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Cole, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Subramaniam, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', & Keller, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', 2014, MNRAS, 442, 1663 Dobbie, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Cole, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Subramaniam, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', & Keller, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', 2014, MNRAS, 442, 1680 Edvardsson, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Andersen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Gustafsson, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 1993, A&A, 500, 391 Fuhr, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Martin, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', & Wiese, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 1988, Journal of Physical and Chemi- cal Reference Data, 17 Fulbright, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2000, AJ, 120, 1841.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1086/301548 Fuhr, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' & Wiese, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2006, Journal of Physical and Chemical Reference Data, 35, 1669.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='2218876 Gaia Collaboration, Prusti, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', de Bruijne, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2016, A&A, 595, A1 Gaia Collaboration, Babusiaux, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', van Leeuwen, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2018, A&A, 616, A10 Gaia Collaboration, Brown, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Vallenari, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2021, A&A, 649, A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1051/0004-6361/202039657 Glatt, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', 2008, AJ, 135, 1106 Article number, page 15 of 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 0 Zr/Fe 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 [Fe/H]0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 [Ba/Fe] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 [Fe/H]1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 0 [La/Fe 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='5 [Fe/H]A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' smc_fld_v7 onzalez, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Mucciarelli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Origlia, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2020, The Messenger, 180, 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='18727/0722-6691/5196 Graczyk, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' etal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', 2014, ApJ, 780, 59 Grady, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Belokurov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', & Evans, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2021, ApJ, 909, 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='3847/1538- 4357/abd4e4 Gratton, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Carretta, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Claudi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2003, A&A, 404, 187.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1051/0004-6361:20030439 Grevesse, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', & Sauval, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 1998, Space Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', 85, 161 Jeˇrábková, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Hasani Zonoozi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Kroupa, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2018, A&A, 620, A39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1051/0004-6361/201833055 Hallakoun, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' & Maoz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2021, MNRAS, 507, 398.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1093/mnras/stab2145 Harris, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', & Zaritsky, D, 2004, AJ, 127, 1531 Harris, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' & Zaritsky, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', 2006, AJ, 131, 2514 Hasselquist, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Hayes, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Lian, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2021, ApJ, 923, 172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='3847/1538-4357/ac25f9 Helmi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Babusiaux, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Koppelman, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2018, Nature, 563, 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1038/s41586-018-0625-x Hill, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Barbuy, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', & Spite, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', 1997, A&A, 323, 461 Hill, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', François, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Spite, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2000, A&A, 364, L19 Hill, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Skúladóttir, Á.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Tolstoy, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2019, A&A, 626, A15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1051/0004-6361/201833950 Hopkins, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2018, PASA, 35, e039.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1017/pasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='29 Iwamoto, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Brachwitz, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Nomoto, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 1999, ApJS, 125, 439.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1086/313278 Kobayashi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Leung, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', & Nomoto, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2020, ApJ, 895, 138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='3847/1538-4357/ab8e44 Kobayashi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Karakas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', & Lugaro, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2020, ApJ, 900, 179.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='3847/1538-4357/abae65 Kallivayalil, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', van der Marel, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Besla, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2013, ApJ, 764, 161.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1088/0004-637X/764/2/161 Kurucz, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', 2005, MSAIS, 8, 14 Lach, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Röpke, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Seitenzahl, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2020, A&A, 644, A118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1051/0004-6361/202038721 Lapenna, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Mucciarelli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Origlia, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2012, ApJ, 761, 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1088/0004-637X/761/1/33 Leung, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' & Nomoto, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2018, ApJ, 861, 143.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='3847/1538-4357/aac2df Limongi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' & Chieffi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2003, ApJ, 592, 404.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1086/375703 Lind, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Asplund, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Barklem, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', & Belyaev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', 2011, A&A, 528, 103 Martin, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Fuhr, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', & Wiese, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 1988, New York: American Institute of Physics (AIP) and American Chemical Society, 1988 Massana, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Ruiz-Lara, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Noël, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2022, MNRAS, 513, L40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1093/mnrasl/slac030 Matteucci, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' & Greggio, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 1986, A&A, 154, 279 McCall, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2004, AJ, 128, 2144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1086/424933 Minelli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Mucciarelli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Romano, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2021, ApJ, 910, 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='3847/1538-4357/abe3f9 Mishenina, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Pignatari, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Korotin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2013, A&A, 552, A128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1051/0004-6361/201220687 Mucciarelli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Origlia, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Maraston, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', & Ferraro, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', 2009, ApJ, 690, 288 Mucciarelli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Origlia, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', & Ferraro, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2010, ApJ, 717, 277.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1088/0004-637X/717/1/277 Mucciarelli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Pancino, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Lovisi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Ferraro, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', & Lapenna, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', 2013, ApJ, 766, 78 Mucciarelli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', 2013, arXiv1311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1403 Mucciarelli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Monaco, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Bonifacio, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', & Saviane, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', 2017, A&A, 603L, 7 Mucciarelli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' & Bonifacio, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2020, A&A, 640, A87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1051/0004- 6361/202037703 Mucciarelli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Bellazzini, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', & Massari, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2021, A&A, 653, A90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1051/0004-6361/202140979 Mucciarelli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Massari, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Minelli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2021, Nature Astronomy, 5, 1247.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1038/s41550-021-01493-y Nidever, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Hasselquist, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Hayes, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2020, ApJ, 895, 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='3847/1538-4357/ab7305 Noel, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' E D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Gallart, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Costa, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', & Mendez, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', 2007, AJ, 133, 2037 Pagel, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' & Tautvaisiene, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 1998, MNRAS, 299, 535.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1046/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1365- 8711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='01792.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='x Parisi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Geisler, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Carraro, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2016, AJ, 152, 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='3847/0004- 6256/152/3/58 Parisi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Gramajo, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Geisler, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2022, A&A, 662, A75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1051/0004-6361/202142597 Pasquini, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', 2000, SPIE, 4008, 129 Pompéia, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Hill, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Spite, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2008, A&A, 480, 379.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1051/0004- 6361:20064854 Reddy, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Tomkin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Lambert, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2003, MNRAS, 340, 304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1046/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1365-8711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='06305.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='x Reddy, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Lambert, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', & Allende Prieto, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2006, MNRAS, 367, 1329.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1111/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1365-2966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='10148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='x Reggiani, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Schlaufman, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Casey, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2021, AJ, 162, 229.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='3847/1538-3881/ac1f9a Ripepi, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Cioni, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Moretti, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2017, MNRAS, 472, 808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1093/mnras/stx2096 Roederer, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Preston, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Thompson, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2014, AJ, 147, 136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1088/0004-6256/147/6/136 Romaniello, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Primas, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Mottini, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Pedicelli, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Lemasle, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Bono, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Francois, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Groenewegen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', & Laney, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', 2008, A&A, 488, 731 Romano, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' & Matteucci, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2007, MNRAS, 378, L59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1111/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1745- 3933.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='00320.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='x Romano, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Karakas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Tosi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2010, A&A, 522, A32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1051/0004-6361/201014483 Romano, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Matteucci, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2017, MNRAS, 470, 401.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1093/mnras/stx1197 Rubele, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Pastorelli, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Girardi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2018, MNRAS, 478, 5017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1093/mnras/sty1279 Sabbi, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', 2009, ApJ, 703, 721 Sbordone, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Bonifacio, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Castelli, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', & Kurucz, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', 2004, MSAIS, 5, 93 Sbordone, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Bonifacio, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Buonanno, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2007, A&A, 465, 815.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1051/0004-6361:20066385 Schlafly, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', & Finkbeiner, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2011, ApJ, 737, 103 Scrucca, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Fop, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Murphy, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', & Raftery, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2016, The R Journal, 8, 289, https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='32614/RJ-2016-021 Skrutskie, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', 2006, AJ, 131, 1163 Smith, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' & Raggett, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 1981, Journal of Physics B Atomic Molecular Physics, 14, 4015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1088/0022-3700/14/21/016 Smith, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2020, ARA&A, 58, 577.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1146/annurev-astro-032620-020217 Spite, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Spite M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', & Francois, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', 1989a, A&A, 210, 25 Spite, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Spite M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', & Barbuy, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', 1989b, A&A, 222, 35 Stanimirovi´c, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Staveley-Smith, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', & Jones, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2004, ApJ, 604, 176.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1086/381869 Stephens, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' & Boesgaard, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2002, AJ, 123, 1647.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1086/338898 Stetson, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', & Pancino, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', PASP, 120, 1332 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2005, PASJ, 57, 27 Subramanian, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' & Subramaniam, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2009, A&A, 496, 399.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1051/0004- 6361/200811029 Subramanian, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Rubele, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Sun, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2017, MNRAS, 467, 2980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1093/mnras/stx205 Taibi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Battaglia, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Leaman, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2022, A&A, 665, A92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1051/0004-6361/202243508 Tinsley, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 1979, ApJ, 229, 1046.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1086/157039 Tolstoy, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Hill, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', & Tosi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2009, ARA&A, 47, 371.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1146/annurev- astro-082708-101650 Travaglio, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Gallino, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Arnone, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2004, ApJ, 601, 864.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1086/380507 Truran, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 1981, A&A, 97, 391 Tsujimoto, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' & Bekki, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2009, ApJ, 700, L69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1088/0004- 637X/700/2/L69 van der Marel, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Kallivayalil, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', & Besla, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2009, The Magellanic System: Stars, Gas, and Galaxies, 256, 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1017/S1743921308028299 Van der Swaelmen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Hill, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Primas, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2013, A&A, 560, A44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1051/0004-6361/201321109 Yan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Shi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', & Zhao, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2015, ApJ, 802, 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1088/0004- 637X/802/1/36 Yan, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', Jerabkova, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=', & Kroupa, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' 2020, A&A, 637, A68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} +page_content='1051/0004- 6361/202037567 Article number, page 16 of 16' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFAT4oBgHgl3EQf9x57/content/2301.08758v1.pdf'} diff --git a/iNFLT4oBgHgl3EQfai-M/content/tmp_files/2301.12075v1.pdf.txt b/iNFLT4oBgHgl3EQfai-M/content/tmp_files/2301.12075v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..eefdc0fc3e00fcf9f369cfd8418fe59705243ba4 --- /dev/null +++ b/iNFLT4oBgHgl3EQfai-M/content/tmp_files/2301.12075v1.pdf.txt @@ -0,0 +1,1078 @@ +arXiv:2301.12075v1 [econ.GN] 28 Jan 2023 +AN EXAMINATION OF RANKED CHOICE VOTING IN THE +UNITED STATES, 2004-2022 +ADAM GRAHAM-SQUIRE AND DAVID MCCUNE +Abstract. From the perspective of social choice theory, ranked-choice vot- +ing (RCV) is known to have many flaws. RCV can fail to elect a Condorcet +winner and is susceptible to monotonicity paradoxes and the spoiler effect, for +example. We use a database of 182 American ranked-choice elections for polit- +ical office from the years 2004-2022 to investigate empirically how frequently +RCV’s deficiencies manifest in practice. +Our general finding is that RCV’s +weaknesses are rarely observed in real-world elections, with the exception that +ballot exhaustion frequently causes majoritarian failures. +1. Introduction +The use of ranked-choice voting (RCV) has greatly increased in the United States +during the last few years. New York City first used RCV for city primary elections +in 2021, the state of Maine has used it for state primary elections and elections for +federal office since 2018, and the state of Alaska implemented RCV for federal and +state offices in 2022. The voting method is well-known to have many deficiencies +which receive attention in the social choice literature. The deficiencies with which +we are concerned are: +• RCV can fail to elect the Condorcet winner. +• RCV is susceptible to the spoiler effect. +• RCV is susceptible to downward and upward monotonicity paradoxes. +• RCV is susceptible to the truncation paradox, the most extreme version of +which is the no-show paradox. +• RCV is susceptible to compromise strategic voting. +• RCV is not truly “majoritarian” because of ballot exhaustion. +The purpose of this article is to examine how often these issues occur in actual +elections, where we focus on the single-winner case. To that end, we collected the +ballot data for as many single-winner ranked-choice American political elections as +we could, resulting in a database of 182 elections. The flaws of RCV listed above +can manifest only in elections without a majority candidate (i.e., elections in which +RCV goes to at least a second round), and thus our database consists only of such +elections. Our general finding is that these flaws occur rarely in actual ranked-choice +elections, except for majoritarian failures caused by ballot exhaustion. While much +of our analysis is new, perhaps the primary value of this article is that we provide +a complete analysis of “problematic” American RCV elections, summarizing much +of the empirical research of the last 18 years. +2010 Mathematics Subject Classification. Primary 91B10; Secondary 91B14. +Key words and phrases. Condorcet winner, monotonicity paradox, spoiler effect, ballot ex- +haustion, majoritarian, empirical results. +1 + +AN EXAMINATION OF RANKED CHOICE VOTING IN THE UNITED STATES, 2004-2022 2 +Num. Voters +27070 +15478 +11262 +34078 +3659 +21237 +47407 +4647 +23733 +1st choice +Begich +Begich +Begich +Palin +Palin +Palin +Peltola +Peltola +Peltola +2nd choice +Palin +Peltola +− +Begich +Peltola +− +Begich +Palin +− +3rd choice +Peltola +Palin +− +Peltola +Begich +− +Palin +Begich +− +Table 1. The August 2022 Alaska Special Election for US House, +after eliminating write-in candidates. +We note RCV has other potential flaws which are best examined through the +lens of political science. For example, some of RCV’s detractors say that voters find +the method confusing, or that RCV does not deliver on its proponents’ promise of +creating more civil political campaigns. We do not address these claims; because +of our backgrounds, we evaluate RCV only on criteria which have been examined +in the mathematically-oriented social choice literature. +2. Definitions: Ranked-Choice Voting and Its Deficiencies +We begin with a definition of RCV and then formally define the flaws with +which we are concerned, with examples throughout. Note that the social choice +literature generally uses the term RCV to refer to any election that involves ranking +candidates, and the literature uses terms such as instant runoff voting, alternative +vote, the Hare rule, etc., to describe the vote method defined below. We use the +term RCV, as it aligns with how the term used by most municipal and state elections +offices in the US. +In an RCV election, voters cast preference ballots which allow a voter to rank the +candidates in order of preference. Voters often do not provide a complete ranking, +either by choice or because the voters’ jurisdiction limits the number of candidates +that can be ranked on a ballot. For example, the city of Minneapolis, MN allows +voters to rank only three candidates, regardless of how many candidates are in +the race. +After an election the ballots are aggregated into a preference profile, +which shows the number of each type of ballot cast. For example, Table 1 shows +a preference profile for the August 2022 Special Election for the single US House +seat in Alaska, an election involving the three candidates Nick Begich, Sarah Palin, +and Mary Peltola1. The table shows that 27070 voters ranked Begich first, Palin +second, and Peltola third; the other numbers across the top row convey similar +information about the number of voters who cast the corresponding type of ballot2. +We use the notation A > B to denote that a voter ranks candidate A above B, and +thus 27070 voters choose the ranking Begich > Palin > Peltola. To keep the table +a manageable size, we combine ballots of the form A > B and A > B > C, which +has no effect on the winner of the election under RCV. We also combine ballots +of these forms in all subsequent examples. Note that voters are not required to +provide a complete ranking of the candidates, and some voters choose to rank only +a single candidate on their ballots. For example, 11262 voters ranked Begich on +their ballots but did not rank Palin or Peltola. +1The race also included several write-in candidates, all of whom received a relatively trivial +amount of votes. We eliminated these candidates before creating the preference profile. +2There is some ambiguity about how the ballots were processed, and our preference profile +numbers differ by a handful of ballots from the official vote counts. For example, according to +Alaska Division of Elections the 27070 should be 27053. These small differences do not affect our +conclusions. + +AN EXAMINATION OF RANKED CHOICE VOTING IN THE UNITED STATES, 2004-2022 3 +The method of RCV declares a winner as follows: if the election contains a +majority candidate, a candidate who receives a majority of first-place votes, then +that candidate is declared the winner. Otherwise eliminate the candidate(s) with +the fewest first-place votes and transfer that candidate’s votes to other candidates +based on the second-place choices on that eliminated candidate’s ballots. +If a +surviving candidate now has a majority of the remaining votes, that candidate wins; +otherwise, the process of elimination and vote-transfer continues in this manner +until a candidate has secured a majority of (first-place and transferred) votes. +We illustrate the RCV algorithm using the preference profile in Table 1. Begich +receives 53810 first-place votes, Palin receives 58974, and Peltola receives 75799, +and thus Begich is eliminated. As a result, 27070 of his votes are transferred to +Palin, 15478 are transferred to Peltola, and 11262 are dropped from the election +(these ballots are said to be exhausted). After the vote transfer, Peltola wins the +election with 91277 votes to Palin’s 86044. +If an election contains more than three candidates then RCV proceeds similarly, +but the process may take more rounds to select a winner. We now define the flaws +of RCV that we investigate in this article. +Condorcet Failure: In the Alaska House election, if we compare Begich to +Palin in a head-to-head matchup then 101229 voters prefer Begich to Palin whereas +63621 voters prefer Palin to Begich. Similarly, 93052 voters prefer Begich to Peltola +whereas 79558 voters prefer Peltola to Begich. Based on the ranking information +provided by the voters3, Begich wins each of his head-to-head matchups and thus +is the Condorcet winner of the election. Condorcet winners receive much attention +in the social choice literature because they are considered “consensus” or “strong” +candidates, candidates who “should” win (assuming the election contains such a +candidate). We say that an RCV election demonstrates a Condorcet failure when +the election contains a Condorcet winner and RCV fails to select this candidate, +which occurs in the Alaska House election. +Spoiler Effect: An RCV election demonstrates the spoiler effect if there exists +a subset of losing candidates such that removing these candidates from the election +causes the winner of the election to change. In the Alaska House election, if the +losing candidate Palin were removed then Begich would win, and thus this election +demonstrates a spoiler effect under RCV. +Upward Monotonicity Paradox: An RCV election demonstrates an upward +monotonicity paradox if there exists a set of ballots such that shifting the RCV +winner up the rankings on those ballots but keeping the relative rankings of the +other candidates the same, thereby creating a hypothetical second preference profile +in which the winner has more voter support, creates an election in which the original +RCV winner does not win. In the Alaska House election, if 6000 ballots on which +Palin is ranked first and no other candidate is ranked on the ballot were changed to +ballots of the form Peltola > Palin then Peltola would lose the resulting election. +That is, giving the winning candidate Peltola more first place votes by shifting her +up the rankings on some ballots turns her into a loser. The reason is that even +though Peltola picks up an additional 6000 first-place votes in the hypothetical +second election, this additional support causes Palin to be eliminated from the +3We choose to interpret ballots using the weak order model (Popov et al., 2014), in which all +candidates left off a ballot are tied for the lowest ranking on a voter’s ballot. + +AN EXAMINATION OF RANKED CHOICE VOTING IN THE UNITED STATES, 2004-2022 4 +Num.Voters +5237 +3316 +5566 +4050 +5708 +1894 +2251 +6909 +1695 +1st choice +Engardio +Engardio +Engardio +Melgar +Melgar +Melgar +Nguyen +Nguyen +Nguyen +2nd choice +Melgar +Nguyen +− +Engardio +Nguyen +− +Engardio +Melgar +− +3rd choice +Nguyen +Melgar +− +Nguyen +Engardio +− +Melgar +Engardio +− +Table 2. The 2020 District 7 Board of Supervisors election in San +Francisco, CA, after eliminating all but the final three candidates. +election first (in contrast to the original election, in which Begich was eliminated +first) and then Peltola would lose to Begich in the final round. +Downward Monotonicity Paradox: An RCV election demonstrates a down- +ward monotonicity paradox if there exists a set of ballots and a losing candidate +L such that shifting L down the rankings on those ballots but keeping the relative +rankings of the other candidates the same, thereby creating a hypothetical second +preference profile in which this losing candidate has less voter support, creates an +election in which L is the RCV winner. Our running example with the Alaska data +does not demonstrate a downward paradox: if either Begich or Palin were shifted +down on any set of ballots, they would remain losers of the election. +To demonstrate a real-world example of this paradox, consider the 2020 Board +of Supervisors election for the 7th District in San Francisco, CA. The election +contained seven (not write-in) candidates; Table 2 shows the preference profile for +the election after four candidates have been eliminated and their votes transferred. +At this point in the RCV process, Engardio has 14119 first-place votes, Melgar +has 11652, and Nguyen has 10855. Nguyen is eliminated, and after their votes +are transferred Melgar wins with 18561 votes to Engardio’s 16370. If Engardio +were shifted down one ranking on 800 ballots of the form Engardio > Nguyen +> Melgar so that those ballots change to Nguyen > Engardio > Melgar then in +the resulting election Melgar would be eliminated first and, even with less voter +support, Engardio would still have enough votes to defeat Nguyen head-to-head +in the final round. That is, shifting the losing candidate Engardio down on some +voters’ ballots would turn Engardio into the RCV winner. +As with the Alaska +election, this example creates a paradoxical outcome by changing the order in which +candidates are eliminated. +Truncation Paradox: An election demonstrates a truncation paradox if there +exists a set of voters such that the voters could create a more desirable electoral +outcome by ranking fewer candidates on their ballots; i.e., when these voters’ ex- +press less information on their ballots then a candidate whom they prefer more +wins the election than if the voters expressed more information. The most extreme +version of this paradox is a no-show paradox, which occurs when there exists +a set of voters such that removing their ballots from the election creates a more +desirable electoral outcome for those voters (removing the ballots altogether is the +most extreme form of ballot truncation). To be precise, a no-show paradox occurs +if A is the RCV winner and there exists a losing candidate B and a set of voters +that prefer B to A such that if these voters abstain from voting then B would be +the winner in the modified election with these voters’ ballots removed. The Alaska +election in Table 1 demonstrates this paradox: if 5400 voters who cast the ballot +Palin > Begich > Peltola were removed from the election then Palin would be elim- +inated first (recall that in the original election, Begich was eliminated first) and, +despite the removal of these ballots, Begich would have enough support to defeat + +AN EXAMINATION OF RANKED CHOICE VOTING IN THE UNITED STATES, 2004-2022 5 +Peltola head-to-head in the final round. These 5400 voters would have achieved a +more desirable outcome (their second-favorite candidate would have won instead of +their third favorite) if they had abstained. +Compromise Voting Failure: Compromise voting is a form of strategic voting +in which some voters calculate that their favorite candidate cannot win and so +insincerely rank another candidate as their first choice. This plays out regularly +in plurality elections, where voters often cast a vote for a candidate perceived as +viable instead of voting for their favorite. Generalizing the definition from (Green- +Armytage, 2014), we say that an election demonstrates a compromise voting failure +if there exists a losing candidate A and a set of ballots such that A is ranked above +the original RCV winner, and A becomes the RCV winner if we shift them up to +the first ranking on these ballots. That is, this failure occurs if there exists a set of +voters who should have cast a “compromise vote” for A, thereby causing A to win. +The definition in (Green-Armytage, 2014) is much more restrictive: they consider +only elections in which shifting A to the top of all ballots on which A is ranked +over the original RCV winner turns A into the winner. +The Alaska election in Table 1 demonstrates this failure: if 5400 voters who +ranked Palin first and Begich second had ranked Begich first instead then Begich +would have won the election and these voters would have obtained their second +choice instead of their third. To obtain a more desirable electoral outcome, these +voters should have “compromised” for Begich. +This election also fits the more +restrictive definition from (Green-Armytage, 2014), as Begich would become the +winner if all 34078 voters who cast the ballot Palin > Begich > Peltola were to +rank Begich first. +Majoritarian Failure: We say that an election demonstrates a majoritarian +failure if, when the RCV algorithm is run until there are only two candidates left, +the winning candidate in this final round does not achieve a majority of the total +number of votes cast. Both the Alaska and San Francisco elections demonstrate +this failure. Table 1 shows a total of 188583 voters in that election (which increases +slightly if we were to include write-in candidates), yet the winner Peltola earns +91277 in the final round, achieving only 48.4% of the votes of the total electorate. +Similarly, in the San Francisco election there were 39322 ballots cast and the winner +Melgar earns 18561/39322 = 47.2% of the total vote in the final round. The reason +these winners fail to earn a majority is ballot exhaustion (Burnett and Kogan, 2015), +where many partial ballots are discarded before the final round because these ballots +do not rank either of the two final candidates. We use the term winner’s vote share +to refer to the percentage of the total votes earned by the RCV winner in the final +round when there are only two candidates remaining, so that Melgar’s vote share +is 47.2%, for example. +3. Data Sources and Collection +We collected the vote data for as many American single-winner ranked-choice +political elections as we could, with the restriction that we obtained data only for +elections in which no candidate earns an initial majority of first-place votes. In +total our database contains 182 elections, 147 of which were collected by the first +author for (McCune & McCune, 2022a). Most of the elections are for municipal +office such as mayor or city councilor; a handful of elections are for statewide or +federal elections, such as elections for US House and Senate in Maine. The data + +AN EXAMINATION OF RANKED CHOICE VOTING IN THE UNITED STATES, 2004-2022 6 +Jurisdiction +Years for which we have data +Num. Elections +Alaska +2022 +13 +Aspen, CO +2009 +2 +Berkeley, CA +2010, 2014-2020 +6 +Bloomington, MN +2021 +1 +Burlington, VT +2006, 2009 +2 +Corvallis, OR +2022 +2 +Easthampton, MA +2020 +1 +Eastpointe, MI +2020 +1 +Elk Ridge, UT +2021 +1 +Las Cruces, NM +2019 +3 +Maine +2018-2022 +11 +Minneapolis, MN +2009, 2013, 2017, 2021 +26 +Minnetonka, MN +2021 +1 +New York City, NY +2021 +41 +Oakland, CA +2010-2020 +21 +Pierce County, WA +2008-2009 +4 +Portland, ME +2021-2022 +2 +San Francisco, CA +2004-2022 +32 +San Leandro, CA +2010-2014 +5 +Santa Fe, NM +2018 +2 +Springville, UT +2021 +1 +St. Louis Park, MN +2019, 2021 +2 +Telluride, CO +2015 +1 +Woodland Hills, UT +2021 +1 +Table 3. Summary of data sources. +was collected from election office websites, and some was received by request from +election offices when the data was not posted. +After our initial round of data +collection, the RCV advocacy organization FairVote created a publicly accessible +repository of American ranked-choice data (Otis, 2022) and we collected a handful +of additional elections from this source. Table 3 gives a summary of our database, +including the number of elections from each jurisdiction and the years for which +data is available. +4. Prior Social Choice Literature on RCV’s Flaws +There is a vast social choice literature which evaluates RCV as a voting method, +and most of this literature is theoretical. Due to the empirical nature of our work +we do not attempt to survey the theoretical literature; instead, we focus on prior +empirical work. +Condorcet failures: The largest empirical investigation of Condorcet failures in +American political elections occurs in (McCune & McCune, 2022a), which analyzed +147 of the elections in our database. The authors found only one Condorcet failure, +in a previously documented election from Burlington, VT (Gierzynski et al. 2010; +Ornstein & Norman 2014). (McCune & McCune, 2022a) also found only one single- +winner election without a Condorcet winner. +(Graham-Squire & Zayatz, 2021) +analyze 35 elections from our database, again finding a failure in only the Burlington + +AN EXAMINATION OF RANKED CHOICE VOTING IN THE UNITED STATES, 2004-2022 7 +election, and (Song, 2022) analyzed many4 of the pre-2021 elections in our database +for Condorcet failures. RCV advocacy groups such as FairVote also check ranked- +choice data for Condorcet failures (Landsman 2017; Otis 2021). +There is a small literature about Condorcet failures which investigates elections +outside the context of American political elections. For example, (Regenwetter et +al. 2007; Popov et al. 2014) analyse ranked-choice election data from the American +Psychological Association, using the data to generate tens of thousands of election +pseudoprofiles. They find an extremely low rate of Condorcet failures under RCV. +(Darmann et al. 2019) use survey data collected prior to the 2015 parliamentary +elections in the Austrian federal state of Styria, and again find little evidence of +Condorcet failures in the data. +Spoiler Effect: Most of the discussion of spoilers has been limited to the theoreti- +cal literature, with the exception that many non-academic articles discuss potential +spoilers in individual elections. For example, many newspaper articles (for a typ- +ical example example, see (Bokat-Lindell, 2021)) mention the famous case of the +2000 US Presidential election in which Ralph Nader is commonly understood to +have spoiled the election for Al Gore. (McCune & Wilson, 2022) is the largest +empirical study of the spoiler effect in American ranked-choice political elections. +The authors find two ranked-choice elections demonstrating the spoiler effect out +of 170 analyzed, 147 of which are in our database. They also perform bootstrap +analysis to generate pseudoprofiles, similar to the analysis in (Popov et al. 2014), +and find low rates of the spoiler effect in the generated data. A Condorcet failure +is a special case of the spoiler effect as we have defined it, and thus the Condorcet +studies mentioned above also indirectly address the spoiler effect. +Monotonicity and truncation paradoxes: Most of the elections in our database +have not been previously processed by code which searches for monotonicity or +truncation paradoxes. The largest empirical study of monotonicity and truncation +paradoxes in American political ranked-choice elections occurs in (Graham-Squire +& Zayatz, 2021), which analyses elections from Alameda County and San Francisco, +CA, as well as a mayoral election from Burlington, VT. The authors find a single +upward monotonicity paradox and no truncation paradoxes out of 35 elections +without a majority candidate. +(McCune & Graham-Squire, 2022) analyze 1079 +ranked-choice multiwinner elections from Scotland and find low (but non-zero) rates +of upward monotonicity, downward monotonicity, and no-show paradoxes. Many +articles which analyze real-world monotonicity failures focus on single elections. +See (Gierzynski et al. 2010; Ornstein & Norman 2014) for an analysis of the 2009 +Burlington, VT mayoral election and (McCune & McCune, 2022b) for an analysis +of monotonicity failures in a 2021 city council election in Minneapolis, MN. +Some prior work is semi-empirical in that the authors use polling or survey +data to estimate monotonicity failure rates in real-world elections. For examples of +this kind of analysis see (Gallagher, 2013), which studies single-transferable vote +elections in Ireland, and (Miller, 2017), which provides a semi-empirical analysis of +English general elections from 1992-2010. +Compromise voting failures: We are unaware of empirical studies which study +this failure. (Green-Armytage, 2014) and (Green-Armytage et al. 2016) analyze +compromise voting using a variety of models of voter behavior, and generally find +4We are not sure how many of the elections analysed in (Song, 2022) contain majority candi- +dates and so we cannot determine how much of their work overlaps with ours. + +AN EXAMINATION OF RANKED CHOICE VOTING IN THE UNITED STATES, 2004-2022 8 +that RCV is much less susceptible to this issue than other famous voting methods +such as plurality and the Borda count. +Majoritarian failures: +The only empirical studies of majoritarian failures of +which we are aware are (Burnett and Kogan, 2015), which analyze four American +elections in our database, and (Kilgour et al. 2020), which analyze 18 ranked-choice +elections, 4 of which are in our database (the other 14 contain majority candidates +or are elections from the UK or the American Psychological Association). FairVote +also seems to have done substantial analysis of ballot exhaustion5, but it is unclear +which elections were analyzed. +More generally, many studies address the topic of partial ballots without focusing +on majoritarian failures per se. (Coll, 2021) and (Donovan et al. 2022) study which +demographic groups are more likely to cast partial ballots. (Tomlinson et al., 2022) +analyse the effects of ballot truncation on the number of possible winners in ranked- +choice elections. +5. Methodology for Detecting RCV Weaknesses in Each Election +All elections in the database were processed using Python code which searched +for the given flaw. The code is adapted from programs used in (Graham-Squire & +Zayatz 2021), (McCune & McCune 2022a), (McCune & Wilson 2022), and (McCune +& Graham-Squire 2022). Checking if the RCV winner is the Condorcet winner or +if the RCV winner earns a majority of the initial total votes in the final round is +computationally straightforward. However, there are challenges when searching for +the spoiler effect, monotonicity paradoxes, truncation paradoxes, or compromise +voting failures. +If the number of candidates in an election is large enough then, due to limits of +computation time, we cannot check every subset of losing candidates for a change +in the winner when this subset is dropped. For example, the 2013 Minneapolis +mayoral election contained 35 candidates, resulting in billions of possible sets of +candidates to check for the spoiler effect. +For all elections in the database we +checked for individual spoiler candidates, but for elections with large numbers of +candidates we additionally checked only candidate subsets of size two. For elections +with more than twelve candidates, we also ran the RCV algorithm until only ten +candidates remained and then checked the resulting election for the spoiler effect. +Most of the elections were already processed in this manner in (McCune & Wilson, +2022). +To demonstrate a monotonicity or truncation paradox, we must find a set of +ballots such that shifting a candidate up or down the rankings, or truncating the +ballots, causes a change in the order of elimination which causes a paradoxical +change in the winner. Except for the case of three-candidate elections in which +every voter provides a complete ranking, there are no known necessary and suf- +ficient conditions for an election to exhibit one of these paradoxes. Thus, if an +election exhibits such a paradox, we cannot guarantee our code will find it. How- +ever, our monotonicity and truncation code has been thoroughly means-tested in +other projects and has successfully found many elections which demonstrate mono- +tonicity paradoxes. For example, the code found 21 elections demonstrating upward +monotonicity paradoxes in Scottish local government elections which had not been +previously documented (McCune & Graham-Squire, 2022). The code essentially +5See https://fairvote.org/resources/data-on-rcv/#evaluating-rcv-election-outcomesnbsp. + +AN EXAMINATION OF RANKED CHOICE VOTING IN THE UNITED STATES, 2004-2022 9 +Num.Voters +2043 +371 +568 +1332 +767 +455 +495 +1513 +1289 +1st choice +Kiss +Kiss +Kiss +Montroll +Montroll +Montroll +Wright +Wright +Wright +2nd choice +Montroll +Wright +− +Kiss +Wright +− +Kiss +Montroll +− +3rd choice +Wright +Montroll +− +Wright +Kiss +− +Montroll +Kiss +− +Table 4. The 2009 mayoral election in Burlington, VT, after elim- +inating all but the final three candidates. This table is taken from +(Ornstein & Norman, 2014). +works by strategically changing or truncating ballots to achieve a change in the or- +der of elimination or the candidates, and measures if a paradoxical outcome occurs +as a result of the ballot changes. +The code we use to search for compromise voting failures is a straightforward +adaptation of our monotonicity code. +6. Results +We now present our results, separating the issue of ballot exhaustion from the +others. Only eight elections in the database, listed below, exhibit any kind of non- +majoritarian flaw. +Four of the elections demonstrate only a compromise voting +failure; we include the preference profile for only one of these elections as the +dynamics of the other three are similar. +2009 Mayoral Election in Burlington, VT (Table 4). This election demon- +strates the following flaws (Gierzynski et al. 2010; Ornstein & Norman 2014): +• Condorcet failure: the Condorcet winner, Montroll, is not the RCV winner, +Kiss. +• Spoiler effect: If the losing candidate Wright were removed from the elec- +tion, the winner changes from Kiss to Montroll. +• Upward monotonicity paradox: If 450 voters who voted only for Wright and +300 voters who ranked Wright first and Kiss second shift Kiss up to the +first ranking on their ballots, then Kiss would no longer win the election. +• Compromise voting failure: If all voters who ranked Montroll over Kiss but +did not rank Montroll first were to “compromise” and rank Montroll first +then the RCV winner would be Montroll instead of Kiss. (This failure has +not been pointed out previously.) +2009 County Executive Election in Pierce County, WA (Table 5). This +election demonstrates a compromise voting failure. In the actual election McCarthy +defeated Bunney 136346 votes to 132292 in the final round. If 15000 voters who did +not rank Goings first but ranked Goings above McCarthy were to rank Goings first +then Goings would be the RCV winner. To see why, note that in Table 5 the first- +place vote totals for Bunney, Goings, and McCarthy are 118690, 77417, and 92208, +respectively. The vote gap between Bunney and McCarthy, the two candidates +who advance to the final round, is 118690 − 92208 = 26482, while the gap between +McCarthy and Goings is 92208 − 77417 = 14791. Thus, in this round the RCV +winner McCarthy is closer to being eliminated than to having the most votes, and +if we can find a number of voters between 14792 and 26481 who rank Bunney first +and rank Goings over McCarthy (and the data does contain these voters) then +we can shift Goings up to first on these ballots so that McCarthy is eliminated. +If the gap between McCarthy and Bunney were smaller than the gap between +McCarthy and Goings then we could not make this failure occur, as McCarthy + +AN EXAMINATION OF RANKED CHOICE VOTING IN THE UNITED STATES, 2004-2022 10 +Num.Voters +27661 +27375 +63654 +13602 +44138 +19687 +12330 +59502 +20376 +1st choice +Bunney +Bunney +Bunney +Goings +Goings +Goings +McCarthy +McCarthy +McCarthy +2nd choice +Goings +McCarthy +− +Bunney +McCarthy +− +Bunney +Goings +− +3rd choice +McCarthy +Goings +− +McCarthy +Bunney +− +Goings +Bunney +− +Table 5. The 2008 County Executive election in Pierce County, +WA, after eliminating all but the final three candidates. +would advance to the final round no matter how we construct compromise votes. +Thus, the compromise voting failure in this election is “non-monotonic” in some +sense because Goings benefits from the extra support of 15000 voters but does not +benefit from the extra support of more than 26482. +2010 Mayoral Election in Oakland, CA. This election demonstrates a com- +promise voting failure. In the actual election Quan was the RCV winner. If 2400 +voters who did not rank Kaplan first but ranked Kaplan over Quan were to rank +Kaplan first then she would be the RCV winner. +2016 District 2 City Council Election in Berkeley, CA. This election +demonstrates a compromise voting failure. In the actual election Davila was the +RCV winner. If 130 voters who did not rank Armstrong-Temple first but ranked +Armstrong-Temple over Davila were to rank Armstrong-Temple first then she would +be the RCV winner. +2017 Ward 3 City Council Election in Minneapolis. This election demon- +strates a compromise voting failure. In the actual election Fletcher was the RCV +winner. If 370 voters who did not rank Bildsoe first but ranked Bildsoe over Fletcher +were to rank Bildsoe first then he would be the RCV winner. +2020 District 7 Board of Supervisors Election in San Francisco, CA +(Table 2). +This election demonstrates a downward monotonicity paradox (as +shown above), which has not been previously documented. +2021 Ward 2 City Council Election in Minneapolis, MN (Table 6). +This election demonstrates the following flaws (McCune & McCune, 2022b). +• Spoiler effect: Worlobah is the RCV winner of the election. If the losing +candidate Arab were removed from the election, the winner changes to +Gordon. +• Upward monotonicity paradox: If 456 of the voters who ranked Arab first +and Worlobah second shift Worlobah up one ranking, Worlobah would lose +the election. +• Downward monotonicity paradox: If 80 of the voters who ranked Arab +first and Gordon second shift Arab down one ranking, Arab would win the +election. +• Compromise voting failure: If the voters who did not rank Gordon first +but did rank Gordon above Worlobah were to rank Gordon first then Gor- +don would be the RCV winner. (This failure has not been pointed out +previously.) +This election cannot demonstrate a Condorcet failure as we have defined it be- +cause there is no Condorcet winner. All other elections in the database contain a +Condorcet winner. +August 2022 Alaska Special Election for US House (Table 1). +This +election demonstrates the following flaws, as demonstrated above (Graham-Squire +& McCune, 2022). + +AN EXAMINATION OF RANKED CHOICE VOTING IN THE UNITED STATES, 2004-2022 11 +Num.Voters +801 +1177 +822 +908 +756 +1572 +1299 +1088 +492 +1st choice +Gordon +Gordon +Gordon +Arab +Arab +Arab +Worlobah +Worlobah +Worlobah +2nd choice +Arab +Worlobah +− +Gordon +Worlobah +− +Gordon +Arab +− +3rd choice +Worlobah +Arab +− +Worlobah +Gordon +− +Arab +Gordon +− +Table 6. The 2021 Ward 2 city council election in Minneapolis, +MN, after eliminating all but the final three candidates. +• Condorcet failure. +• Spoiler effect. +• Upward monotonicity paradox. +• No-show paradox. +• Compromise voting failure. (This failure has not been pointed out previ- +ously.) +Of the six elections which demonstrate a compromise voting failure, only three +(Alaska, Burlington, and the 2021 Minneapolis election) demonstrate this failure +in the strong sense of the definition from (Green-Armytage, 2014). +Of the 182 elections in our database, 95 demonstrate a majoritarian failure. That +is, 95 elections have the property that when we run the RCV algorithm until only +two candidates remain, the winning candidate does not secure a majority of the +total votes cast in the first round. +7. Discussion +As the results of the previous section suggest, anomalies other than majoritarian +failures occur very rarely in real-world ranked-choice elections (see Table 7). Recall +that none of our elections contain majority candidates; we did not attempt to +count the number of American political ranked-choice elections which contain such +a candidate, but there are easily at least 100 such elections across all jurisdictions. +If we were to include these elections, the failure rates become significantly smaller. +Thus, non-majoritarian failures seem to be of little practical concern in real-world +elections. +We note, however, that even one failure could potentially have large +consequences. The city of Burlington, VT repealed the use of RCV after the 2009 +mayoral election, for example. Also, if a failure were observed in an election for a +very important office then RCV’s overall good performance becomes less important. +For example, the state of Maine uses RCV to allocate its Electoral College votes in +US Presidential elections; a Condorcet failure or monotonicity paradox in such an +election would likely have much more weight than when these failures occurred in +a city council election in Minneapolis. +Monotonicity paradoxes, truncation paradoxes, and compromise voting failures +are all specific cases of RCV being susceptible to strategic voting; i.e., all of these +failures show that RCV is manipulable. Our results show that such manipulability +is rarely a concern in practice; furthermore, we argue the data shows that even in +elections which are manipulable in some way, it would be very difficult for voters +to implement tactical voting successfully. It is hard to believe that voters would +vote insincerely to attempt to engineer a monotonicity or no-show paradox; these +paradoxes occur rarely enough and affect a relatively small enough number of voters +that attempting to manipulate an election in this fashion seems like an absurd +strategy. Compromise voting failures occur more frequently, but the RCV algorithm +is complicated enough that it is not clear that groups of voters could correctly + +AN EXAMINATION OF RANKED CHOICE VOTING IN THE UNITED STATES, 2004-2022 12 +Flaw +Condorcet +Spoiler +Upward +Downward +Truncation +Compromise +Majoritarian +Rate +1.1% +1.6% +1.1% +1.1% +0.5% +3.3% +52.5% +Table 7. The failure rate in our database of 182 elections for the +six non-majoritarian flaws of RCV. For the Condorcet failure rate +we use a denominator of 181 because one of the elections does not +contain a Condorcet winner. +anticipate when to cast a compromise vote. +For example, in the 2009 County +Executive Election in Pierce County, WA (Table 5), voters who ranked Bunney +first and ranked Goings over McCarthy would have to anticipate that Bunney could +not defeat McCarthy head-to-head in the final round and would have to calculate +that in the penultimate round the gap between Goings and McCarthy would be +smaller than the gap between McCarthy and Bunney. Furthermore, if more than +26482 of these voters decide to make this compromise then McCarthy would still +win because this level of compromising would cause Bunney to be eliminated and +McCarthy would advance to the final round and defeat Goings. Thus, there is a +relatively narrow range of vote compromising that would allow Goings to win the +election. It does not seem likely that voters would make these calculations. We +also note that our definition of a compromise voting failure is quite expansive, and +thus it is notable that the failure rate is so low. +The only election in which we think voters might have been able to calculate +correctly that they should cast compromise votes is the August 2022 Alaska House +election, which had unique political dynamics. The election contained only three +candidates, two Republicans (Begich and Palin) and one Democrat (Peltola), and +Palin had a national profile which made her a polarizing figure. Peltola was a mod- +erate, non-polarizing candidate, and thus voters who cast the ballot Palin > Begich +> Peltola could probably anticipate that Palin would not defeat Peltola head-to- +head. Therefore, such voters should have been able to calculate that their only +chance of electing a Republican to the House was to cast a compromise vote for +Begich, but these voters seemingly did not cast compromise votes. Furthermore, +this House election was an off-schedule special election which occurred because of +the death of the sitting congressman and this House seat was up for election again +in November 2022, just three months later. The November election contained four +candidates, the same three from the August election and a fourth candidate Chris +Bye, but Bye received less than 2% of the vote and thus the November election +was essentially just a rerun of the August election. In response to the issue-riddled +August election, supporters of Begich and Palin did not seem to meaningfully alter +their behaviour. Palin still received approximately 5000 more votes than Begich, +but because Peltola increased her support by a substantial relative amount (most +likely due to higher turnout in this general election), she was the Condorcet winner +and defeated Palin by a wider margin in the final round. Furthermore, the No- +vember election contained none of the RCV failures we discuss. Thus, even when +voters are given poll data of the highest quality, an election which occurred three +months prior, they do not seem to react in a strategic fashion (at least, the number +of voters reacting strategically seems relatively small). +In summary of non-majoritarian failures, while RCV is manipulable in theory +it does not seem to be manipulable in practice. Non-majoritarian failures occur + +AN EXAMINATION OF RANKED CHOICE VOTING IN THE UNITED STATES, 2004-2022 13 +infrequently and, when they do occur, it is not clear that these failures are “action- +able” on the part of voters. Issues such as monotonicity paradoxes are undesirable, +but they become offensive only in hindsight; we find little evidence that voters +could vote strategically to engineer such outcomes in actual elections. Our empiri- +cal results are consonant with the theoretical work of (Green-Armytage 2014) and +(Green-Armytage et al. 2016), which show that RCV is less manipulable than most +other voting methods. +Majoritarian failures occur at a much higher rate than the other failures, ac- +counting for more than half the elections in the database. These occur because a +significant portion of the electorate in these elections cast partial ballots, causing +their ballots to become exhausted before the final round. As pointed out in (Bur- +nett and Kogan, 2015), partial ballots occur for two reasons. First, voters may +voluntarily provide an incomplete ranking, choosing not to rank all candidates. +Second, some jurisdictions limit the number of candidates that voters can rank +on their ballots. For example, the city of Minneapolis, MN, allows voters to rank +only three candidates regardless of the number of candidates in the race. In such +elections, voters are often forced to cast partial ballots. +For our discussion, it is useful to distinguish between elections in which voters +can provide a complete ranking of the candidates if they so choose, and elections +in which they cannot. In an election in which the jurisdiction limits the number +of candidates ranked on ballots, we say that the election’s truncation level is the +number of candidates a voter can rank. If the number of candidates in an election +is more than the election’s truncation level plus one, we say the election is truncated +because voters cannot provide a complete ranking of the candidates. For example, +a ranked-choice election in Minneapolis with five or more candidates is truncated; +if an election contains four or fewer candidates, voters can provide a complete +ranking6. In our database 72 elections are truncated and 110 are not. +Because voters are forced to cast partial ballots in truncated elections and a +majoritarian failure is caused by a significant portion of ballots being partial, we +expect these failures to be more common in truncated than non-truncated elec- +tions. Our findings bear this out: 57 of the 72 truncated elections demonstrate a +majoritarian failure, while 38 of the 110 non-truncated elections demonstrate this +flaw. In elections with a majoritarian failure, the winner’s vote share in truncated +elections also tends to be lower (on average, 43.9%) than the winner’s vote share in +the non-truncated elections (on average, 47.7%). Table 8 shows the five elections +with the smallest winner vote share among the truncated elections. These vote +shares are all significantly lower than 40%, quite far from a majority. The 2010 San +Francisco Board of Supervisors election in the 10th district is particularly extreme, +with the winner earning less than 25% of the total vote. By contrast, Table 9 shows +the five elections with the smallest winner’s vote share among the non-truncated +elections. The smallest winner’s vote share among the non-truncated elections is +42.2%; there are 13 truncated elections in which the winner’s vote share is smaller. +6In an election with four candidates and truncation level 3, voters can provide a complete +ranking of the four candidates by providing a complete ranking of their top three; the candidate +left off the ballot is assumed to be the voter’s fourth choice. + +AN EXAMINATION OF RANKED CHOICE VOTING IN THE UNITED STATES, 2004-2022 14 +Election +Winner’s Vote Share +2010 San Francisco Board of Supervisors Dist. 10 +24.3% +2021 NYC Dem. Primary City Council Dist. 9 +35.2% +2020 Minneapolis City Council Ward 6 +36.1% +2021 NYC Dem. Primary Borough President Kings +37.3% +2008 Pierce County, WA, County Treasurer +37.5% +Table 8. The five truncated elections with the smallest winner +vote shares. +Election +Winner’s Vote Share +2021 NYC Rep. Primary Dist. 50 +42.2% +2021 Portland, ME City Council At-Large +44.5% +2019 San Francisco District Attorney +44.9% +2021 NYC Dem. Primary Dist. 32 +45.6% +2008 Pierce County, WA, County Executive +45.6% +Table 9. The five non-truncated elections with the smallest win- +ner vote shares. +The high rate of majoritarian failure in the data seems concerning. There are a +few potential solutions to address this failure rate. First, jurisdictions with trun- +cated elections could remove the limit on the number of candidates, allowing voters +to provide a complete ranking. Our results suggest this would have an effect on +the failure rate, and we are unaware of mathematical reasons for including a trun- +cation level in an election (although there may be political reasons for doing so). +Second, some have argued (Kilgour et al. 2020) that voters may cast partial ballots +due to the cognitive load of trying to rank multiple candidates. If this is the case, +jurisdictions could try to alleviate this load by finding ways to limit the number of +candidates on the ballot. For example, ranked-choice elections in Alaska contain at +most four candidates because prior to the RCV election there is a primary election +which uses plurality voting to whittle down the field of candidates. However, in +the 2022 ranked-choice elections in Alaska we still see a high rate of majoritar- +ian failure, with six out of thirteen elections demonstrating this issue (although the +smallest of the winner vote shares is 47.2%, and so these failures are not particularly +egregious). It is possible that voters will be willing to rank more candidates over +time as they become more comfortable with RCV, in which case Alaska’s strategy +of limiting the number of candidates to four could significantly lower the rate of +majoritarian failures. +Of course, depending on one’s values it is possible that majoritarian failures are +not important, and therefore the high failure rates are irrelevant. In non-truncated +elections, if voters who cast partial ballots are voting sincerely then it is possible that +there does not exist a candidate who could earn majority support in a final round of +RCV. For example, if enough voters care only about one or two candidates and are +indifferent among the rest, then no voting method will be able to find a “majority +winner,” and it is not the fault of RCV that the election produces a majoritarian +failure. However, even if one finds majoritarian failures unimportant, given the high +rate of such failures (even in non-truncated elections) it is likely advisable that RCV +advocates adjust their rhetoric around RCV. For example, (Lair 2022) states that + +AN EXAMINATION OF RANKED CHOICE VOTING IN THE UNITED STATES, 2004-2022 15 +“the majoritarian principle is an axiom of democratic government” and uses this +statement (among others) to justify the adoption of RCV. Similarly, (Lavin 2019) +states: “[W]e need majority rule in elections—not only as a principle or best practice +but as a practical assurance to legitimize outcomes and give elected officials strong +mandates to govern. By requiring winners to earn a majority of votes—if not in first +choices alone then with backup choices—RCV meets both of these critical needs.” +Our results suggest that RCV does not live up to such statements in practice. +8. Conclusion +When evaluated based on criteria important to the social choice literature, RCV +mostly performs well in practice. +In the American ranked-choice political elec- +tions in our database, RCV almost always selects the Condorcet winner and avoids +the spoiler effect, while also demonstrating practical resistance to strategic voting. +Paradoxes which feature prominently in the theoretical literature such as mono- +tonicity and no-show paradoxes seem to occur on the order of 0.5-1.1% for real- +world elections without a majority candidate, and these failure rates would decrease +considerably if we also included ranked-choice elections which do not advance to a +second round. The percentage of elections in which the winner does not receive a +majority in the final round is very high, which should give pause to RCV advocates. +Since a perfect voting method seemingly does not exist, choosing a method +involves trade-offs. The weaknesses of RCV are mostly not observed in real-world +ranked-choice data available in the US. Of course, the failure rates are not zero, and +it is reasonable to insist on a method which always chooses the Condorcet winner or +is not susceptible to monotonicity paradoxes. We have contributed to the literature +by providing a comprehensive empirical analysis of the social-choice weaknesses of +RCV in the US, but whether RCV’s benefits outweigh its costs is, in our view, still +an open question. +Acknowledgements +Thank you to Deb Otis for showing us the FairVote data repository. +References +[1] Bokat-Lindell, +S. +(2021). +“Can +Ranked-Choice +Voting +Cure +American +Politics?,” +New +York +Times, +June +24, +2021, +https://www.nytimes.com/2021/06/24/opinion/ +ranked-choice-new-york.html.AccessedJan.14,2023. +[2] Burnett C. & Kogan V. (2015). Ballot (and voter) “exhaustion” under Instant Runoff Voting: +An examination of four ranked-choice elections. Electoral Studies, 37, 41-49. +[3] Coll J. (2021). Demographic Disparities Using Ranked-Choice Voting? Ranking Difficulty, +Under-Voting, and the 2020 Democratic Primary. Politics and Governance, 9 (2), 293-305. +[4] Darmann A., Grundner J., & Klamler C. (2019). Evaluative voting or classical voting rules: +Does it make a difference? Empirical evidence for consensus among voting rules. European +Journal of Political Economy, 59, 345-353. +[5] Donovan T., Tolbert C., & Harper S. (2022). Demographic differences in understanding and +utilization of ranked choice voting. Social Science Quarterly, 103 (7), 1539-1550. +[6] Gallagher M. (2013). Monotonicity and non-monotonicity at PR-STV elections. Paper pre- +sented at the annual conference of the elections, public opinion and parties (EPOP) specialist +group. +[7] Gierzynski A., Hamilton W., & Smith W. Burlington Vermont 2009 IRV mayor election. +https://rangevoting.org/Burlington.html. Accessed 12/20/22. +[8] Graham-Squire A. & McCune D. (2022). A Mathematical Analysis of the 2022 Alaska Special +Election for US House. Preprint: https://arxiv.org/abs/2209.04764. + +AN EXAMINATION OF RANKED CHOICE VOTING IN THE UNITED STATES, 2004-2022 16 +[9] Graham-Squire A. & Zayatz N. (2021). Lack of Monotonicity Anomalies in Empirical Data +of Instant-runoff Elections. Representation, 57 (4), 565-573. +[10] Green-Armytage J. (2014). Strategic voting and nomination. Social Choice and Welfare, 42, +111-138. +[11] Green-Armytage J., Tideman T.N., & Cosman R. (2016). Statistical evaluation of voting +rules. Social Choice and Welfare, 46, 183-212. +[12] Kilgour D.M., Gr´egoire J.C., & A.M. Foley. (2020). The prevalence and consequences of ballot +truncation in ranked-choice elections. Public Choice 184: 197-218. +[13] Lair +S. +(2022). +Ranked-choice +voting +is +needed, +along +with +open +primaries. +https://thenevadaindependent.com/article/ +ranked-choice-voting-is-needed-along-with-open-primaries/. Accessed 12/26/22. +[14] Landsman +T. +(2017). +All +RCV +Elections +in +the +Bay +Area +So +Far +Have +Pro- +duced Condorcet Winners. https://fairvote.org/every_rcv_election_in_the_bay_area_ +so_far_has_produced_condorcet_winners/. Accessed 12/26/22. +[15] Lavin N. (2019). Majority Rule: more than just a principle for successful elections. https:// +fairvote.org/majority_rule_more_than_just_a_principle_for_successful_elections/. +Accessed 12/26/22. +[16] McCune D. and Graham-Squire A. (2022). Monotonicity Anomalies in Scottish Local Gov- +ernment Elections. Preprint. +[17] McCune D. & McCune L. (2022). Does the Choice of Preferential Voting Method Matter? +An Empirical Study Using Ranked Choice Elections in the United States. Representation. +https://doi.org/10.1080/00344893.2022.2133003 +[18] McCune D. & McCune L. (2022). The Curious Case of the 2021 Minneapolis Ward 2 City +Council Election, to appear in The College Mathematics Journal. +[19] Miller N.R. (2017). Closeness matters: Monotonicity failure in IRV elections with three can- +didates. Public Choice 173 (1-2): 91-108. +[20] Otis D. RCV in New York City. (2021, October 14). Retrieved from https://www.fairvote. +org/rcv_in_new_york_city#candidate_analysis. +[21] Otis D. (2022). Single winner ranked choice voting CVRs. https://doi.org10.7910/DVN/ +AMK8PJ, Harvard Dataverse, V5. +[22] Ornstein J. & Norman R. (2014). Frequency of monotonicity failure under Instant Runoff +Voting: estimates based on a spatial model of elections. Public Choice, 161 (1-2), 1-9. +[23] Popov S., Popova A., & Regenwetter M. (2014). Consensus in Organizations: Hunting for +the Social Choice Conundrum in APA Elections. Decision 1 (2), 123-146. +[24] Regenwetter M., Kim A., & Ho M. (2007). The Unexpected Empirical Consensus Among +Consensus Methods. Psychological Science 18 (7), 629-635. +[25] Song C.G. (2022). Three Empirical Analyses of Voting [Unpublished doctoral dissertation]. +Virginia Tech. +[26] Tomlinson K., Ugander J., & Kleinberg J. Ballot Length in Instant Runoff Voting. +Preprint:https://arxiv.org/abs/2207.08958. +Adam Graham-Squire, Department of Mathematical Sciences, High Point University, +1 University Parkway, High Point, NC, 27268 +Email address: agrahams@highpoint.edu +David McCune, Department of Physics and Mathematics, William Jewell College, +500 College Hill, Liberty, MO, 64068-1896 +Email address: mccuned@william.jewell.edu + diff --git a/iNFLT4oBgHgl3EQfai-M/content/tmp_files/load_file.txt b/iNFLT4oBgHgl3EQfai-M/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c9dc869222920193f51f9b954d32f612d2b128f1 --- /dev/null +++ b/iNFLT4oBgHgl3EQfai-M/content/tmp_files/load_file.txt @@ -0,0 +1,585 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf,len=584 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='12075v1 [econ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='GN] 28 Jan 2023 AN EXAMINATION OF RANKED CHOICE VOTING IN THE UNITED STATES, 2004-2022 ADAM GRAHAM-SQUIRE AND DAVID MCCUNE Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' From the perspective of social choice theory, ranked-choice vot- ing (RCV) is known to have many flaws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' RCV can fail to elect a Condorcet winner and is susceptible to monotonicity paradoxes and the spoiler effect, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' We use a database of 182 American ranked-choice elections for polit- ical office from the years 2004-2022 to investigate empirically how frequently RCV’s deficiencies manifest in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Our general finding is that RCV’s weaknesses are rarely observed in real-world elections, with the exception that ballot exhaustion frequently causes majoritarian failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Introduction The use of ranked-choice voting (RCV) has greatly increased in the United States during the last few years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' New York City first used RCV for city primary elections in 2021, the state of Maine has used it for state primary elections and elections for federal office since 2018, and the state of Alaska implemented RCV for federal and state offices in 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' The voting method is well-known to have many deficiencies which receive attention in the social choice literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' The deficiencies with which we are concerned are: RCV can fail to elect the Condorcet winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' RCV is susceptible to the spoiler effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' RCV is susceptible to downward and upward monotonicity paradoxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' RCV is susceptible to the truncation paradox, the most extreme version of which is the no-show paradox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' RCV is susceptible to compromise strategic voting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' RCV is not truly “majoritarian” because of ballot exhaustion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' The purpose of this article is to examine how often these issues occur in actual elections, where we focus on the single-winner case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' To that end, we collected the ballot data for as many single-winner ranked-choice American political elections as we could, resulting in a database of 182 elections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' The flaws of RCV listed above can manifest only in elections without a majority candidate (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=', elections in which RCV goes to at least a second round), and thus our database consists only of such elections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Our general finding is that these flaws occur rarely in actual ranked-choice elections, except for majoritarian failures caused by ballot exhaustion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' While much of our analysis is new, perhaps the primary value of this article is that we provide a complete analysis of “problematic” American RCV elections, summarizing much of the empirical research of the last 18 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Primary 91B10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Secondary 91B14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Condorcet winner, monotonicity paradox, spoiler effect, ballot ex- haustion, majoritarian, empirical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' 1 AN EXAMINATION OF RANKED CHOICE VOTING IN THE UNITED STATES, 2004-2022 2 Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Voters 27070 15478 11262 34078 3659 21237 47407 4647 23733 1st choice Begich Begich Begich Palin Palin Palin Peltola Peltola Peltola 2nd choice Palin Peltola − Begich Peltola − Begich Palin − 3rd choice Peltola Palin − Peltola Begich − Palin Begich − Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' The August 2022 Alaska Special Election for US House, after eliminating write-in candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' We note RCV has other potential flaws which are best examined through the lens of political science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' For example, some of RCV’s detractors say that voters find the method confusing, or that RCV does not deliver on its proponents’ promise of creating more civil political campaigns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' We do not address these claims;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' because of our backgrounds, we evaluate RCV only on criteria which have been examined in the mathematically-oriented social choice literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Definitions: Ranked-Choice Voting and Its Deficiencies We begin with a definition of RCV and then formally define the flaws with which we are concerned, with examples throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Note that the social choice literature generally uses the term RCV to refer to any election that involves ranking candidates, and the literature uses terms such as instant runoff voting, alternative vote, the Hare rule, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=', to describe the vote method defined below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' We use the term RCV, as it aligns with how the term used by most municipal and state elections offices in the US.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' In an RCV election, voters cast preference ballots which allow a voter to rank the candidates in order of preference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Voters often do not provide a complete ranking, either by choice or because the voters’ jurisdiction limits the number of candidates that can be ranked on a ballot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' For example, the city of Minneapolis, MN allows voters to rank only three candidates, regardless of how many candidates are in the race.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' After an election the ballots are aggregated into a preference profile, which shows the number of each type of ballot cast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' For example, Table 1 shows a preference profile for the August 2022 Special Election for the single US House seat in Alaska, an election involving the three candidates Nick Begich, Sarah Palin, and Mary Peltola1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' The table shows that 27070 voters ranked Begich first, Palin second, and Peltola third;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' the other numbers across the top row convey similar information about the number of voters who cast the corresponding type of ballot2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' We use the notation A > B to denote that a voter ranks candidate A above B, and thus 27070 voters choose the ranking Begich > Palin > Peltola.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' To keep the table a manageable size, we combine ballots of the form A > B and A > B > C, which has no effect on the winner of the election under RCV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' We also combine ballots of these forms in all subsequent examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Note that voters are not required to provide a complete ranking of the candidates, and some voters choose to rank only a single candidate on their ballots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' For example, 11262 voters ranked Begich on their ballots but did not rank Palin or Peltola.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' 1The race also included several write-in candidates, all of whom received a relatively trivial amount of votes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' We eliminated these candidates before creating the preference profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' 2There is some ambiguity about how the ballots were processed, and our preference profile numbers differ by a handful of ballots from the official vote counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' For example, according to Alaska Division of Elections the 27070 should be 27053.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' These small differences do not affect our conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' AN EXAMINATION OF RANKED CHOICE VOTING IN THE UNITED STATES, 2004-2022 3 The method of RCV declares a winner as follows: if the election contains a majority candidate, a candidate who receives a majority of first-place votes, then that candidate is declared the winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Otherwise eliminate the candidate(s) with the fewest first-place votes and transfer that candidate’s votes to other candidates based on the second-place choices on that eliminated candidate’s ballots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' If a surviving candidate now has a majority of the remaining votes, that candidate wins;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' otherwise, the process of elimination and vote-transfer continues in this manner until a candidate has secured a majority of (first-place and transferred) votes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' We illustrate the RCV algorithm using the preference profile in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Begich receives 53810 first-place votes, Palin receives 58974, and Peltola receives 75799, and thus Begich is eliminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' As a result, 27070 of his votes are transferred to Palin, 15478 are transferred to Peltola, and 11262 are dropped from the election (these ballots are said to be exhausted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' After the vote transfer, Peltola wins the election with 91277 votes to Palin’s 86044.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' If an election contains more than three candidates then RCV proceeds similarly, but the process may take more rounds to select a winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' We now define the flaws of RCV that we investigate in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Condorcet Failure: In the Alaska House election, if we compare Begich to Palin in a head-to-head matchup then 101229 voters prefer Begich to Palin whereas 63621 voters prefer Palin to Begich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Similarly, 93052 voters prefer Begich to Peltola whereas 79558 voters prefer Peltola to Begich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Based on the ranking information provided by the voters3, Begich wins each of his head-to-head matchups and thus is the Condorcet winner of the election.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Condorcet winners receive much attention in the social choice literature because they are considered “consensus” or “strong” candidates, candidates who “should” win (assuming the election contains such a candidate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' We say that an RCV election demonstrates a Condorcet failure when the election contains a Condorcet winner and RCV fails to select this candidate, which occurs in the Alaska House election.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Spoiler Effect: An RCV election demonstrates the spoiler effect if there exists a subset of losing candidates such that removing these candidates from the election causes the winner of the election to change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' In the Alaska House election, if the losing candidate Palin were removed then Begich would win, and thus this election demonstrates a spoiler effect under RCV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Upward Monotonicity Paradox: An RCV election demonstrates an upward monotonicity paradox if there exists a set of ballots such that shifting the RCV winner up the rankings on those ballots but keeping the relative rankings of the other candidates the same, thereby creating a hypothetical second preference profile in which the winner has more voter support, creates an election in which the original RCV winner does not win.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' In the Alaska House election, if 6000 ballots on which Palin is ranked first and no other candidate is ranked on the ballot were changed to ballots of the form Peltola > Palin then Peltola would lose the resulting election.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' That is, giving the winning candidate Peltola more first place votes by shifting her up the rankings on some ballots turns her into a loser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' The reason is that even though Peltola picks up an additional 6000 first-place votes in the hypothetical second election, this additional support causes Palin to be eliminated from the 3We choose to interpret ballots using the weak order model (Popov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=', 2014), in which all candidates left off a ballot are tied for the lowest ranking on a voter’s ballot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' AN EXAMINATION OF RANKED CHOICE VOTING IN THE UNITED STATES, 2004-2022 4 Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='Voters 5237 3316 5566 4050 5708 1894 2251 6909 1695 1st choice Engardio Engardio Engardio Melgar Melgar Melgar Nguyen Nguyen Nguyen 2nd choice Melgar Nguyen − Engardio Nguyen − Engardio Melgar − 3rd choice Nguyen Melgar − Nguyen Engardio − Melgar Engardio − Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' The 2020 District 7 Board of Supervisors election in San Francisco, CA, after eliminating all but the final three candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' election first (in contrast to the original election, in which Begich was eliminated first) and then Peltola would lose to Begich in the final round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Downward Monotonicity Paradox: An RCV election demonstrates a down- ward monotonicity paradox if there exists a set of ballots and a losing candidate L such that shifting L down the rankings on those ballots but keeping the relative rankings of the other candidates the same, thereby creating a hypothetical second preference profile in which this losing candidate has less voter support, creates an election in which L is the RCV winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Our running example with the Alaska data does not demonstrate a downward paradox: if either Begich or Palin were shifted down on any set of ballots, they would remain losers of the election.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' To demonstrate a real-world example of this paradox, consider the 2020 Board of Supervisors election for the 7th District in San Francisco, CA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' The election contained seven (not write-in) candidates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Table 2 shows the preference profile for the election after four candidates have been eliminated and their votes transferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' At this point in the RCV process, Engardio has 14119 first-place votes, Melgar has 11652, and Nguyen has 10855.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Nguyen is eliminated, and after their votes are transferred Melgar wins with 18561 votes to Engardio’s 16370.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' If Engardio were shifted down one ranking on 800 ballots of the form Engardio > Nguyen > Melgar so that those ballots change to Nguyen > Engardio > Melgar then in the resulting election Melgar would be eliminated first and, even with less voter support, Engardio would still have enough votes to defeat Nguyen head-to-head in the final round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' That is, shifting the losing candidate Engardio down on some voters’ ballots would turn Engardio into the RCV winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' As with the Alaska election, this example creates a paradoxical outcome by changing the order in which candidates are eliminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Truncation Paradox: An election demonstrates a truncation paradox if there exists a set of voters such that the voters could create a more desirable electoral outcome by ranking fewer candidates on their ballots;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=', when these voters’ ex- press less information on their ballots then a candidate whom they prefer more wins the election than if the voters expressed more information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' The most extreme version of this paradox is a no-show paradox, which occurs when there exists a set of voters such that removing their ballots from the election creates a more desirable electoral outcome for those voters (removing the ballots altogether is the most extreme form of ballot truncation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' To be precise, a no-show paradox occurs if A is the RCV winner and there exists a losing candidate B and a set of voters that prefer B to A such that if these voters abstain from voting then B would be the winner in the modified election with these voters’ ballots removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' The Alaska election in Table 1 demonstrates this paradox: if 5400 voters who cast the ballot Palin > Begich > Peltola were removed from the election then Palin would be elim- inated first (recall that in the original election, Begich was eliminated first) and, despite the removal of these ballots, Begich would have enough support to defeat AN EXAMINATION OF RANKED CHOICE VOTING IN THE UNITED STATES, 2004-2022 5 Peltola head-to-head in the final round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' These 5400 voters would have achieved a more desirable outcome (their second-favorite candidate would have won instead of their third favorite) if they had abstained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Compromise Voting Failure: Compromise voting is a form of strategic voting in which some voters calculate that their favorite candidate cannot win and so insincerely rank another candidate as their first choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' This plays out regularly in plurality elections, where voters often cast a vote for a candidate perceived as viable instead of voting for their favorite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Generalizing the definition from (Green- Armytage, 2014), we say that an election demonstrates a compromise voting failure if there exists a losing candidate A and a set of ballots such that A is ranked above the original RCV winner, and A becomes the RCV winner if we shift them up to the first ranking on these ballots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' That is, this failure occurs if there exists a set of voters who should have cast a “compromise vote” for A, thereby causing A to win.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' The definition in (Green-Armytage, 2014) is much more restrictive: they consider only elections in which shifting A to the top of all ballots on which A is ranked over the original RCV winner turns A into the winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' The Alaska election in Table 1 demonstrates this failure: if 5400 voters who ranked Palin first and Begich second had ranked Begich first instead then Begich would have won the election and these voters would have obtained their second choice instead of their third.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' To obtain a more desirable electoral outcome, these voters should have “compromised” for Begich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' This election also fits the more restrictive definition from (Green-Armytage, 2014), as Begich would become the winner if all 34078 voters who cast the ballot Palin > Begich > Peltola were to rank Begich first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Majoritarian Failure: We say that an election demonstrates a majoritarian failure if, when the RCV algorithm is run until there are only two candidates left, the winning candidate in this final round does not achieve a majority of the total number of votes cast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Both the Alaska and San Francisco elections demonstrate this failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Table 1 shows a total of 188583 voters in that election (which increases slightly if we were to include write-in candidates), yet the winner Peltola earns 91277 in the final round, achieving only 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='4% of the votes of the total electorate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Similarly, in the San Francisco election there were 39322 ballots cast and the winner Melgar earns 18561/39322 = 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='2% of the total vote in the final round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' The reason these winners fail to earn a majority is ballot exhaustion (Burnett and Kogan, 2015), where many partial ballots are discarded before the final round because these ballots do not rank either of the two final candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' We use the term winner’s vote share to refer to the percentage of the total votes earned by the RCV winner in the final round when there are only two candidates remaining, so that Melgar’s vote share is 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='2%, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Data Sources and Collection We collected the vote data for as many American single-winner ranked-choice political elections as we could, with the restriction that we obtained data only for elections in which no candidate earns an initial majority of first-place votes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' In total our database contains 182 elections, 147 of which were collected by the first author for (McCune & McCune, 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Most of the elections are for municipal office such as mayor or city councilor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' a handful of elections are for statewide or federal elections, such as elections for US House and Senate in Maine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' The data AN EXAMINATION OF RANKED CHOICE VOTING IN THE UNITED STATES, 2004-2022 6 Jurisdiction Years for which we have data Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Elections Alaska 2022 13 Aspen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' CO 2009 2 Berkeley,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' CA 2010,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' 2014-2020 6 Bloomington,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' MN 2021 1 Burlington,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' VT 2006,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' 2009 2 Corvallis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' OR 2022 2 Easthampton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' MA 2020 1 Eastpointe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' MI 2020 1 Elk Ridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' UT 2021 1 Las Cruces,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' NM 2019 3 Maine 2018-2022 11 Minneapolis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' MN 2009,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' 2013,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' 2017,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' 2021 26 Minnetonka,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' MN 2021 1 New York City,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' NY 2021 41 Oakland,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' CA 2010-2020 21 Pierce County,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' WA 2008-2009 4 Portland,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' ME 2021-2022 2 San Francisco,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' CA 2004-2022 32 San Leandro,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' CA 2010-2014 5 Santa Fe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' NM 2018 2 Springville,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' UT 2021 1 St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Louis Park, MN 2019, 2021 2 Telluride, CO 2015 1 Woodland Hills, UT 2021 1 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Summary of data sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' was collected from election office websites, and some was received by request from election offices when the data was not posted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' After our initial round of data collection, the RCV advocacy organization FairVote created a publicly accessible repository of American ranked-choice data (Otis, 2022) and we collected a handful of additional elections from this source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Table 3 gives a summary of our database, including the number of elections from each jurisdiction and the years for which data is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Prior Social Choice Literature on RCV’s Flaws There is a vast social choice literature which evaluates RCV as a voting method, and most of this literature is theoretical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Due to the empirical nature of our work we do not attempt to survey the theoretical literature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' instead, we focus on prior empirical work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Condorcet failures: The largest empirical investigation of Condorcet failures in American political elections occurs in (McCune & McCune, 2022a), which analyzed 147 of the elections in our database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' The authors found only one Condorcet failure, in a previously documented election from Burlington, VT (Gierzynski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Ornstein & Norman 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' (McCune & McCune, 2022a) also found only one single- winner election without a Condorcet winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' (Graham-Squire & Zayatz, 2021) analyze 35 elections from our database, again finding a failure in only the Burlington AN EXAMINATION OF RANKED CHOICE VOTING IN THE UNITED STATES, 2004-2022 7 election, and (Song, 2022) analyzed many4 of the pre-2021 elections in our database for Condorcet failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' RCV advocacy groups such as FairVote also check ranked- choice data for Condorcet failures (Landsman 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Otis 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' There is a small literature about Condorcet failures which investigates elections outside the context of American political elections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' For example, (Regenwetter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Popov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' 2014) analyse ranked-choice election data from the American Psychological Association, using the data to generate tens of thousands of election pseudoprofiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' They find an extremely low rate of Condorcet failures under RCV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' (Darmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' 2019) use survey data collected prior to the 2015 parliamentary elections in the Austrian federal state of Styria, and again find little evidence of Condorcet failures in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Spoiler Effect: Most of the discussion of spoilers has been limited to the theoreti- cal literature, with the exception that many non-academic articles discuss potential spoilers in individual elections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' For example, many newspaper articles (for a typ- ical example example, see (Bokat-Lindell, 2021)) mention the famous case of the 2000 US Presidential election in which Ralph Nader is commonly understood to have spoiled the election for Al Gore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' (McCune & Wilson, 2022) is the largest empirical study of the spoiler effect in American ranked-choice political elections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' The authors find two ranked-choice elections demonstrating the spoiler effect out of 170 analyzed, 147 of which are in our database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' They also perform bootstrap analysis to generate pseudoprofiles, similar to the analysis in (Popov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' 2014), and find low rates of the spoiler effect in the generated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' A Condorcet failure is a special case of the spoiler effect as we have defined it, and thus the Condorcet studies mentioned above also indirectly address the spoiler effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Monotonicity and truncation paradoxes: Most of the elections in our database have not been previously processed by code which searches for monotonicity or truncation paradoxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' The largest empirical study of monotonicity and truncation paradoxes in American political ranked-choice elections occurs in (Graham-Squire & Zayatz, 2021), which analyses elections from Alameda County and San Francisco, CA, as well as a mayoral election from Burlington, VT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' The authors find a single upward monotonicity paradox and no truncation paradoxes out of 35 elections without a majority candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' (McCune & Graham-Squire, 2022) analyze 1079 ranked-choice multiwinner elections from Scotland and find low (but non-zero) rates of upward monotonicity, downward monotonicity, and no-show paradoxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Many articles which analyze real-world monotonicity failures focus on single elections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' See (Gierzynski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Ornstein & Norman 2014) for an analysis of the 2009 Burlington, VT mayoral election and (McCune & McCune, 2022b) for an analysis of monotonicity failures in a 2021 city council election in Minneapolis, MN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Some prior work is semi-empirical in that the authors use polling or survey data to estimate monotonicity failure rates in real-world elections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' For examples of this kind of analysis see (Gallagher, 2013), which studies single-transferable vote elections in Ireland, and (Miller, 2017), which provides a semi-empirical analysis of English general elections from 1992-2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Compromise voting failures: We are unaware of empirical studies which study this failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' (Green-Armytage, 2014) and (Green-Armytage et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' 2016) analyze compromise voting using a variety of models of voter behavior, and generally find 4We are not sure how many of the elections analysed in (Song, 2022) contain majority candi- dates and so we cannot determine how much of their work overlaps with ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' AN EXAMINATION OF RANKED CHOICE VOTING IN THE UNITED STATES, 2004-2022 8 that RCV is much less susceptible to this issue than other famous voting methods such as plurality and the Borda count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Majoritarian failures: The only empirical studies of majoritarian failures of which we are aware are (Burnett and Kogan, 2015), which analyze four American elections in our database, and (Kilgour et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' 2020), which analyze 18 ranked-choice elections, 4 of which are in our database (the other 14 contain majority candidates or are elections from the UK or the American Psychological Association).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' FairVote also seems to have done substantial analysis of ballot exhaustion5, but it is unclear which elections were analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' More generally, many studies address the topic of partial ballots without focusing on majoritarian failures per se.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' (Coll, 2021) and (Donovan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' 2022) study which demographic groups are more likely to cast partial ballots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' (Tomlinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=', 2022) analyse the effects of ballot truncation on the number of possible winners in ranked- choice elections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Methodology for Detecting RCV Weaknesses in Each Election All elections in the database were processed using Python code which searched for the given flaw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' The code is adapted from programs used in (Graham-Squire & Zayatz 2021), (McCune & McCune 2022a), (McCune & Wilson 2022), and (McCune & Graham-Squire 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Checking if the RCV winner is the Condorcet winner or if the RCV winner earns a majority of the initial total votes in the final round is computationally straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' However, there are challenges when searching for the spoiler effect, monotonicity paradoxes, truncation paradoxes, or compromise voting failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' If the number of candidates in an election is large enough then, due to limits of computation time, we cannot check every subset of losing candidates for a change in the winner when this subset is dropped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' For example, the 2013 Minneapolis mayoral election contained 35 candidates, resulting in billions of possible sets of candidates to check for the spoiler effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' For all elections in the database we checked for individual spoiler candidates, but for elections with large numbers of candidates we additionally checked only candidate subsets of size two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' For elections with more than twelve candidates, we also ran the RCV algorithm until only ten candidates remained and then checked the resulting election for the spoiler effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Most of the elections were already processed in this manner in (McCune & Wilson, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' To demonstrate a monotonicity or truncation paradox, we must find a set of ballots such that shifting a candidate up or down the rankings, or truncating the ballots, causes a change in the order of elimination which causes a paradoxical change in the winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Except for the case of three-candidate elections in which every voter provides a complete ranking, there are no known necessary and suf- ficient conditions for an election to exhibit one of these paradoxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Thus, if an election exhibits such a paradox, we cannot guarantee our code will find it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' How- ever, our monotonicity and truncation code has been thoroughly means-tested in other projects and has successfully found many elections which demonstrate mono- tonicity paradoxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' For example, the code found 21 elections demonstrating upward monotonicity paradoxes in Scottish local government elections which had not been previously documented (McCune & Graham-Squire, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' The code essentially 5See https://fairvote.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='org/resources/data-on-rcv/#evaluating-rcv-election-outcomesnbsp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' AN EXAMINATION OF RANKED CHOICE VOTING IN THE UNITED STATES, 2004-2022 9 Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='Voters 2043 371 568 1332 767 455 495 1513 1289 1st choice Kiss Kiss Kiss Montroll Montroll Montroll Wright Wright Wright 2nd choice Montroll Wright − Kiss Wright − Kiss Montroll − 3rd choice Wright Montroll − Wright Kiss − Montroll Kiss − Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' The 2009 mayoral election in Burlington, VT, after elim- inating all but the final three candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' This table is taken from (Ornstein & Norman, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' works by strategically changing or truncating ballots to achieve a change in the or- der of elimination or the candidates, and measures if a paradoxical outcome occurs as a result of the ballot changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' The code we use to search for compromise voting failures is a straightforward adaptation of our monotonicity code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Results We now present our results, separating the issue of ballot exhaustion from the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Only eight elections in the database, listed below, exhibit any kind of non- majoritarian flaw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Four of the elections demonstrate only a compromise voting failure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' we include the preference profile for only one of these elections as the dynamics of the other three are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' 2009 Mayoral Election in Burlington, VT (Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' This election demon- strates the following flaws (Gierzynski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Ornstein & Norman 2014): Condorcet failure: the Condorcet winner, Montroll, is not the RCV winner, Kiss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Spoiler effect: If the losing candidate Wright were removed from the elec- tion, the winner changes from Kiss to Montroll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Upward monotonicity paradox: If 450 voters who voted only for Wright and 300 voters who ranked Wright first and Kiss second shift Kiss up to the first ranking on their ballots, then Kiss would no longer win the election.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Compromise voting failure: If all voters who ranked Montroll over Kiss but did not rank Montroll first were to “compromise” and rank Montroll first then the RCV winner would be Montroll instead of Kiss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' (This failure has not been pointed out previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=') 2009 County Executive Election in Pierce County, WA (Table 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' This election demonstrates a compromise voting failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' In the actual election McCarthy defeated Bunney 136346 votes to 132292 in the final round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' If 15000 voters who did not rank Goings first but ranked Goings above McCarthy were to rank Goings first then Goings would be the RCV winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' To see why, note that in Table 5 the first- place vote totals for Bunney, Goings, and McCarthy are 118690, 77417, and 92208, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' The vote gap between Bunney and McCarthy, the two candidates who advance to the final round, is 118690 − 92208 = 26482, while the gap between McCarthy and Goings is 92208 − 77417 = 14791.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Thus, in this round the RCV winner McCarthy is closer to being eliminated than to having the most votes, and if we can find a number of voters between 14792 and 26481 who rank Bunney first and rank Goings over McCarthy (and the data does contain these voters) then we can shift Goings up to first on these ballots so that McCarthy is eliminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' If the gap between McCarthy and Bunney were smaller than the gap between McCarthy and Goings then we could not make this failure occur, as McCarthy AN EXAMINATION OF RANKED CHOICE VOTING IN THE UNITED STATES, 2004-2022 10 Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='Voters 27661 27375 63654 13602 44138 19687 12330 59502 20376 1st choice Bunney Bunney Bunney Goings Goings Goings McCarthy McCarthy McCarthy 2nd choice Goings McCarthy − Bunney McCarthy − Bunney Goings − 3rd choice McCarthy Goings − McCarthy Bunney − Goings Bunney − Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' The 2008 County Executive election in Pierce County, WA, after eliminating all but the final three candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' would advance to the final round no matter how we construct compromise votes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Thus, the compromise voting failure in this election is “non-monotonic” in some sense because Goings benefits from the extra support of 15000 voters but does not benefit from the extra support of more than 26482.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' 2010 Mayoral Election in Oakland, CA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' This election demonstrates a com- promise voting failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' In the actual election Quan was the RCV winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' If 2400 voters who did not rank Kaplan first but ranked Kaplan over Quan were to rank Kaplan first then she would be the RCV winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' 2016 District 2 City Council Election in Berkeley, CA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' This election demonstrates a compromise voting failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' In the actual election Davila was the RCV winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' If 130 voters who did not rank Armstrong-Temple first but ranked Armstrong-Temple over Davila were to rank Armstrong-Temple first then she would be the RCV winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' 2017 Ward 3 City Council Election in Minneapolis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' This election demon- strates a compromise voting failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' In the actual election Fletcher was the RCV winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' If 370 voters who did not rank Bildsoe first but ranked Bildsoe over Fletcher were to rank Bildsoe first then he would be the RCV winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' 2020 District 7 Board of Supervisors Election in San Francisco, CA (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' This election demonstrates a downward monotonicity paradox (as shown above), which has not been previously documented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' 2021 Ward 2 City Council Election in Minneapolis, MN (Table 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' This election demonstrates the following flaws (McCune & McCune, 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Spoiler effect: Worlobah is the RCV winner of the election.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' If the losing candidate Arab were removed from the election, the winner changes to Gordon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Upward monotonicity paradox: If 456 of the voters who ranked Arab first and Worlobah second shift Worlobah up one ranking, Worlobah would lose the election.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Downward monotonicity paradox: If 80 of the voters who ranked Arab first and Gordon second shift Arab down one ranking, Arab would win the election.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Compromise voting failure: If the voters who did not rank Gordon first but did rank Gordon above Worlobah were to rank Gordon first then Gor- don would be the RCV winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' (This failure has not been pointed out previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=') This election cannot demonstrate a Condorcet failure as we have defined it be- cause there is no Condorcet winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' All other elections in the database contain a Condorcet winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' August 2022 Alaska Special Election for US House (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' This election demonstrates the following flaws, as demonstrated above (Graham-Squire & McCune, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' AN EXAMINATION OF RANKED CHOICE VOTING IN THE UNITED STATES, 2004-2022 11 Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='Voters 801 1177 822 908 756 1572 1299 1088 492 1st choice Gordon Gordon Gordon Arab Arab Arab Worlobah Worlobah Worlobah 2nd choice Arab Worlobah − Gordon Worlobah − Gordon Arab − 3rd choice Worlobah Arab − Worlobah Gordon − Arab Gordon − Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' The 2021 Ward 2 city council election in Minneapolis, MN, after eliminating all but the final three candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Condorcet failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Spoiler effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Upward monotonicity paradox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' No-show paradox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Compromise voting failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' (This failure has not been pointed out previ- ously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=') Of the six elections which demonstrate a compromise voting failure, only three (Alaska, Burlington, and the 2021 Minneapolis election) demonstrate this failure in the strong sense of the definition from (Green-Armytage, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Of the 182 elections in our database, 95 demonstrate a majoritarian failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' That is, 95 elections have the property that when we run the RCV algorithm until only two candidates remain, the winning candidate does not secure a majority of the total votes cast in the first round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Discussion As the results of the previous section suggest, anomalies other than majoritarian failures occur very rarely in real-world ranked-choice elections (see Table 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Recall that none of our elections contain majority candidates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' we did not attempt to count the number of American political ranked-choice elections which contain such a candidate, but there are easily at least 100 such elections across all jurisdictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' If we were to include these elections, the failure rates become significantly smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Thus, non-majoritarian failures seem to be of little practical concern in real-world elections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' We note, however, that even one failure could potentially have large consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' The city of Burlington, VT repealed the use of RCV after the 2009 mayoral election, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Also, if a failure were observed in an election for a very important office then RCV’s overall good performance becomes less important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' For example, the state of Maine uses RCV to allocate its Electoral College votes in US Presidential elections;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' a Condorcet failure or monotonicity paradox in such an election would likely have much more weight than when these failures occurred in a city council election in Minneapolis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Monotonicity paradoxes, truncation paradoxes, and compromise voting failures are all specific cases of RCV being susceptible to strategic voting;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=', all of these failures show that RCV is manipulable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Our results show that such manipulability is rarely a concern in practice;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' furthermore, we argue the data shows that even in elections which are manipulable in some way, it would be very difficult for voters to implement tactical voting successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' It is hard to believe that voters would vote insincerely to attempt to engineer a monotonicity or no-show paradox;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' these paradoxes occur rarely enough and affect a relatively small enough number of voters that attempting to manipulate an election in this fashion seems like an absurd strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Compromise voting failures occur more frequently, but the RCV algorithm is complicated enough that it is not clear that groups of voters could correctly AN EXAMINATION OF RANKED CHOICE VOTING IN THE UNITED STATES, 2004-2022 12 Flaw Condorcet Spoiler Upward Downward Truncation Compromise Majoritarian Rate 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='1% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='6% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='1% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='1% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='5% 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='3% 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='5% Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' The failure rate in our database of 182 elections for the six non-majoritarian flaws of RCV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' For the Condorcet failure rate we use a denominator of 181 because one of the elections does not contain a Condorcet winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' anticipate when to cast a compromise vote.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' For example, in the 2009 County Executive Election in Pierce County, WA (Table 5), voters who ranked Bunney first and ranked Goings over McCarthy would have to anticipate that Bunney could not defeat McCarthy head-to-head in the final round and would have to calculate that in the penultimate round the gap between Goings and McCarthy would be smaller than the gap between McCarthy and Bunney.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Furthermore, if more than 26482 of these voters decide to make this compromise then McCarthy would still win because this level of compromising would cause Bunney to be eliminated and McCarthy would advance to the final round and defeat Goings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Thus, there is a relatively narrow range of vote compromising that would allow Goings to win the election.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' It does not seem likely that voters would make these calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' We also note that our definition of a compromise voting failure is quite expansive, and thus it is notable that the failure rate is so low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' The only election in which we think voters might have been able to calculate correctly that they should cast compromise votes is the August 2022 Alaska House election, which had unique political dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' The election contained only three candidates, two Republicans (Begich and Palin) and one Democrat (Peltola), and Palin had a national profile which made her a polarizing figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Peltola was a mod- erate, non-polarizing candidate, and thus voters who cast the ballot Palin > Begich > Peltola could probably anticipate that Palin would not defeat Peltola head-to- head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Therefore, such voters should have been able to calculate that their only chance of electing a Republican to the House was to cast a compromise vote for Begich, but these voters seemingly did not cast compromise votes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Furthermore, this House election was an off-schedule special election which occurred because of the death of the sitting congressman and this House seat was up for election again in November 2022, just three months later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' The November election contained four candidates, the same three from the August election and a fourth candidate Chris Bye, but Bye received less than 2% of the vote and thus the November election was essentially just a rerun of the August election.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' In response to the issue-riddled August election, supporters of Begich and Palin did not seem to meaningfully alter their behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Palin still received approximately 5000 more votes than Begich, but because Peltola increased her support by a substantial relative amount (most likely due to higher turnout in this general election), she was the Condorcet winner and defeated Palin by a wider margin in the final round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Furthermore, the No- vember election contained none of the RCV failures we discuss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Thus, even when voters are given poll data of the highest quality, an election which occurred three months prior, they do not seem to react in a strategic fashion (at least, the number of voters reacting strategically seems relatively small).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' In summary of non-majoritarian failures, while RCV is manipulable in theory it does not seem to be manipulable in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Non-majoritarian failures occur AN EXAMINATION OF RANKED CHOICE VOTING IN THE UNITED STATES, 2004-2022 13 infrequently and, when they do occur, it is not clear that these failures are “action- able” on the part of voters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Issues such as monotonicity paradoxes are undesirable, but they become offensive only in hindsight;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' we find little evidence that voters could vote strategically to engineer such outcomes in actual elections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Our empiri- cal results are consonant with the theoretical work of (Green-Armytage 2014) and (Green-Armytage et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' 2016), which show that RCV is less manipulable than most other voting methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Majoritarian failures occur at a much higher rate than the other failures, ac- counting for more than half the elections in the database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' These occur because a significant portion of the electorate in these elections cast partial ballots, causing their ballots to become exhausted before the final round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' As pointed out in (Bur- nett and Kogan, 2015), partial ballots occur for two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' First, voters may voluntarily provide an incomplete ranking, choosing not to rank all candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Second, some jurisdictions limit the number of candidates that voters can rank on their ballots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' For example, the city of Minneapolis, MN, allows voters to rank only three candidates regardless of the number of candidates in the race.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' In such elections, voters are often forced to cast partial ballots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' For our discussion, it is useful to distinguish between elections in which voters can provide a complete ranking of the candidates if they so choose, and elections in which they cannot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' In an election in which the jurisdiction limits the number of candidates ranked on ballots, we say that the election’s truncation level is the number of candidates a voter can rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' If the number of candidates in an election is more than the election’s truncation level plus one, we say the election is truncated because voters cannot provide a complete ranking of the candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' For example, a ranked-choice election in Minneapolis with five or more candidates is truncated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' if an election contains four or fewer candidates, voters can provide a complete ranking6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' In our database 72 elections are truncated and 110 are not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Because voters are forced to cast partial ballots in truncated elections and a majoritarian failure is caused by a significant portion of ballots being partial, we expect these failures to be more common in truncated than non-truncated elec- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Our findings bear this out: 57 of the 72 truncated elections demonstrate a majoritarian failure, while 38 of the 110 non-truncated elections demonstrate this flaw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' In elections with a majoritarian failure, the winner’s vote share in truncated elections also tends to be lower (on average, 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='9%) than the winner’s vote share in the non-truncated elections (on average, 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='7%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Table 8 shows the five elections with the smallest winner vote share among the truncated elections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' These vote shares are all significantly lower than 40%, quite far from a majority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' The 2010 San Francisco Board of Supervisors election in the 10th district is particularly extreme, with the winner earning less than 25% of the total vote.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' By contrast, Table 9 shows the five elections with the smallest winner’s vote share among the non-truncated elections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' The smallest winner’s vote share among the non-truncated elections is 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='2%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' there are 13 truncated elections in which the winner’s vote share is smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' 6In an election with four candidates and truncation level 3, voters can provide a complete ranking of the four candidates by providing a complete ranking of their top three;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' the candidate left off the ballot is assumed to be the voter’s fourth choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' AN EXAMINATION OF RANKED CHOICE VOTING IN THE UNITED STATES, 2004-2022 14 Election Winner’s Vote Share 2010 San Francisco Board of Supervisors Dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' 10 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='3% 2021 NYC Dem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Primary City Council Dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' 9 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='2% 2020 Minneapolis City Council Ward 6 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='1% 2021 NYC Dem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Primary Borough President Kings 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='3% 2008 Pierce County, WA, County Treasurer 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='5% Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' The five truncated elections with the smallest winner vote shares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Election Winner’s Vote Share 2021 NYC Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Primary Dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' 50 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='2% 2021 Portland, ME City Council At-Large 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='5% 2019 San Francisco District Attorney 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='9% 2021 NYC Dem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Primary Dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' 32 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='6% 2008 Pierce County, WA, County Executive 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='6% Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' The five non-truncated elections with the smallest win- ner vote shares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' The high rate of majoritarian failure in the data seems concerning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' There are a few potential solutions to address this failure rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' First, jurisdictions with trun- cated elections could remove the limit on the number of candidates, allowing voters to provide a complete ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Our results suggest this would have an effect on the failure rate, and we are unaware of mathematical reasons for including a trun- cation level in an election (although there may be political reasons for doing so).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Second, some have argued (Kilgour et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' 2020) that voters may cast partial ballots due to the cognitive load of trying to rank multiple candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' If this is the case, jurisdictions could try to alleviate this load by finding ways to limit the number of candidates on the ballot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' For example, ranked-choice elections in Alaska contain at most four candidates because prior to the RCV election there is a primary election which uses plurality voting to whittle down the field of candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' However, in the 2022 ranked-choice elections in Alaska we still see a high rate of majoritar- ian failure, with six out of thirteen elections demonstrating this issue (although the smallest of the winner vote shares is 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='2%, and so these failures are not particularly egregious).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' It is possible that voters will be willing to rank more candidates over time as they become more comfortable with RCV, in which case Alaska’s strategy of limiting the number of candidates to four could significantly lower the rate of majoritarian failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Of course, depending on one’s values it is possible that majoritarian failures are not important, and therefore the high failure rates are irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' In non-truncated elections, if voters who cast partial ballots are voting sincerely then it is possible that there does not exist a candidate who could earn majority support in a final round of RCV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' For example, if enough voters care only about one or two candidates and are indifferent among the rest, then no voting method will be able to find a “majority winner,” and it is not the fault of RCV that the election produces a majoritarian failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' However, even if one finds majoritarian failures unimportant, given the high rate of such failures (even in non-truncated elections) it is likely advisable that RCV advocates adjust their rhetoric around RCV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' For example, (Lair 2022) states that AN EXAMINATION OF RANKED CHOICE VOTING IN THE UNITED STATES, 2004-2022 15 “the majoritarian principle is an axiom of democratic government” and uses this statement (among others) to justify the adoption of RCV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Similarly, (Lavin 2019) states: “[W]e need majority rule in elections—not only as a principle or best practice but as a practical assurance to legitimize outcomes and give elected officials strong mandates to govern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' By requiring winners to earn a majority of votes—if not in first choices alone then with backup choices—RCV meets both of these critical needs.” Our results suggest that RCV does not live up to such statements in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Conclusion When evaluated based on criteria important to the social choice literature, RCV mostly performs well in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' In the American ranked-choice political elec- tions in our database, RCV almost always selects the Condorcet winner and avoids the spoiler effect, while also demonstrating practical resistance to strategic voting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Paradoxes which feature prominently in the theoretical literature such as mono- tonicity and no-show paradoxes seem to occur on the order of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='5-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='1% for real- world elections without a majority candidate, and these failure rates would decrease considerably if we also included ranked-choice elections which do not advance to a second round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' The percentage of elections in which the winner does not receive a majority in the final round is very high, which should give pause to RCV advocates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Since a perfect voting method seemingly does not exist, choosing a method involves trade-offs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' The weaknesses of RCV are mostly not observed in real-world ranked-choice data available in the US.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Of course, the failure rates are not zero, and it is reasonable to insist on a method which always chooses the Condorcet winner or is not susceptible to monotonicity paradoxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' We have contributed to the literature by providing a comprehensive empirical analysis of the social-choice weaknesses of RCV in the US, but whether RCV’s benefits outweigh its costs is, in our view, still an open question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Acknowledgements Thank you to Deb Otis for showing us the FairVote data repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' References [1] Bokat-Lindell, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' “Can Ranked-Choice Voting Cure American Politics?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=',” New York Times, June 24, 2021, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='nytimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='com/2021/06/24/opinion/ ranked-choice-new-york.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='AccessedJan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='14,2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' [2] Burnett C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' & Kogan V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Ballot (and voter) “exhaustion” under Instant Runoff Voting: An examination of four ranked-choice elections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Electoral Studies, 37, 41-49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' [3] Coll J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Demographic Disparities Using Ranked-Choice Voting?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Ranking Difficulty, Under-Voting, and the 2020 Democratic Primary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Politics and Governance, 9 (2), 293-305.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' [4] Darmann A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=', Grundner J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=', & Klamler C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Evaluative voting or classical voting rules: Does it make a difference?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Empirical evidence for consensus among voting rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' European Journal of Political Economy, 59, 345-353.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' [5] Donovan T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=', Tolbert C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=', & Harper S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Demographic differences in understanding and utilization of ranked choice voting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Social Science Quarterly, 103 (7), 1539-1550.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' [6] Gallagher M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Monotonicity and non-monotonicity at PR-STV elections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Paper pre- sented at the annual conference of the elections, public opinion and parties (EPOP) specialist group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' [7] Gierzynski A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=', Hamilton W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=', & Smith W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Burlington Vermont 2009 IRV mayor election.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' https://rangevoting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='org/Burlington.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Accessed 12/20/22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' [8] Graham-Squire A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' & McCune D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' A Mathematical Analysis of the 2022 Alaska Special Election for US House.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Preprint: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='org/abs/2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='04764.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' AN EXAMINATION OF RANKED CHOICE VOTING IN THE UNITED STATES, 2004-2022 16 [9] Graham-Squire A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' & Zayatz N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Lack of Monotonicity Anomalies in Empirical Data of Instant-runoff Elections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Representation, 57 (4), 565-573.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' [10] Green-Armytage J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Strategic voting and nomination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Social Choice and Welfare, 42, 111-138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' [11] Green-Armytage J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=', Tideman T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=', & Cosman R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Statistical evaluation of voting rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Social Choice and Welfare, 46, 183-212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' [12] Kilgour D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=', Gr´egoire J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=', & A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Foley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' The prevalence and consequences of ballot truncation in ranked-choice elections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Public Choice 184: 197-218.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' [13] Lair S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Ranked-choice voting is needed, along with open primaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' https://thenevadaindependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='com/article/ ranked-choice-voting-is-needed-along-with-open-primaries/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Accessed 12/26/22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' [14] Landsman T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' All RCV Elections in the Bay Area So Far Have Pro- duced Condorcet Winners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' https://fairvote.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='org/every_rcv_election_in_the_bay_area_ so_far_has_produced_condorcet_winners/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Accessed 12/26/22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' [15] Lavin N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Majority Rule: more than just a principle for successful elections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' https:// fairvote.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='org/majority_rule_more_than_just_a_principle_for_successful_elections/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Accessed 12/26/22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' [16] McCune D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' and Graham-Squire A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Monotonicity Anomalies in Scottish Local Gov- ernment Elections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' [17] McCune D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' & McCune L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Does the Choice of Preferential Voting Method Matter?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' An Empirical Study Using Ranked Choice Elections in the United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='1080/00344893.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='2133003 [18] McCune D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' & McCune L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' The Curious Case of the 2021 Minneapolis Ward 2 City Council Election, to appear in The College Mathematics Journal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' [19] Miller N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Closeness matters: Monotonicity failure in IRV elections with three can- didates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Public Choice 173 (1-2): 91-108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' [20] Otis D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' RCV in New York City.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' (2021, October 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Retrieved from https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='fairvote.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' org/rcv_in_new_york_city#candidate_analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' [21] Otis D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Single winner ranked choice voting CVRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='org10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='7910/DVN/ AMK8PJ, Harvard Dataverse, V5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' [22] Ornstein J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' & Norman R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Frequency of monotonicity failure under Instant Runoff Voting: estimates based on a spatial model of elections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Public Choice, 161 (1-2), 1-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' [23] Popov S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=', Popova A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=', & Regenwetter M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Consensus in Organizations: Hunting for the Social Choice Conundrum in APA Elections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Decision 1 (2), 123-146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' [24] Regenwetter M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=', Kim A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=', & Ho M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' The Unexpected Empirical Consensus Among Consensus Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Psychological Science 18 (7), 629-635.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' [25] Song C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Three Empirical Analyses of Voting [Unpublished doctoral dissertation].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Virginia Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' [26] Tomlinson K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=', Ugander J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=', & Kleinberg J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Ballot Length in Instant Runoff Voting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Preprint:https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='org/abs/2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='08958.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content=' Adam Graham-Squire, Department of Mathematical Sciences, High Point University, 1 University Parkway, High Point, NC, 27268 Email address: agrahams@highpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='edu David McCune, Department of Physics and Mathematics, William Jewell College, 500 College Hill, Liberty, MO, 64068-1896 Email address: mccuned@william.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='jewell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} +page_content='edu' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNFLT4oBgHgl3EQfai-M/content/2301.12075v1.pdf'} diff --git 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Nanotechnology 9170124, Estaci´on Central, Santiago, Chile. +We study the tunable control of the non-Markovianity of a bosonic mode due to its coupling to a set of +auxiliary qubits, both embedded in a thermal reservoir. Specifically, we consider a cavity mode coupled to +auxiliary qubits described by the Tavis-Cummings model. As a figure of merit, we define the dynamical non- +Markovianity as the tendency of a system to return to its initial state, instead of evolving monotonically to its +steady state. We study how this dynamical non-Markovianity can be manipulated in terms of the qubit frequency. +We find that the control of the auxiliary systems affects the cavity dynamics as an effective time-dependent decay +rate. Finally, we show how this tunable time-dependent decay rate can be tuned to engineer bosonic quantum +memristors, involving memory effects that are fundamental for developing neuromorphic quantum technologies. +I. +INTRODUCTION +In open quantum systems, the Markovian approximation is +widely used due to its mathematical simplicity and the good +description of the phenomenology observed in the lab. The +Markovian approximation, from a pedagogical perspective, +considers that the state of the reservoir is not correlated at +different times, which can be interpreted as a memoryless +bath [1, 2]. Nevertheless, in many real-world systems, mem- +ory effects emerge, causing non-trivial dynamics, where trans- +port effects in biological systems are one of the most paradig- +matic ones [3–5]. It suggests that the manipulation of non- +Markovian systems and the control of the open system dy- +namics is important for several applications, such as quan- +tum metrology [6, 7], quantum simulation [8, 9], and quan- +tum memdevices [10–15]. In this context, the manipulation +of the non-Markovianity looks promising for implementing +new technological devices, particularly for quantum memris- +tive systems useful for the realization of neuromorphic com- +puting at a quantum level [16–18]. +On the other hand, the definition and quantification of the +non-Markovianity in quantum systems is still an open ques- +tion, as can be checked in the recent literature [19–23]. Nev- +ertheless, there are two widely accepted cases by the scientific +community. The first is based on the distinguishability of a +quantum system [22] under a dissipative evolution. This def- +inition considers that if a system interacts with a Markovian +environment, the system’s information will flow unidirection- +ally to the environment. It means that the system loses distin- +guishability between different initial states during the dynam- +ics. In other words, the system monotonically forgets the ini- +tial condition. Oppositely, in a system with a non-Markovian +environment, the distinguishability between two evolutions +with different initial conditions will increase at some time, re- +covering the lost information from the environment. This def- +inition allows us to quantify the non-Markovianity of a dissi- +pative channel by the sum of the regions where some distance +∗ F. Albarr´an-Arriagada +francisco.albarran@usach.cl +measure increases in time for some pair of initial conditions. +The second definition of non-Markovianity is based on entan- +glement with the auxiliary system [23]. The system and the +auxiliary system are in maximum entanglement at the initial +moment. If the entanglement gradually decreases as the sys- +tem evolves, the system is in a Markovian environment. Now, +if the entanglement does not decrease monotonically, it means +that it increases for some time, then the system interacts with +a non-Markovian environment. Even if a method exists to +quantify the non-Markovianity of a channel in both cases, the +degree of non-Markovianity involves an optimization process +over all the possible evolutions (initial state), quantifying the +increments of the distinguishability or entanglement as a cost +function. +Recently, the non-Markovian dynamics has been actived re- +searched both in theory and experiment, driven by the wide +interest in quantum technologies [24–30]. For example, from +a theoretical point of view, with a harmonic oscillator coupled +to both non-Markovian and Markovian baths, many charac- +teristics of the system can be explored, like spectral proper- +ties [25]. In experiments using an all-optical experiment, the +transitions between Markovian and non-Markovian regimes +can be reached, controlling the information backflow of the +system [26] as well as the observation of the called weak non- +markovinity regime [27]. +In this article, we focus on the dynamical non-Markovianity +(DnM), which means the degree of non-Markovianity pre- +sented in a given dynamics. Specifically, we will focus on +a system composed of a cavity mode (main system) coupled +to a set of qubits (auxiliary systems) described by the Tavis- +Cummings model [31, 32] embedded in a Markovian bath. +We are interested in studying the DnM that arises in the main +system dynamics by tracing the auxiliary qubits, creating a +tunable bosonic quantum memristor. Also, we will explore +how the DnM can be manipulated by external control over the +auxiliary systems. We find that by tuning the energy gap of +the set of qubits, we can simulate a time-dependent decay rate +in the cavity going from a regime with maximal DnM and +another with minimal DnM and Markovian evolution. This +tunable dynamical Non-Markovianity, allow us to define vari- +ables that follow a memristive behavior, obtaining an experi- +arXiv:2301.13365v1 [quant-ph] 31 Jan 2023 + +2 +Environment +FIG. 1. Diagram of the model: a cavity (bosonic mode) coupled +to a set of qubits embedded in a Markovian reservoir. Each auxil- +iary qubit can be dynamically tuned and the cavity can be classically +driven. +mental feasible, scalable and general framework to implement +switchable memory devices useful for neuromorphic quantum +computing. +II. +MODEL AND METHODS +We consider a system consisting of a single bosonic mode +(resonator) coupled to a set of n qubits in contact with a ther- +mal reservoir at zero temperature as shown in Fig. 1. The +interaction between the qubits and the resonator is described +by the Tavis-Cumming model +ˆHTC = ˆHR + ˆHQ + ˆHR−Q +(1) +where +ˆHR = ℏωRˆa†ˆa, +ˆHQ = ℏ +2 +n +� +j=1 +ωQ ˆσz,j, +ˆHR−Q = ℏ +n +� +j=1 +g( ˆσ− +j ˆa† + ˆσ+ +j ˆa)), +(2) +are the Hamiltonians for the bosonic mode, the qubits, and the +resonator-qubits interaction, respectively. Here, ωR, ωQ, g, +and ℏ represent the resonator frequency, the qubit frequency, +the qubit-resonator coupling strength, and the Planck con- +stant. The operator ˆa(ˆa†) is the annihilation(creation) oper- +ator for the bosonic mode, ˆσz,j is the Pauli z matrix for the jth +qubit, and ˆσ−(+) +j +is the lowering(raising) operator for the jth +qubits. In order to ensure the validity of our model, we con- +sider ωQ/ωR ∼ 1 and g/ωR < 0.1. From now we will consider +ℏ = 1. +We consider that the total system undergoes Markovian dy- +namics described by the following master equation, +˙ρ(t) = −i[ ˆHTC, ρ(t)] + +n +� +j=0 +ΓjD[ ˆO j]ρ, +(3) +with +D[ ˆOj]ρ = ˆO jρ ˆO† +j − 1 +2{ ˆO† +j ˆO j, ρ}, +(4) +where ˆO0 = a and ˆOj = σ− +j for j > 0 and Γj is the decay rate +of the jth channel. We are interested only in the dynamics of +the resonator, thus we focus on its reduced state by tracing out +the qubits, ρR(t) = TrQ(ρ(t)). In this way, the set of qubits act +as an auxiliary system that introduces non-Markovian prop- +erties to the dissipative evolution of the resonator. We want +to characterize the degree of non-Markovianity of a particular +evolution of our system (the resonator) determined by its ini- +tial state. We look for a figure of merit that can be understood +as a degree of non-Markovianity of the particular dynamics +of the system that result from a given initial condition. To +this end, we notice that when the dynamics of the system are +Markovian and purely dissipative, then its quantum state will +monotonically approach the corresponding steady state of the +environment. We can characterize this behavior by calculat- +ing the trace distance between the instantaneous state of the +system and the steady state of the evolution, +DS (ρR(t)) = 1 +2|ρR(t) − ρS S |, +(5) +where the subindex S denotes that the trace distance is taken +with respect to the steady state. For a Markovian evolution, +this quantity will decrease monotonically to zero [22], where +|ρ| = Tr[ +� +ρ†ρ] and ρS S is the steady state of the system. In +our case, the temperature of the environment is zero and there- +fore ρS S = |0⟩⟨0|. Now, the quantity D(t) allows us to de- +tect when the evolution deviates from Markovian behavior +whenever it is no longer monotonically decreasing. There- +fore, we can characterize the non-Markovianity of a particu- +lar system evolution by considering all the time intervals with +non-monotonic behavior. In this way, we define the DnM as +ND = +� +ζ>0 +ζ(t)dt, +(6) +where ζ(t) = (d/dt)DS (ρR(t)), for an evolution long enough to +reach the steady state. We note that this definition is closely +related to the non-Markovianity measure for dissipative chan- +nels based on distinguishability [22]. However, our defini- +tion considers only the dynamics under study and not an op- +timization over all the initial conditions. In our system, the +qubit-resonator coupling is the factor that introduces non- +Markovian behavior into the resonator dynamics due to in- +formation backflow from the set of qubits. In the next section, +we will characterize how a given configuration of the set of +qubits affects the behavior of the DnM for such bosonic quan- +tum memristor. +III. +RESULTS +A. +Dynamical non-Markovianity (DnM) +For our first case, we will focus on the resonator interact- +ing with one qubit (Jaynes-Cummings model) and interacting + +3 +with n = 5 qubits. We will analyze how the DnM depends +on the qubit frequency and coupling strength. It is important +to mention that the set of auxiliary qubits is always initialized +in the ground state in order not to introduce energy into the +resonator since it would undermine the interpretation of the +DnM. First, we consider the initial state |ψ0⟩ = |1R0Q⟩. In +Fig. 2 (a), we show the DnM of the resonator when varying +the coupling strength g/ωR and the frequency ratio ωQ/ωR. +We can see that the DnM is largest when qubit and resonator +are in resonance and when g increases. Notice that for larger +values of g/ωR, the qubit-resonator detuning can yield sig- +nificant values of DnM. Figure 2 (b) shows the DnM for the +case of five auxiliary qubits. We can observe that the effect of +enlarging the set of auxiliary qubits is relaxing the resonance +condition and increasing the value for the DnM. +This behavior is to be expected since the resonance con- +dition allows for maximal information transfer and informa- +tion backflow due to the complete Rabi oscillations (in the +case of n = 1). In addition, the coupling strength g/ωR is +related to the speed of the information transfer and informa- +tion backflow. Then, for small g (slow information transfer), +a stronger resonance condition is needed to have information +backflow before the system reaches the stationary state. If +g increases the communication between the auxiliary qubits +and the bosonic mode is faster, and a more relaxed resonance +condition will still have information backflow. +Increasing +the number of qubits, increases the channels for information +backflow, which leads to larger values of DnM even at higher +detuning. +Next, we study the scaling of the DnM with the number of +qubits under fixed conditions. In Figure 3, we study ND as +a function of the number of qubits (until n = 8) for different +coupling strengths when the resonator is initialized with one +excitation. Figure 3 (a) shows how ND scales with the num- +ber of particles (n), with a monotonically increasing behavior +reminiscent of a power law. In Fig. 3 (b), we do a log-log plot +of the quantities of Figure 3 (a) which confirms the power-law +dependence. In all instances the R2 coefficient is larger than +0.995 which means that the scaling of the DnM is very well +approximated by +ND ∝ nk, +(7) +(a) +(b) +FIG. 2. Dynamical non-Markovainity of the resonator. (a) one-qubit +case. (b) five-qubit case. In both cases, the decay rate of qubit and +resonator is ΓQ = ΓR = 0.005. We consider all qubits are in res- +onance (ωQ = ωR), the coupling strength g/ωR ∈ [0, 0.1] and the +initial state |ψ0⟩ = |1R0Q⟩. +(a) +(b) +(c) +(d) +FIG. 3. (a) The DnM of the resonator in terms of the number of +qubits n. (b) the log-log plot of DnM and the number of qubits. +For (a) and (b), we consider three cases g = 0.01, 0.05, 0.1, with +the resonant condition ωQ = ωR = 1. (c) The exponent k of the +power-law dependence as a function of the coupling strength g/ωR, +with qubit and resonator in resonance. (d) The exponent k of the +power-law dependence as a function of the frequency of the qubit +ωQ/ωR and a fixed coupling strength g/ωR = 0.05. For all cases, +we consider decay rates ΓQ = ΓR = 0.005ωR, the initial state of the +resonator |ψ0⟩ = |1R⟩, and all the qubits initialized in the ground state. +(a) +(b) +FIG. 4. The non-Markovainity of the resonator. (a) one-qubit case. +(b) five-qubit case. Parameters: In both cases, the decaying rate of +qubit and resonator is ΓQ = ΓR = 0.005ωR. The driving frequency +of qubit µQ/ωR ∈ [0, 1], the driving amplitude of the qubit ΩQ/ωR ∈ +[0, 1], the qubit frequency ωQ/ωR = 1, the coupling strength g/ωR = +0.1. The initial state is |ψ0⟩ = |1R0Q⟩. +where the exponent k depends on the coupling strength g/ωR +and the qubit frequency ωQ/ωR, as shown in Fig. 3 (c) and +Fig. 3 (d), respectively. We can see that when the DnM is +maximal, that is, for large g and ωQ = ωR, the value of k is +minimum. This is a finite size effect of the auxiliary system as +we would expect that if the number of qubits increases to the +thermodynamic limit they would induce Markovian dynamics +for the resonator. +We have seen that the DnM of the resonator strongly de- +pends on the parameters of the set of auxiliary qubits. It is then +interesting to consider whether we can have dynamic control +over the DnM by manipulating the set of auxiliary qubits. In + +1.5 +5 +4 +wR +/03 +3 +2 +1 +0.5 +0 +0.05 +0 +0.1 +/WR1.5 +14 +12 +10 +8 +93 +6 +4 +2 +0.5 +0 +0 +0.05 +0.1 +g/wR0.75 +0.70 +0.65 +0.60 +0.55 +0.02 +0.04 +0.06 +0.08 +0.10 +000.9 +0.8 +K +0.7 +0.6 +0.5 +0.6 +0.8 +1.0 +1.2 +1.4 +9320 +3 +g=0.01 +g=0.05 +15 +2 +g=0.1 +10 +0 +5 +2 +3 +4 +5 +6 +7 +8 +0.0 +0.5 +1.0 +1.5 +2.0 +log(n) +ng/wRR +3 +Q +35 +4 +0.5 +3 +2 +0.5 +1.0 +μQ12 +10 +0.5 +8 +6 +0.5 +1.0 +0 +μQ4 +(b) +(a) +FIG. 5. The minimum (a) and maximum (b) of DnM for the resonator +with a different number of auxiliary qubits. In both cases, the decay- +ing rate is ΓQ = ΓR = 0.005ωR. The frequency of qubit ωQ/ωR = 1, +and the initial state is of resonator |ψ0⟩ = |1R⟩. The qubits are all +initialized in the ground state. +what follows, we apply a driving term in the z-direction to the +set of auxiliary qubits in order to dynamically modulate the +qubit gap and control the degree of DnM in the evolution. The +driving is chosen so that it does not introduce energy into the +qubits which could excite the resonator and be interpreted as +information backflow by the DnM. This situation is described +by the following Hamiltonian +ˆH = HTC + ℏΩQ +n +� +j +sin(µQt) ˆσz +j, +(8) +where HTC is the Hamiltonian of eq.(1), ΩQ and µQ are the +amplitude and frequency of the driving over the qubits, re- +spectively. Notice that we consider that each qubit is driven +by the same signal. +We numerically calculate ND for different values of the +driving frequency and amplitude which we show in Fig. 4. In +Fig. 4 (a) we show the case of one auxiliary qubit. Here, there +is non-zero DnM over the whole range of parameters, how- +ever it is interesting to notice the dark lines that are spanned +from near the origin where the DnM is almost completely +suppressed. A similar behavior occurs when we increase the +number of qubits, as is shown in Fig. 4 (b) for the five qubits +case, where the DnM is suppressed over thin lines in the fre- +quency/amplitude plane. Although the suppression is not as +strong as in the one-qubit case, these lines show significant +decrease in the DnM. This indicates that by modulating either +the frequency or the amplitude of the driving we can enhance +or suppress the DnM of the resonator. +Similarly, we study the DnM in terms of the coupling +strength for different number of qubits in the auxiliary system. +Here, we will consider two cases, the parameters of the driv- +ing that yield the maximum and minimum DnM. In Fig. 5 (a), +we plot the minimal DnM for different coupling strength g. +We can see that up to g = 0.02 we can essentially completely +suppress the non-Markovian behavior by a suitable choice of +driving parameters. Increasing the number of qubits decreases +the necessary value of coupling strength that allows for com- +pletely supressed DnM. On the other hand, in Fig. 4 (b), we +plot the maximal DnM for different coupling strength g. Here +we can see that ND has linear dependence on the coupling +strength except for a small range around zero. From these +FIG. 6. Transition from non-Markovian to Markovian dynamics by +changing the driving frequency over the auxiliary qubits. Parameters: +the driving amplitude of qubit is Ωq = 0.5, the number qubit n = 1, +decaying rate is ΓQ = ΓR = 0.005, frequency ωQ = ωR, coupling +strength g = 0.05. +results we have that provided we choose a suitable value of +the coupling strength, we can switch between Markovian and +non-Markovian dynamics for the resonator by just controlling +the auxiliary set of qubits. +In Fig. 6 we show how we can dynamically switch the non- +Markovian behavior on and off by just changing the driving +frequency of the qubits. Here, we plot the trace distance as +a function of time. At the start of the evolution, we choose +a driving frequency that yields maximum non-Markovianity +(µQ = ΩR), later at t = 350ω−1 +R we switch the driving fre- +quency to µQ = 0.75ωR which yields minimum DnM. As +can be seen in the figure, at t = 350ω−1 +R , the trace distance +switches from non-monotonic to monotonically decreasing +behavior, which characterizes Markovian evolution. +B. +Time-dependent decay rate +The observed memory effect can be understood as the sys- +tem effectively interacting with an environment with a time- +dependent decay rate, which becomes negative during some +time intervals, favoring the information back-flow [33]. To +understand this statement, consider a resonator, with state ˜ρ, +undergoing dissipative dynamics with a time-dependent decay +rate and without any interaction with an auxiliary system, the +system is then described by the following master equation +˙˜ρ(t) = −i[H, ˜ρ] + Γ(t) +� +a˜ρa† − 1 +2{˜ρ, a†a} +� +(9) +where H = ℏωRa†a, and Γ(t) is a time-dependent decay rate +that can be negative. Here, ˜ρ represents the state of the res- +onator undergoing dynamics as described above, and is dif- +ferent from ρR which is the reduced state of the resonator as +described by Hamiltonian (1). For Γ(t) > 0, the energy of +the resonator dissipates to the environment, meaning that the +information in the resonator is continuously lost. Meanwhile, +for Γ(t) < 0 there is energy entering the resonator, giving place +to information back-flow and therefore to a non-Markovian +process. + +12.5 +n=1 +n=2 +10.0 +n=3 +n=4 +M +7.5 +n=5 +5.0 +2.5 +0.0 +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +g-——— +n=1 +n=2 +n=3 +3 +n=4 +N +n=5 +2 +1 +0 +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +gg/wRg/wR1.0 +μ3=1 +u1=0.75 +0.8 +distance +0.6 +Trace +0.4 +0.2 +0.0 +0 +200 +400 +600 +800 +1000 +t (unit of w)5 +(a) +(b) +(c) +(d) +(d) +FIG. 7. Trace distance of the resonator and the DnM under dif- +ferent driving frequencies. +Top, (a) the blue line, the frequency +of driving µ1 = 0.419, the green line, the resonator’s decaying +rate is constant Γ(t) = 0.005. (b) the blue line, the frequency of +driving µ2 = 0.20, the green line, the resonator’s decaying rate is +Γ1(t) = 0.05 (sin(0.023t) + 0.09). (c) the blue line, the frequency +of driving µ3 = 1. the green line, the resonator’s decaying rate is +Γ1(t) = 0.25 (sin(0.079t) + 0.021). Bottom, (d) the DnM of resonator +in different driving frequency µQ ∈ (0, 1). Parameters: the number +qubit n = 1, decaying rate is ΓQ = ΓR = 0.005, frequency ωQ = ωR, +coupling strength g = 0.05, the driving amplitude ΩQ = 0.5ωR. +We consider the time-dependent decay rate parametrized +as Γ1(t) = A (sin(Bt) + C). Notice that the master equation +of Eq. (9) has the same steady state as that of our original +system in Eq. (3). Therefore, for a given dynamics induced +by the set of auxiliary qubits, we can find the closest non- +Markovian dynamics corresponding to negative decay rate by +finding A, B, and C that minimize the difference of trace dis- +tance |D(ρR(t)) − D(˜ρ(t))|. In Fig. 7, we plot DS (ρR(t)) and +DS (˜ρopt(t)) where ρR(t) is for one qubit case and ˜ρopt(t) is the +resonator evolved with the optimal parameters for the decay +rate. +We consider 3 cases, in Fig. 7 (a), the effective decay rate +is time-independent Γ = 0.005 corresponding to Markovian +behavior; whereas for time-dependent decay rate we have in +Fig. 7 (b) where Γ(t) = 0.05[sin(0.023t) + 0.09] and Fig. 7 +(c) where Γ(t) = 0.25[sin(0.079t) + 0.021]. Finally, Fig. 7 (d) +shows the DnM as a function of the qubit-driving frequency +where it displays the qubit driving frequency corresponding +to Fig. 7 (a) - (c). We can see that for time-dependent de- +cay the behavior of both trace distance is very similar, which +means that the set of auxiliary qubits is inducing highly non- +Markovian dynamics. +Finally, it is interesting to study how this simulated time- +dependent decay rate can affect the response of the cavity over +external driving, in order to control the memristive properties +of the dynamics. +C. +Bosonic quantum memristor +One interesting application of our results is to induce mem- +ristive behavior into the bosonic mode, which can be tuned by +FIG. 8. Memristive behavior, the green line shows the dynamics +when the auxiliary qubits are not driven and off-resonant and the blue +curve is when we add a driving over the auxiliary qubits. (a) larger- +DnM case, the number of qubits n = 1, driving frequency µc = 1. (b) +medium-DnM case, the number of qubits n = 1, driving frequency +µc = 0.2. (c) larger-DnM case, the number of qubits n = 5, driving +frequency µc = 1. (d) medium-DnM case, the number of qubits +n = 5, driving frequency µc = 0.2. Parameters: the driving amplitude +of qubit is Ωq = 0.5, decaying rate is ΓQ = ΓR = 0.005, frequency +ωQ = ωR, coupling strength g = 0.05, the driving amplitude of cavity +Ωc = 0.2, frequency µc = 0.5. +the set of auxiliary qubits. In Ref. [34], it was shown that a +kind of time-dependent decay rate produces a quantum mem- +ristor, which could be reached in a superconducting circuits +platform. Later, in Ref. [35], a memristive dynamics was +obtained in a quantum computer by the simulation of a non- +Markovian bath. In this line, we analyze the response of the +cavity under an external driving, obtaining a Hamiltonian of +the form: +ˆH = HTC + ℏΩQ +n +� +j +sin(µQt) ˆσz +j + F(t)(a + a†). +(10) +Now, if we define the variables I = −⟨i(a − a†)⟩ and O = +⟨ ˙N⟩ + α⟨N⟩, with α a constant. If we consider α = Γc, it is the +natural decay rate of the cavity, we have that +O = F(t)I + G(t), +(11) +for more details see appendix ??. The function G(t) depends +on the DnM of the system, which means that we can control +the memristive relation +O = F(t)I, +(12) +by controlling the value of G(t). Now, if we choose F(t) = +Ωc[1−sin(cos µct)], it is possible to obtain the typical pinched +hysteresis loop that characterizes a quantum memristor. +This situation is shown in Fig. +8, where we obtain the +pinched hysteresis loop (green curve), in a similar way that +it is obtained for previous proposals of quantum memris- +tos [12, 14, 34] as a signature of memristive behavior. It is + +1.0 +Qubit environment +Time-dependent decay +distance +0.8 +0.6 +Irace +0.4 +0.2 +0.0 +0 +250 +500 +t (unit of wrl1.0 +Qubit environment +Time-dependent decay +distance +0.8 +0.6 +Trace +0.4 +0.2 +0.0 +0 +250 +500 750 1000 1250 1500 1750 2000 +t (unit of wRl1.0 +Qubit environment +Time-dependent decay +distance +0.8 +0.6 +Irace +0.4 +0.2 +0.0 +0 +250 +t (unit of wrl)2.5 +2.0 +μ3 +1.5 +U2 +1.0 +0.5 +μ1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +μQ0.20 +0.20 +(a) +(b) +0.15 +0.15 +0.10 +0.10 +0.05 +0.05 +0.00 +0.00 +-0.05 +-0.05 +-0.10 +-0.10 +-0.15 +-0.15 +-0.20 +0.20 +-0.10 +-0.05 +0.00 +0.05 +0.10 +-0.10 +-0.05 +0.00 +0.05 +0.10 +<1) +<1) +0.3 +0.20 +(c) +(d) +0.2 +0.15 +0.10 +0.1 F +0.05 +0.0 +0 +0.00 +-0.1 +-0.05 +-0.2 +-0.10 +-0.15 +-0.3 +0.20 +-0.15 +-0.10 +-0.05 +0.00 +0.05 +0.10 +0.15 +0.20 +-0.10 +-0.05 +0.00 +0.05 +0.10 +<1) +<1)6 +interesting to note that such bosonic quantum memristor dy- +namics appear when the auxiliary qubits are not driven and +off-resonant with the cavity, which means that in an effective +way, the qubits are decoupled from the cavity. In contrast, we +can observe that when we drive the qubits, the memristive be- +havior can be destroyed for different cases, obtaining a way to +go from memristive dynamics to non-memristive dynamics. It +means that we also can control the memory properties induced +by the decay rate in the cavity, which can be helpful in neu- +romorphic computing, considering that the proposed system +can be implemented in many platforms like trapped ions, op- +tical devices, and superconducting circuits, among others. We +also need to remark that our proposal can work as a switchable +bosonic quantum memristor. This suggests that our formalism +allows implementing devices with controllable and switchable +memory properties, only by tuning the energy gap of auxil- +iary qubits. This proposal opens the door for the experimental +implementation of memristive devices, providing a general, +platform-free, and scalable model for the next generation of +neuromorphic quantum computing technology. +IV. +CONCLUSIONS +We consider a cavity coupled to a set of auxiliary qubits, +which induce a controllable dynamical non-Markovianity +(DnM). We show that by dynamical tuning of the energy gap +of the auxiliary qubits, we can go from high DnM to low DnM, +which can be considered as an effective time-dependent decay +rate. We also show that the induced DnM in the cavity mode +follows a power-law dependence with the number of auxil- +iary qubits. Finally, we show as an application that we can +define memristive dynamics in the bosonic mode, which can +be switched off by controlling the energy gap of the auxil- +iary qubits. This means that we can control the memristive +dynamics in the cavity by external control over the auxiliary +system, obtaining a switchable bosonic quantum memristor. +These results provide a general protocol to obtain controllable +bosonic quantum memristors which can be useful in neuro- +morphic quantum computing. This proposal is experimentally +feasible since it only uses a bosonic mode coupled to a set of +qubits, a ubiquitous setup in hardware platforms like trapped +ions, superconducting circuit quantum electrodynamics, and +atomic devices. +Appendix A: The derivative of expectation the number of photons +d⟨ˆa†ˆa⟩ +dt += Tr +� +− i +ℏ +�H, ρ� ˆa†ˆa +� ++ Tr +� +L(ρ)a†a +� += S1 + S2 +(A1) +we set ℏ = 1, S1 and S2 is +S1 = −iTr +�������� +��������ωca†a + ωq +n +� +j=1 +σz +j + +n +� +j=1 +g(a†σ j + aσ+ +j ) + +n +� +j=1 +Ωq sin(µqt)σz +j + F(t)(a + a†), ρ +�������� ˆa†ˆa +�������� += −iTr +�������� +�������� +n +� +j=1 +g(a†σj + aσ+ +j )ρ − ρ +n +� +j=1 +g(a†σj + aσ+ +j ) + F(t)(a + a†)ρ − ρF(t)(a + a†) +�������� ˆa†ˆa +�������� += −iTr +�������� +n +� +j=1 +g +� +σjρa†aa† − σ+ +j ρa†aa − ρa†σ ja†a − ρaσ+ +j a†a +� ++ F(t)ρ +� +ρa†a + ρa†aa†a − ρaa†a − ρa†a†a +��������� += −iTr +�������� +n +� +j=1 +gρ +� +σ+ � +a†a − aa†� +a + σa† � +aa† − a†a +�� ++ F(t)ρ +�� +a†a − aa†� +a + a† � +aa† − a†a +���������� += −iTr +�������� +n +� +j=1 +gρ +� +−σ+a + σa†� ++ Ωc sin(νct)ρ +� +−a + a†��������� += +� n +� +j=1 +ig +� +−σ+a + σa†�� ++ F(t) +� +i(−a + a†) +� += +n +� +j=1 +g +� +i +� +−σ+a + σa†�� +− F(t) +� ˆP +� +(A2) + +7 +S2 = Tr +� +Γc +� +aρa† − 1 +2 +� +a†aρ + ρa†a +�� +a†a + Γq +� +σρσ+ − 1 +2 +�σ+a + ρσ+σ�� +a†a +� += Tr +� +Γcρ +� +a†a†aa − 1 +2a†aa†a − 1 +2a†aa†a +�� += Tr +� +Γcρ +� +a†(aa† − 1)a − 1 +2a†aa†a − 1 +2a†aa†a +�� += −Γc +� +(a†a) +� +(A3) +so the derivative of expectation photon number is +d⟨ˆa†ˆa⟩ +dt += +n +� +j=1 +g +� +i +� +−σ+a + σa†�� +− F(t) +� ˆP +� +− Γc +� +(a†a) +� +(A4) +The input and output is +⟨ˆI⟩ = −⟨ ˆP⟩ +(A5) +⟨ ˆO⟩ = d⟨ˆa†ˆa⟩ +dt ++ Γc +� +(a†a) +� +(A6) +and the relation between input and output +ˆO = F(t)ˆI +(A7) +where F(t) is Ωc ((1 − sin(cos µct)), due to the auxiliary qubits being one part of the environment, we discard the second term of +the output, now the output is +⟨ ˆO⟩ = d⟨ˆa†ˆa⟩ +dt ++ Γc +� +(a†a) +� ++ G(t) +(A8) +with G(t) = +n� +j=1 +g +� +i +� +−σ+a + σa†�� +. As G(t) depends on the interaction between the qubits and the cavity, it can be controlled +by the external driving over the set of auxiliary qubits, which means that G(t) will be close to zero then the qubits are off of +resonance with the cavity. +[1] Li, C.-F.; Guo, G.-C.; Pillo, J. Non-Markovian quantum dynamics: What does it mean? EPL 2019, 127, 50001. +[2] Li, C.-F.; Guo, G.-C.; Pillo, J. Non-Markovian quantum dynamics: What is it good for? EPL 2020, 128, 30001. +[3] Lee, H.; Cheng, Y.-C.; Fleming, G. R. Coherence Dynamics in Photosynthesis: Protein Protection of Excitonic Coherence Science 2007, +316, 1462. +[4] Chin, A. W.; Datta, A.; Caruso, F.; Huelga, S. F.; Plenio, M. B. Noise-assisted energy transfer in quantum networks and light-harvesting +complexes? New J. Phys. 2010, 12, 065002. +[5] Fleming, G. R.; Huelga, S. F.; Plenio, M. B. Focus on quantum effects and noise in biomolecules New J. Phys. 2011, 13, 115002. +[6] Alex, W. C; Susana, F.H; Martin, B. P. Quantum Metrology in Non-Markovian Environments Phys. Rev. Lett. 2012, 109, 233601. +[7] Nicol´as, M; Mart´ı, L, F.H; Diego, W Quantum metrology in a non-Markovian quantum evolution Phys. Rev. A. 2020, 102, 022618. +[8] Sweke, R; Sanz, M; Sinayskiy, I; Petruccione, F; Solano, E. Digital quantum simulation of many-body non-Markovian dynamics Phys. +Rev. A 2016, 94, 022317. +[9] Julio, T.B; Markus, M; Philipp, S; Daniel, N; Thomas, M; Michael, C; Markus, H; Christian F. Roos; Peter, Z; Rainer, B. An open-system +quantum simulator with trapped ions Nature 2011, 470, 486–491. +[10] Pfeiffer, P; Egusquiza, I. L.; Di Ventra, M; Sanz, M; Solano, E. Quantum memristors. Sci Rep. 2016, 6, 29507. +[11] Michele, S; Joshua, M; Simone, P; Michael, A; Francesco, M; Andrea, C; Francesco, C; Roberto, O; Philip, Walther Experimental +photonic quantum memristor. Nat. Photon. 2022, 16, 318–323. +[12] M. Sanz, L. Lamata, and E. Solano. Quantum memristors in quantum photonics APL Photonics. 2018, 3, 080801 +[13] Shevchenko, S. N.; Pershin, Y. V.; Nori, F. Qubit-Based Memcapacitors and Meminductors Phys. Rev. Appl. 2016, 6, 014006. +[14] Norambuena, S.; Torres, F.; Di Ventra, M.; Coto, R. Polariton-Based Quantum Memristors Phys. Rev. Appl. 2022, 17, 024056. + +8 +[15] Sweke, R; Sanz, M; Sinayskiy, I; Petruccione, F; Solano, E. Digital quantum simulation of many-body non-Markovian dynamics. Phys. +Rev. A 2016, 94, 022317. +[16] Pershin, Y. V.; Di Ventra, M. Neuromorphic quantum computing. Proc. IEEE 2012, 100, 2071. +[17] Pehle, C.; Wetterich, C. Digital quantum simulation of many-body non-Markovian dynamics. Phys. Rev. E 2022, 106, 045311. +[18] Xu, H.; Krisnanda, T.; Verstraelen, W.; Liew, T. C. H.; Ghosh, S. Superpolynomial quantum enhancement in polaritonic neuromorphic +computing. Phys. Rev. B 2021, 103, 195302. +[19] de Vega, I.; Aloso, D. Dynamics of non-Markovian open quantum systems. Rev. Mod. Phys. 2017, 89, 015001. +[20] Rivas, ´A; F Huelga, S; B Plenio, M; Quantum non-Markovianity: characterization, quantification and detection. Rep. Prog. Phys. 2014, +77, 094001. +[21] Breuer, H.P.; Foundations and measures of quantum non-Markovianity. J. Phys. B: At. Mol. Opt. Phys. 2012, 45, 154001 +[22] Breuer, H.P.; Laine, E.M; Piilo, J.; Measure for the Degree of Non-Markovian Behavior of Quantum Processes in Open Systems. Phys. +Rev. Lett. 2009, 103, 210401. +[23] Rivas, ´A; Huelga, S. F.; Plenio, M. B.; Entanglement and Non-Markovianity of Quantum Evolutions. Phys. Rev. Lett. 2010, 105, 050403. +[24] Luchnikov, I.A.; Vintskevich, S.V.; Grigoriev, D.A.; Filippov, S.N.; Machine Learning Non-Markovian Quantum Dynamics. Phys. Rev. +Lett. 2020, 124, 140502. +[25] M Bastidas, V; Kyaw, T.H.; Tangpanitanon, J; Romero, G; Kwek, L-C; G Angelakis, D; Floquet stroboscopic divisibility in non- +Markovian dynamics. New J. Phys. 2018, 20, 093004. +[26] Liu, B-H; Li, L.; Huang, Y-F; Li, C-F; Guo, G-C; Laine, E-M; Breuer, H-P; Piilo, J. Experimental control of the transition from Markovian +to non-Markovian dynamics of open quantum systems. Nat. Phys. 2011, 7, 931–934. +[27] Bernardes, N.K.; Cuevas, A; Orieux, A; Monken, C. H.; Mataloni, P; Sciarrino, F; Santos M. F.; Experimental observation of weak +non-Markovianity. Sci Rep. 2015, 5, 17520. +[28] Li, B.-W; Mei, Q.-X; Wu, Y.-K.; Cai, M.-L.; Wang, L.; Yao, L.; Zhou, Z.-C.; Duan, L.-M; Observation of Non-Markovian Spin Dynamics +in a Jaynes-Cummings-Hubbard Model using a Trapped-Ion Quantum Simulator Phys. Rev. Lett. 2022, 129, 140501. +[29] Garc´ıa-P´erez, G; Rossi, Matteo A. C. Maniscalco, S; IBM Q Experience as a versatile experimental testbed for simulating open quantum +systems. npj Quantum Inf. 2020, 6, 1. +[30] Chen, X-Y; Zhang, N-N; He, W-T; Kong, X-Y; Tao, M-J; Deng, F-G; Ai, Q; Long, G-L; Global correlation and local information flows +in controllable non-Markovian open quantum dynamics. npj Quantum Inf. 2022, 8, 22. +[31] Retzker, A.; Solano, E.; and Reznik, E. Tavis-Cummings model and collective multiqubit entanglement in trapped ions. Phys. Rev. A. +2007, 75, 022312 +[32] J¨ager, S. B.; Schmit, T; Morigi, G; Holland, M. J.; Betzholz, R; Lindblad Master Equations for Quantum Systems Coupled to Dissipative +Bosonic Modes. Phys. Rev. Lett. 2022, 129, 063601. +[33] Hu, M.-L.; Lian, H.-L.; Geometric quantum discord and non-Markovianity of structured reservoirs. Ann. Phys. 2015, 362, 795. +[34] Salmiletho, J.; Deppe, F.; Di Ventra, M.; Sanz, M.; Solano, E.; Quantum Memristors with Superconducting Circuits. Sci. Rep. 2017, 7, +42044. +[35] Guo, Y.-M.; Albarr´an-Arriagada, F.; Alaeian, H.; Solano, E.; Barrios, G. A.; Quantum Memristors with Quantum Computers. Phys. Rev. +Appl. 2022, 18, 024082. + diff --git a/kNFQT4oBgHgl3EQfmTaL/content/tmp_files/load_file.txt b/kNFQT4oBgHgl3EQfmTaL/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ace1a85eb548711c86f2ee3babf6e82e7d203ca9 --- /dev/null +++ b/kNFQT4oBgHgl3EQfmTaL/content/tmp_files/load_file.txt @@ -0,0 +1,999 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf,len=998 +page_content='Tunable Non-Markovianity for Bosonic Quantum Memristors Jia-Liang Tang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='1 Gabriel Alvarado Barrios,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='2 Enrique Solano,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='2 and Francisco Albarr´an-Arriagada3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' ∗ 1nternational Center of Quantum Artificial Intelligence for Science and Technology (QuArtist) and Physics Department,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Shanghai University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 200444 Shanghai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' China 2Kipu Quantum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Greifswalderstrasse 226,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 10405 Berlin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Germany 3Departamento de F´ısica,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Universidad de Santiago de Chile (USACH),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Avenida V´ıctor Jara 3493,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 9170124,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Santiago,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Chile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 4Center for the Development of Nanoscience and Nanotechnology 9170124, Estaci´on Central, Santiago, Chile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' We study the tunable control of the non-Markovianity of a bosonic mode due to its coupling to a set of auxiliary qubits, both embedded in a thermal reservoir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Specifically, we consider a cavity mode coupled to auxiliary qubits described by the Tavis-Cummings model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' As a figure of merit, we define the dynamical non- Markovianity as the tendency of a system to return to its initial state, instead of evolving monotonically to its steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' We study how this dynamical non-Markovianity can be manipulated in terms of the qubit frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' We find that the control of the auxiliary systems affects the cavity dynamics as an effective time-dependent decay rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Finally, we show how this tunable time-dependent decay rate can be tuned to engineer bosonic quantum memristors, involving memory effects that are fundamental for developing neuromorphic quantum technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' INTRODUCTION In open quantum systems, the Markovian approximation is widely used due to its mathematical simplicity and the good description of the phenomenology observed in the lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' The Markovian approximation, from a pedagogical perspective, considers that the state of the reservoir is not correlated at different times, which can be interpreted as a memoryless bath [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Nevertheless, in many real-world systems, mem- ory effects emerge, causing non-trivial dynamics, where trans- port effects in biological systems are one of the most paradig- matic ones [3–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' It suggests that the manipulation of non- Markovian systems and the control of the open system dy- namics is important for several applications, such as quan- tum metrology [6, 7], quantum simulation [8, 9], and quan- tum memdevices [10–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' In this context, the manipulation of the non-Markovianity looks promising for implementing new technological devices, particularly for quantum memris- tive systems useful for the realization of neuromorphic com- puting at a quantum level [16–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' On the other hand, the definition and quantification of the non-Markovianity in quantum systems is still an open ques- tion, as can be checked in the recent literature [19–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Nev- ertheless, there are two widely accepted cases by the scientific community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' The first is based on the distinguishability of a quantum system [22] under a dissipative evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' This def- inition considers that if a system interacts with a Markovian environment, the system’s information will flow unidirection- ally to the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' It means that the system loses distin- guishability between different initial states during the dynam- ics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' In other words, the system monotonically forgets the ini- tial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Oppositely, in a system with a non-Markovian environment, the distinguishability between two evolutions with different initial conditions will increase at some time, re- covering the lost information from the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' This def- inition allows us to quantify the non-Markovianity of a dissi- pative channel by the sum of the regions where some distance ∗ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Albarr´an-Arriagada francisco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='albarran@usach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='cl measure increases in time for some pair of initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' The second definition of non-Markovianity is based on entan- glement with the auxiliary system [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' The system and the auxiliary system are in maximum entanglement at the initial moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' If the entanglement gradually decreases as the sys- tem evolves, the system is in a Markovian environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Now, if the entanglement does not decrease monotonically, it means that it increases for some time, then the system interacts with a non-Markovian environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Even if a method exists to quantify the non-Markovianity of a channel in both cases, the degree of non-Markovianity involves an optimization process over all the possible evolutions (initial state), quantifying the increments of the distinguishability or entanglement as a cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Recently, the non-Markovian dynamics has been actived re- searched both in theory and experiment, driven by the wide interest in quantum technologies [24–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' For example, from a theoretical point of view, with a harmonic oscillator coupled to both non-Markovian and Markovian baths, many charac- teristics of the system can be explored, like spectral proper- ties [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' In experiments using an all-optical experiment, the transitions between Markovian and non-Markovian regimes can be reached, controlling the information backflow of the system [26] as well as the observation of the called weak non- markovinity regime [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' In this article, we focus on the dynamical non-Markovianity (DnM), which means the degree of non-Markovianity pre- sented in a given dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Specifically, we will focus on a system composed of a cavity mode (main system) coupled to a set of qubits (auxiliary systems) described by the Tavis- Cummings model [31, 32] embedded in a Markovian bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' We are interested in studying the DnM that arises in the main system dynamics by tracing the auxiliary qubits, creating a tunable bosonic quantum memristor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Also, we will explore how the DnM can be manipulated by external control over the auxiliary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' We find that by tuning the energy gap of the set of qubits, we can simulate a time-dependent decay rate in the cavity going from a regime with maximal DnM and another with minimal DnM and Markovian evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' This tunable dynamical Non-Markovianity, allow us to define vari- ables that follow a memristive behavior, obtaining an experi- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='13365v1 [quant-ph] 31 Jan 2023 2 Environment FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Diagram of the model: a cavity (bosonic mode) coupled to a set of qubits embedded in a Markovian reservoir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Each auxil- iary qubit can be dynamically tuned and the cavity can be classically driven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' mental feasible, scalable and general framework to implement switchable memory devices useful for neuromorphic quantum computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' MODEL AND METHODS We consider a system consisting of a single bosonic mode (resonator) coupled to a set of n qubits in contact with a ther- mal reservoir at zero temperature as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' The interaction between the qubits and the resonator is described by the Tavis-Cumming model ˆHTC = ˆHR + ˆHQ + ˆHR−Q (1) where ˆHR = ℏωRˆa†ˆa, ˆHQ = ℏ 2 n � j=1 ωQ ˆσz,j, ˆHR−Q = ℏ n � j=1 g( ˆσ− j ˆa† + ˆσ+ j ˆa)), (2) are the Hamiltonians for the bosonic mode, the qubits, and the resonator-qubits interaction, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Here, ωR, ωQ, g, and ℏ represent the resonator frequency, the qubit frequency, the qubit-resonator coupling strength, and the Planck con- stant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' The operator ˆa(ˆa†) is the annihilation(creation) oper- ator for the bosonic mode, ˆσz,j is the Pauli z matrix for the jth qubit, and ˆσ−(+) j is the lowering(raising) operator for the jth qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' In order to ensure the validity of our model, we con- sider ωQ/ωR ∼ 1 and g/ωR < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' From now we will consider ℏ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' We consider that the total system undergoes Markovian dy- namics described by the following master equation, ˙ρ(t) = −i[ ˆHTC, ρ(t)] + n � j=0 ΓjD[ ˆO j]ρ, (3) with D[ ˆOj]ρ = ˆO jρ ˆO† j − 1 2{ ˆO† j ˆO j, ρ}, (4) where ˆO0 = a and ˆOj = σ− j for j > 0 and Γj is the decay rate of the jth channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' We are interested only in the dynamics of the resonator, thus we focus on its reduced state by tracing out the qubits, ρR(t) = TrQ(ρ(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' In this way, the set of qubits act as an auxiliary system that introduces non-Markovian prop- erties to the dissipative evolution of the resonator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' We want to characterize the degree of non-Markovianity of a particular evolution of our system (the resonator) determined by its ini- tial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' We look for a figure of merit that can be understood as a degree of non-Markovianity of the particular dynamics of the system that result from a given initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' To this end, we notice that when the dynamics of the system are Markovian and purely dissipative, then its quantum state will monotonically approach the corresponding steady state of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' We can characterize this behavior by calculat- ing the trace distance between the instantaneous state of the system and the steady state of the evolution, DS (ρR(t)) = 1 2|ρR(t) − ρS S |, (5) where the subindex S denotes that the trace distance is taken with respect to the steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' For a Markovian evolution, this quantity will decrease monotonically to zero [22], where |ρ| = Tr[ � ρ†ρ] and ρS S is the steady state of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' In our case, the temperature of the environment is zero and there- fore ρS S = |0⟩⟨0|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Now, the quantity D(t) allows us to de- tect when the evolution deviates from Markovian behavior whenever it is no longer monotonically decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' There- fore, we can characterize the non-Markovianity of a particu- lar system evolution by considering all the time intervals with non-monotonic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' In this way, we define the DnM as ND = � ζ>0 ζ(t)dt, (6) where ζ(t) = (d/dt)DS (ρR(t)), for an evolution long enough to reach the steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' We note that this definition is closely related to the non-Markovianity measure for dissipative chan- nels based on distinguishability [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' However, our defini- tion considers only the dynamics under study and not an op- timization over all the initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' In our system, the qubit-resonator coupling is the factor that introduces non- Markovian behavior into the resonator dynamics due to in- formation backflow from the set of qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' In the next section, we will characterize how a given configuration of the set of qubits affects the behavior of the DnM for such bosonic quan- tum memristor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Dynamical non-Markovianity (DnM) For our first case, we will focus on the resonator interact- ing with one qubit (Jaynes-Cummings model) and interacting 3 with n = 5 qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' We will analyze how the DnM depends on the qubit frequency and coupling strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' It is important to mention that the set of auxiliary qubits is always initialized in the ground state in order not to introduce energy into the resonator since it would undermine the interpretation of the DnM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' First, we consider the initial state |ψ0⟩ = |1R0Q⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 2 (a), we show the DnM of the resonator when varying the coupling strength g/ωR and the frequency ratio ωQ/ωR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' We can see that the DnM is largest when qubit and resonator are in resonance and when g increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Notice that for larger values of g/ωR, the qubit-resonator detuning can yield sig- nificant values of DnM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Figure 2 (b) shows the DnM for the case of five auxiliary qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' We can observe that the effect of enlarging the set of auxiliary qubits is relaxing the resonance condition and increasing the value for the DnM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' This behavior is to be expected since the resonance con- dition allows for maximal information transfer and informa- tion backflow due to the complete Rabi oscillations (in the case of n = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' In addition, the coupling strength g/ωR is related to the speed of the information transfer and informa- tion backflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Then, for small g (slow information transfer), a stronger resonance condition is needed to have information backflow before the system reaches the stationary state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' If g increases the communication between the auxiliary qubits and the bosonic mode is faster, and a more relaxed resonance condition will still have information backflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Increasing the number of qubits, increases the channels for information backflow, which leads to larger values of DnM even at higher detuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Next, we study the scaling of the DnM with the number of qubits under fixed conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' In Figure 3, we study ND as a function of the number of qubits (until n = 8) for different coupling strengths when the resonator is initialized with one excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Figure 3 (a) shows how ND scales with the num- ber of particles (n), with a monotonically increasing behavior reminiscent of a power law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 3 (b), we do a log-log plot of the quantities of Figure 3 (a) which confirms the power-law dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' In all instances the R2 coefficient is larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='995 which means that the scaling of the DnM is very well approximated by ND ∝ nk, (7) (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Dynamical non-Markovainity of the resonator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' (a) one-qubit case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' (b) five-qubit case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' In both cases, the decay rate of qubit and resonator is ΓQ = ΓR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' We consider all qubits are in res- onance (ωQ = ωR), the coupling strength g/ωR ∈ [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='1] and the initial state |ψ0⟩ = |1R0Q⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' (a) (b) (c) (d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' (a) The DnM of the resonator in terms of the number of qubits n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' (b) the log-log plot of DnM and the number of qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' For (a) and (b), we consider three cases g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='1, with the resonant condition ωQ = ωR = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' (c) The exponent k of the power-law dependence as a function of the coupling strength g/ωR, with qubit and resonator in resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' (d) The exponent k of the power-law dependence as a function of the frequency of the qubit ωQ/ωR and a fixed coupling strength g/ωR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' For all cases, we consider decay rates ΓQ = ΓR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='005ωR, the initial state of the resonator |ψ0⟩ = |1R⟩, and all the qubits initialized in the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' The non-Markovainity of the resonator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' (a) one-qubit case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' (b) five-qubit case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Parameters: In both cases, the decaying rate of qubit and resonator is ΓQ = ΓR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='005ωR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' The driving frequency of qubit µQ/ωR ∈ [0, 1], the driving amplitude of the qubit ΩQ/ωR ∈ [0, 1], the qubit frequency ωQ/ωR = 1, the coupling strength g/ωR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' The initial state is |ψ0⟩ = |1R0Q⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' where the exponent k depends on the coupling strength g/ωR and the qubit frequency ωQ/ωR, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 3 (c) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 3 (d), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' We can see that when the DnM is maximal, that is, for large g and ωQ = ωR, the value of k is minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' This is a finite size effect of the auxiliary system as we would expect that if the number of qubits increases to the thermodynamic limit they would induce Markovian dynamics for the resonator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' We have seen that the DnM of the resonator strongly de- pends on the parameters of the set of auxiliary qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' It is then interesting to consider whether we can have dynamic control over the DnM by manipulating the set of auxiliary qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' In 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='5 5 4 wR /03 3 2 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='1 /WR1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='5 14 12 10 8 93 6 4 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='5 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='1 g/wR0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='10 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='8 K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='4 9320 3 g=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='01 g=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='05 15 2 g=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='1 10 0 5 2 3 4 5 6 7 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='0 log(n) ng/wRR 3 Q 35 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='5 3 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='0 μQ12 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='5 8 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='0 0 μQ4 (b) (a) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' The minimum (a) and maximum (b) of DnM for the resonator with a different number of auxiliary qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' In both cases, the decay- ing rate is ΓQ = ΓR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='005ωR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' The frequency of qubit ωQ/ωR = 1, and the initial state is of resonator |ψ0⟩ = |1R⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' The qubits are all initialized in the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' what follows, we apply a driving term in the z-direction to the set of auxiliary qubits in order to dynamically modulate the qubit gap and control the degree of DnM in the evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' The driving is chosen so that it does not introduce energy into the qubits which could excite the resonator and be interpreted as information backflow by the DnM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' This situation is described by the following Hamiltonian ˆH = HTC + ℏΩQ n � j sin(µQt) ˆσz j, (8) where HTC is the Hamiltonian of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' (1), ΩQ and µQ are the amplitude and frequency of the driving over the qubits, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Notice that we consider that each qubit is driven by the same signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' We numerically calculate ND for different values of the driving frequency and amplitude which we show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 4 (a) we show the case of one auxiliary qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Here, there is non-zero DnM over the whole range of parameters, how- ever it is interesting to notice the dark lines that are spanned from near the origin where the DnM is almost completely suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' A similar behavior occurs when we increase the number of qubits, as is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 4 (b) for the five qubits case, where the DnM is suppressed over thin lines in the fre- quency/amplitude plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Although the suppression is not as strong as in the one-qubit case, these lines show significant decrease in the DnM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' This indicates that by modulating either the frequency or the amplitude of the driving we can enhance or suppress the DnM of the resonator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Similarly, we study the DnM in terms of the coupling strength for different number of qubits in the auxiliary system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Here, we will consider two cases, the parameters of the driv- ing that yield the maximum and minimum DnM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 5 (a), we plot the minimal DnM for different coupling strength g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' We can see that up to g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='02 we can essentially completely suppress the non-Markovian behavior by a suitable choice of driving parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Increasing the number of qubits decreases the necessary value of coupling strength that allows for com- pletely supressed DnM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' On the other hand, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 4 (b), we plot the maximal DnM for different coupling strength g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Here we can see that ND has linear dependence on the coupling strength except for a small range around zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' From these FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Transition from non-Markovian to Markovian dynamics by changing the driving frequency over the auxiliary qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Parameters: the driving amplitude of qubit is Ωq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='5, the number qubit n = 1, decaying rate is ΓQ = ΓR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='005, frequency ωQ = ωR, coupling strength g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' results we have that provided we choose a suitable value of the coupling strength, we can switch between Markovian and non-Markovian dynamics for the resonator by just controlling the auxiliary set of qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 6 we show how we can dynamically switch the non- Markovian behavior on and off by just changing the driving frequency of the qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Here, we plot the trace distance as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' At the start of the evolution, we choose a driving frequency that yields maximum non-Markovianity (µQ = ΩR), later at t = 350ω−1 R we switch the driving fre- quency to µQ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='75ωR which yields minimum DnM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' As can be seen in the figure, at t = 350ω−1 R , the trace distance switches from non-monotonic to monotonically decreasing behavior, which characterizes Markovian evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Time-dependent decay rate The observed memory effect can be understood as the sys- tem effectively interacting with an environment with a time- dependent decay rate, which becomes negative during some time intervals, favoring the information back-flow [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' To understand this statement, consider a resonator, with state ˜ρ, undergoing dissipative dynamics with a time-dependent decay rate and without any interaction with an auxiliary system, the system is then described by the following master equation ˙˜ρ(t) = −i[H, ˜ρ] + Γ(t) � a˜ρa† − 1 2{˜ρ, a†a} � (9) where H = ℏωRa†a, and Γ(t) is a time-dependent decay rate that can be negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Here, ˜ρ represents the state of the res- onator undergoing dynamics as described above, and is dif- ferent from ρR which is the reduced state of the resonator as described by Hamiltonian (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' For Γ(t) > 0, the energy of the resonator dissipates to the environment, meaning that the information in the resonator is continuously lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Meanwhile, for Γ(t) < 0 there is energy entering the resonator, giving place to information back-flow and therefore to a non-Markovian process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='5 n=1 n=2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='0 n=3 n=4 M 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='5 n=5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='10 g-——— n=1 n=2 n=3 3 n=4 N n=5 2 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='10 gg/wRg/wR1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='0 μ3=1 u1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='8 distance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='6 Trace 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='0 0 200 400 600 800 1000 t (unit of w)5 (a) (b) (c) (d) (d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Trace distance of the resonator and the DnM under dif- ferent driving frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Top, (a) the blue line, the frequency of driving µ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='419, the green line, the resonator’s decaying rate is constant Γ(t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' (b) the blue line, the frequency of driving µ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='20, the green line, the resonator’s decaying rate is Γ1(t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='05 (sin(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='023t) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='09).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' (c) the blue line, the frequency of driving µ3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' the green line, the resonator’s decaying rate is Γ1(t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='25 (sin(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='079t) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Bottom, (d) the DnM of resonator in different driving frequency µQ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Parameters: the number qubit n = 1, decaying rate is ΓQ = ΓR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='005, frequency ωQ = ωR, coupling strength g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='05, the driving amplitude ΩQ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='5ωR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' We consider the time-dependent decay rate parametrized as Γ1(t) = A (sin(Bt) + C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Notice that the master equation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' (9) has the same steady state as that of our original system in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Therefore, for a given dynamics induced by the set of auxiliary qubits, we can find the closest non- Markovian dynamics corresponding to negative decay rate by finding A, B, and C that minimize the difference of trace dis- tance |D(ρR(t)) − D(˜ρ(t))|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 7, we plot DS (ρR(t)) and DS (˜ρopt(t)) where ρR(t) is for one qubit case and ˜ρopt(t) is the resonator evolved with the optimal parameters for the decay rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' We consider 3 cases, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 7 (a), the effective decay rate is time-independent Γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='005 corresponding to Markovian behavior;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' whereas for time-dependent decay rate we have in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 7 (b) where Γ(t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='05[sin(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='023t) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='09] and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 7 (c) where Γ(t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='25[sin(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='079t) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Finally, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 7 (d) shows the DnM as a function of the qubit-driving frequency where it displays the qubit driving frequency corresponding to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 7 (a) - (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' We can see that for time-dependent de- cay the behavior of both trace distance is very similar, which means that the set of auxiliary qubits is inducing highly non- Markovian dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Finally, it is interesting to study how this simulated time- dependent decay rate can affect the response of the cavity over external driving, in order to control the memristive properties of the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Bosonic quantum memristor One interesting application of our results is to induce mem- ristive behavior into the bosonic mode, which can be tuned by FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Memristive behavior, the green line shows the dynamics when the auxiliary qubits are not driven and off-resonant and the blue curve is when we add a driving over the auxiliary qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' (a) larger- DnM case, the number of qubits n = 1, driving frequency µc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' (b) medium-DnM case, the number of qubits n = 1, driving frequency µc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' (c) larger-DnM case, the number of qubits n = 5, driving frequency µc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' (d) medium-DnM case, the number of qubits n = 5, driving frequency µc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Parameters: the driving amplitude of qubit is Ωq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='5, decaying rate is ΓQ = ΓR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='005, frequency ωQ = ωR, coupling strength g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='05, the driving amplitude of cavity Ωc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='2, frequency µc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' the set of auxiliary qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' [34], it was shown that a kind of time-dependent decay rate produces a quantum mem- ristor, which could be reached in a superconducting circuits platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Later, in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' [35], a memristive dynamics was obtained in a quantum computer by the simulation of a non- Markovian bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' In this line, we analyze the response of the cavity under an external driving, obtaining a Hamiltonian of the form: ˆH = HTC + ℏΩQ n � j sin(µQt) ˆσz j + F(t)(a + a†).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' (10) Now, if we define the variables I = −⟨i(a − a†)⟩ and O = ⟨ ˙N⟩ + α⟨N⟩, with α a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' If we consider α = Γc, it is the natural decay rate of the cavity, we have that O = F(t)I + G(t), (11) for more details see appendix ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='. The function G(t) depends on the DnM of the system, which means that we can control the memristive relation O = F(t)I, (12) by controlling the value of G(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Now, if we choose F(t) = Ωc[1−sin(cos µct)], it is possible to obtain the typical pinched hysteresis loop that characterizes a quantum memristor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' This situation is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 8, where we obtain the pinched hysteresis loop (green curve), in a similar way that it is obtained for previous proposals of quantum memris- tos [12, 14, 34] as a signature of memristive behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' It is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='0 Qubit environment Time-dependent decay distance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='6 Irace 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='0 0 250 500 t (unit of wrl1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='0 Qubit environment Time-dependent decay distance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='6 Trace 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='0 0 250 500 750 1000 1250 1500 1750 2000 t (unit of wRl1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='0 Qubit environment Time-dependent decay distance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='6 Irace 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='0 0 250 t (unit of wrl)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='0 μ3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='5 U2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='5 μ1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='9 1 μQ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='20 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='10 <1) <1)6 interesting to note that such bosonic quantum memristor dy- namics appear when the auxiliary qubits are not driven and off-resonant with the cavity, which means that in an effective way, the qubits are decoupled from the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' In contrast, we can observe that when we drive the qubits, the memristive be- havior can be destroyed for different cases, obtaining a way to go from memristive dynamics to non-memristive dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' It means that we also can control the memory properties induced by the decay rate in the cavity, which can be helpful in neu- romorphic computing, considering that the proposed system can be implemented in many platforms like trapped ions, op- tical devices, and superconducting circuits, among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' We also need to remark that our proposal can work as a switchable bosonic quantum memristor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' This suggests that our formalism allows implementing devices with controllable and switchable memory properties, only by tuning the energy gap of auxil- iary qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' This proposal opens the door for the experimental implementation of memristive devices, providing a general, platform-free, and scalable model for the next generation of neuromorphic quantum computing technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' CONCLUSIONS We consider a cavity coupled to a set of auxiliary qubits, which induce a controllable dynamical non-Markovianity (DnM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' We show that by dynamical tuning of the energy gap of the auxiliary qubits, we can go from high DnM to low DnM, which can be considered as an effective time-dependent decay rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' We also show that the induced DnM in the cavity mode follows a power-law dependence with the number of auxil- iary qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Finally, we show as an application that we can define memristive dynamics in the bosonic mode, which can be switched off by controlling the energy gap of the auxil- iary qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' This means that we can control the memristive dynamics in the cavity by external control over the auxiliary system, obtaining a switchable bosonic quantum memristor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' These results provide a general protocol to obtain controllable bosonic quantum memristors which can be useful in neuro- morphic quantum computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' This proposal is experimentally feasible since it only uses a bosonic mode coupled to a set of qubits, a ubiquitous setup in hardware platforms like trapped ions, superconducting circuit quantum electrodynamics, and atomic devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Appendix A: The derivative of expectation the number of photons d⟨ˆa†ˆa⟩ dt = Tr � − i ℏ �H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' ρ� ˆa†ˆa � + Tr � L(ρ)a†a � = S1 + S2 (A1) we set ℏ = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' S1 and S2 is S1 = −iTr �������� ��������ωca†a + ωq n � j=1 σz j + n � j=1 g(a†σ j + aσ+ j ) + n � j=1 Ωq sin(µqt)σz j + F(t)(a + a†),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' ρ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='�������� ˆa†ˆa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='�������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='= −iTr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='�������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='�������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='j=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='g(a†σj + aσ+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='j )ρ − ρ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='j=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='g(a†σj + aσ+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='j ) + F(t)(a + a†)ρ − ρF(t)(a + a†) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='�������� ˆa†ˆa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='�������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='= −iTr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='�������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='j=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='σjρa†aa† − σ+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='j ρa†aa − ρa†σ ja†a − ρaσ+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='j a†a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='+ F(t)ρ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='ρa†a + ρa†aa†a − ρaa†a − ρa†a†a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='��������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='= −iTr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='�������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='j=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='gρ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='σ+ � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='a†a − aa†� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='a + σa† � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='aa† − a†a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='+ F(t)ρ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='a†a − aa†� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='a + a† � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='aa† − a†a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='���������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='= −iTr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='�������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='j=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='gρ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='−σ+a + σa†� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='+ Ωc sin(νct)ρ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='−a + a†��������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='� n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='j=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='ig ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='−σ+a + σa†�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='+ F(t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='i(−a + a†) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='j=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='−σ+a + σa†�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='− F(t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='� ˆP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='(A2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='S2 = Tr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='Γc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='aρa† − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='a†aρ + ρa†a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='a†a + Γq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='σρσ+ − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='�σ+a + ρσ+σ�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='a†a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='= Tr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='Γcρ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='a†a†aa − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='2a†aa†a − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='2a†aa†a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='= Tr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='Γcρ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='a†(aa† − 1)a − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='2a†aa†a − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='2a†aa†a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='= −Γc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='(a†a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='(A3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='so the derivative of expectation photon number is ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='d⟨ˆa†ˆa⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='dt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='j=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='−σ+a + σa†�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='− F(t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='� ˆP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='− Γc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='(a†a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='(A4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='The input and output is ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='⟨ˆI⟩ = −⟨ ˆP⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='(A5) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='⟨ ˆO⟩ = d⟨ˆa†ˆa⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='dt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='+ Γc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='(a†a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='(A6) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='and the relation between input and output ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='ˆO = F(t)ˆI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='(A7) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='where F(t) is Ωc ((1 − sin(cos µct)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' due to the auxiliary qubits being one part of the environment,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' we discard the second term of the output,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' now the output is ⟨ ˆO⟩ = d⟨ˆa†ˆa⟩ dt + Γc � (a†a) � + G(t) (A8) with G(t) = n� j=1 g � i � −σ+a + σa†�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' As G(t) depends on the interaction between the qubits and the cavity, it can be controlled by the external driving over the set of auxiliary qubits, which means that G(t) will be close to zero then the qubits are off of resonance with the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' [1] Li, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Guo, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Pillo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Non-Markovian quantum dynamics: What does it mean?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' EPL 2019, 127, 50001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' [2] Li, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Guo, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Pillo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Non-Markovian quantum dynamics: What is it good for?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' EPL 2020, 128, 30001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' [3] Lee, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Cheng, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Fleming, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Coherence Dynamics in Photosynthesis: Protein Protection of Excitonic Coherence Science 2007, 316, 1462.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' [4] Chin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Datta, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Caruso, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Huelga, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Plenio, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Noise-assisted energy transfer in quantum networks and light-harvesting complexes?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' New J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 2010, 12, 065002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' [5] Fleming, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Huelga, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Plenio, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Focus on quantum effects and noise in biomolecules New J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 2011, 13, 115002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' [6] Alex, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Susana, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Martin, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Quantum Metrology in Non-Markovian Environments Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 2012, 109, 233601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' [7] Nicol´as, M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Mart´ı, L, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Diego, W Quantum metrology in a non-Markovian quantum evolution Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 2020, 102, 022618.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' [8] Sweke, R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Sanz, M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Sinayskiy, I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Petruccione, F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Solano, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Digital quantum simulation of many-body non-Markovian dynamics Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' A 2016, 94, 022317.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' [9] Julio, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Markus, M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Philipp, S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Daniel, N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Thomas, M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Michael, C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Markus, H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Christian F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Roos;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Peter, Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Rainer, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' An open-system quantum simulator with trapped ions Nature 2011, 470, 486–491.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' [10] Pfeiffer, P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Egusquiza, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Di Ventra, M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Sanz, M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Solano, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Quantum memristors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Sci Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 2016, 6, 29507.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' [11] Michele, S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Joshua, M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Simone, P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Michael, A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Francesco, M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Andrea, C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Francesco, C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Roberto, O;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Philip, Walther Experimental photonic quantum memristor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 2022, 16, 318–323.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' [12] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Sanz, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Lamata, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Solano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Quantum memristors in quantum photonics APL Photonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 2018, 3, 080801 [13] Shevchenko, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Pershin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Nori, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Qubit-Based Memcapacitors and Meminductors Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 2016, 6, 014006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' [14] Norambuena, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Torres, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Di Ventra, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Coto, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Polariton-Based Quantum Memristors Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 2022, 17, 024056.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 8 [15] Sweke, R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Sanz, M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Sinayskiy, I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Petruccione, F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Solano, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Digital quantum simulation of many-body non-Markovian dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' A 2016, 94, 022317.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' [16] Pershin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Di Ventra, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Neuromorphic quantum computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' IEEE 2012, 100, 2071.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' [17] Pehle, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Wetterich, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Digital quantum simulation of many-body non-Markovian dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' E 2022, 106, 045311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' [18] Xu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Krisnanda, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Verstraelen, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Liew, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Ghosh, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Superpolynomial quantum enhancement in polaritonic neuromorphic computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' B 2021, 103, 195302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' [19] de Vega, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Aloso, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Dynamics of non-Markovian open quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 2017, 89, 015001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' [20] Rivas, ´A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' F Huelga, S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' B Plenio, M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Quantum non-Markovianity: characterization, quantification and detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 2014, 77, 094001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' [21] Breuer, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Foundations and measures of quantum non-Markovianity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' B: At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 2012, 45, 154001 [22] Breuer, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Laine, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Piilo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Measure for the Degree of Non-Markovian Behavior of Quantum Processes in Open Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 2009, 103, 210401.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' [23] Rivas, ´A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Huelga, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Plenio, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Entanglement and Non-Markovianity of Quantum Evolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 2010, 105, 050403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' [24] Luchnikov, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Vintskevich, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Grigoriev, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Filippov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Machine Learning Non-Markovian Quantum Dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 2020, 124, 140502.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' [25] M Bastidas, V;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Kyaw, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Tangpanitanon, J;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Romero, G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Kwek, L-C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' G Angelakis, D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Floquet stroboscopic divisibility in non- Markovian dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' New J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 2018, 20, 093004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' [26] Liu, B-H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Li, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Huang, Y-F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Li, C-F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Guo, G-C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Laine, E-M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Breuer, H-P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Piilo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Experimental control of the transition from Markovian to non-Markovian dynamics of open quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 2011, 7, 931–934.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' [27] Bernardes, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Cuevas, A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Orieux, A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Monken, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Mataloni, P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Sciarrino, F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Santos M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Experimental observation of weak non-Markovianity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Sci Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 2015, 5, 17520.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' [28] Li, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='-W;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Mei, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='-X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Wu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Cai, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Wang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Yao, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Zhou, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Duan, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='-M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Observation of Non-Markovian Spin Dynamics in a Jaynes-Cummings-Hubbard Model using a Trapped-Ion Quantum Simulator Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 2022, 129, 140501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' [29] Garc´ıa-P´erez, G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Rossi, Matteo A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Maniscalco, S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' IBM Q Experience as a versatile experimental testbed for simulating open quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' npj Quantum Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 2020, 6, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' [30] Chen, X-Y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Zhang, N-N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' He, W-T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Kong, X-Y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Tao, M-J;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Deng, F-G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Ai, Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Long, G-L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Global correlation and local information flows in controllable non-Markovian open quantum dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' npj Quantum Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 2022, 8, 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' [31] Retzker, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Solano, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' and Reznik, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Tavis-Cummings model and collective multiqubit entanglement in trapped ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 2007, 75, 022312 [32] J¨ager, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Schmit, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Morigi, G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Holland, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Betzholz, R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Lindblad Master Equations for Quantum Systems Coupled to Dissipative Bosonic Modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 2022, 129, 063601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' [33] Hu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Lian, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Geometric quantum discord and non-Markovianity of structured reservoirs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 2015, 362, 795.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' [34] Salmiletho, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Deppe, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Di Ventra, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Sanz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Solano, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Quantum Memristors with Superconducting Circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 2017, 7, 42044.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' [35] Guo, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Albarr´an-Arriagada, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Alaeian, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Solano, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Barrios, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Quantum Memristors with Quantum Computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} +page_content=' 2022, 18, 024082.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFQT4oBgHgl3EQfmTaL/content/2301.13365v1.pdf'} diff --git a/lNE3T4oBgHgl3EQfKAkA/content/tmp_files/2301.04348v1.pdf.txt b/lNE3T4oBgHgl3EQfKAkA/content/tmp_files/2301.04348v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..df22339306f183548e8e44b294a8d4d43ac0c39d --- /dev/null +++ b/lNE3T4oBgHgl3EQfKAkA/content/tmp_files/2301.04348v1.pdf.txt @@ -0,0 +1,810 @@ +arXiv:2301.04348v1 [eess.IV] 11 Jan 2023 +On the Influence of Clipping in Lossless Predictive +and Wavelet Coding of Noisy Images +Wolfgang Schnurrer, Jürgen Seiler, Michael Schöberl, and André Kaup +Multimedia Communications and Signal Processing +University of Erlangen-Nuremberg, Cauerstr. 7, 91058 Erlangen, Germany +Email: {schnurrer, seiler, schoeberl, kaup}@lnt.de +Abstract—Especially in lossless image coding the obtainable +compression ratio strongly depends on the amount of noise +included in the data as all noise has to be coded, too. Different +approaches exist for lossless image coding. We analyze the +compression performance of three kinds of approaches, namely +direct entropy, predictive and wavelet-based coding. The results +from our theoretical model are compared to simulated results +from standard algorithms that base on the three approaches. As +long as no clipping occurs with increasing noise more bits are +needed for lossless compression. We will show that for very noisy +signals it is more advantageous to directly use an entropy coder +without advanced preprocessing steps. +I. INTRODUCTION +Lossless compression is an important task in all areas where +any modification of information is not allowed or at least not +acceptable. Examples are among many others measurements +for quality assurance, archiving, surveillance, conservation of +evidence material or medical data. Noise that is contained in +the data has also to be coded in this case. In the medical +environment lossy compression is often not acceptable as the +correct diagnosis cannot be guaranteed for the lossy coded +images. But medical images contain a lot of noise because on +the one hand radiation has to be kept low to reduce the risks +for the patients and on the other hand the acquisition time is +kept short to avoid motion artifacts. +Several different approaches exist for lossless coding. We +observed that the performance of the different approaches +varies significantly when the data contains different amounts +of noise. Prediction-based methods like lossless JPEG [3], +[4] and wavelet-based methods like JPEG 2000 [5], [6] are +advantageous to code the structural information but become +less effective when the images contain a lot of noise. We will +provide a theoretical analysis on the behavior of the energy +of the noise when a wavelet transform is applied. We will +compare this to direct entropy coding and a predictive coding +scheme. +Figure 1 shows a block diagram of our signal model. Noise +n [k] with a standard deviation σ is added to the signal s [k] +that contains the structural information. The signal f [k] results +from quantizing the noisy signal. One of the methods, i.e., +direct, predictive and wavelet, is then applied in the gray +box and the output is analyzed. In our study we assume that +clipping occurs mainly when the additive noise leads to values +that exceed the limits of the quantizer Q. +PSfrag replacements +σ +f[k] +n[k] +s[k] +Q +H ++ +noise +direct +wavelet +predictive +Figure 1. +Signal model +0 +50 +100 +150 +200 +250 +0 +1000 +2000 +3000 +4000 +5000 +code value +frequency +noisy +original +(a) ↑ / (b) ↓ +(c) +Figure 2. +Two detail images from the Big Buck Bunny sequence (a), (b). (a) +is shown with additive noise with σ = 30, (b) is shown original. (c) shows +histograms of the image (a) with additive noise (solid red) and original of +image (a), i.e., without additive noise (dashed blue) +In Section 2 we present the analysis. The description of our +simulation and results are given in Section 3. Section 4 will +conclude this study. +II. THEORETICAL ANALYSIS OF NOISE IMPACT +We compare three different approaches for coding a signal +without loss. The first method is called direct as the samples +are directly entropy coded without any preprocessing. The +second method is called predictive. Before entropy coding it +is possible to subtract a prediction where the predictor for the +current sample is computed from already decoded samples. +The various predictors differ by the number and the weight of +the incorporated samples. We analyze the prediction from one +previous sample. The combination of more samples leads to +a noise variance reduction due to averaging. But the overall +noise variance of the predictor will stay greater than zero. The +third class of methods in our analysis is based on the wavelet + +transform. Instead of subtracting a prediction, the samples +are transformed and the coefficients from the sub-bands are +then entropy coded. In our analysis we compare two different +wavelets, the Haar wavelet and the LeGall 5/3 wavelet. +We assume that the input signal f [k] consists of the +structural information s [k] with additive noise n [k] after +quantization as shown in Figure 1. For simplicity we show +the analysis for the one dimensional case only. At first we +neglect the structural part and quantization step. We analyze +and compare the output of the different methods. We then show +how to calculate the entropy and finally add the structural +information of the signal and the quantization step to our +modeling. +A. Noise Variance for Different Coding Methods +For our analysis we are mainly interested in the noise +part. We consider the structural signal s [k] = 0 in this +subsection and assume Gaussian noise n [k] with zero mean +and a variance σ2. We analyze the influence of the different +methods on the noise by comparing the noise variance at their +output. +The direct method does not apply preprocessing and so the +error distribution does not change and stays equal to +E +� +n2 +direct[k] +� += σ2. +(1) +The second method based on prediction uses one previous +sample for the calculation of the predictor p[k] = f[k − 1] +and will be called predictive. The residuum of the predictor +r[k] = f[k]− p [k] is then coded and the variance of the noise +in the resulting sample doubles to +E +� +n2 +p[k] +� += E +� +(n[k] − n[k − 1])2� += σ2 + σ2 = 2σ2. (2) +Wavelet transforms consist of a filter pair for the compu- +tation of the high HP- and the low LP-band. For the Haar +wavelet, the samples of the HPHaar- and the LPHaar-band are +calculated to HPHaar [k] = f [2k] − f [2k − 1] and LPHaar [k] = +� 1 +2 (f [2k] − f [2k − 1]) +� +. The rounding operation is due to +the integer wavelet transform [1] for lossless coding. The error +variance for the coefficients of the HPHaar band calculates to +E +� +n2 +HPHaar[k] +� += E +� +(n[2k] − n[2k − 1])2� += 2σ2. +(3) +For the LPHaar-band the error variance is equal to +E +� +n2 +LPHaar[k] +� += 1 +4σ2 + 1 +4σ2 = 1 +2σ2. +(4) +For the LeGall 5/3 wavelet the samples of the HPLeGall- and +the LPLeGall- band are calculated to +HPLeGall [k] = f [2k] − +�1 +2 (f [2k − 1] + f [2k + 1]) +� +(5) +and +LPLeGall[k]=f[2k−1]+ +�1 +4(HPLeGall[k]+HPLeGall[k−1]) +� +(6) +as given in [1]. The error variance for the coefficients of the +HPLeGall-band calculates to +E +� +n2 +HPLeGall[k] +� += 1 +4σ2 + σ2 + 1 +4σ2 = 3 +2σ2. +(7) +For the LPLeGall-band the error variance needs to be calculated +from the filter representation of the wavelet to preserve the +signal independence. The error variance is then equal to +E +� +n2 +LPLeGall[k] +� += 12 + 22 + 62 + 22 + 12 +64 +σ2 = 46 +64σ2. +(8) +The combined error variance cannot be computed by aver- +aging the two values E +� +n2 +LP +� +and E +� +n2 +HP +� +from the subbands. +The two subbands are calculated from the same signal values +so the independence assumption does not hold anymore. In the +next subsection we will show how the values can be combined. +B. From Noise Variance to Bits +For the wavelet approach a combined variance cannot be +computed by simply averaging the variances from the HP- +and LP-bands. In order to get an estimation for the number of +bits needed for coding the signal processed by the different +methods with an optimum entropy coder we use the entropy +H = − +� +i +pilog2 (pi) [bit per sample] . +(9) +The probabilities pi result from integrals over the probability +density function with +pi = +ˆ i+ 1 +2 +i− 1 +2 +1 +√ +2πσ2 e− 1 +2( x +σ) +2 +dx. +(10) +Without considering clipping on the upper and lower end +of the co-domain i ∈ Z holds. For this case (9) can be +solved analytically and be approximated by a formula that +is dependent on the standard deviation σ of the noise +Hdirect (σ) ≈ log2 +� +σ +√ +2πe +� +. +(11) +Now we can calculate the entropy for the different methods +by evaluating (11) for the variances derived in the previous +subsection. +For the predictive scheme the resulting entropy is given by +Hpredictive (σ) ≈ H( +√ +2σ) = 1 +2 bit + Hdirect (σ) . +(12) +After the wavelet transform, the signal is represented by +coefficients in the high and the low band in a ratio of 1:1. +So the overall entropy can be computed by summing up the +entropy of the high and the low band containing half the +samples each. For the Haar wavelet this results in +HHaar (σ) +≈ +1 +2 +� +H +�� +1 +2σ +� ++ H +�√ +2σ +�� +(13) += +log2 +� +σ +√ +2πe +� += Hdirect (σ) +and for the LeGall 5/3 wavelet in +HLeGall (σ) +≈ +1 +2 +� +H +�� +3 +2σ +� ++ H +�� +46 +64σ +�� += +0.027 bit + Hdirect (σ) . +(14) + +Even though the value range doubles for the HHaar-band +that contains half of all coefficients, the entropy of the whole +signal stays the same for the Haar wavelet. Comparing the +two wavelets, the entropy increases slightly for the LeGall 5/3 +wavelet by an offset of 0.027 bit per sample. +Comparing the entropy of the different methods we can +conclude that without clipping all the methods need the same +rate for coding the noise up to an offset that does not depend +on σ. While this offset is very small for the wavelet-based +methods for the predictive scheme that uses one previous +sample the coding of the noise is more expensive by 0.5 bit +per sample. +C. The Influence of Clipping +We now extend the modeling by the structural information +and the quantization. As we are interested in lossless coding +we are limited to integer values in order to avoid rounding +errors. To analyze the impact of clipping to zero mean noise +we use a structural signal s [k] with a constant value in the +center of the co-domain µ = +� +28−1 +2 +� +. For this we extend (9) +by a limitation of i. The probabilities pi, i ∈ 0..255 for a co- +domain of 8 bit, of the signal result from integrating of the +probability density function fN (n) of the noise over the bin +size. The calculation of the probabilities pi is given by +pi = + + + + + +´ −127.5 +−∞ +fN (ν) dν +for i = 0 +´ ∞ +126.5 fN (ν) dν +for i = 255 +´ i+0.5 +i−0.5 fN (ν) dν +for 0 < i < 254 +. +(15) +Values outside the co-domain are clipped to the minimum 0 +and maximum 255 code value respectively. This leads to peaks +in the probability distribution of the signal at the borders of +the co-domain as in Figure 2 (c). +In our analysis we assume statistically independent Gaus- +sian noise, so the probability density function of the noise +equals fN (n) = +1 +√ +2πσ2 e− 1 +2( x +σ) +2 +. As we assume a signal with +a constant code value only, the probability distribution of the +signal equals pi. The probability distribution of the input signal +can now be combined according to the presented methods. +Then the entropy can be calculated by inserting the values +of the probabilities from 15 into the entropy formula (9). For +the wavelet-based methods the entropy for the HP-band and +for the LP-band have to be calculated separately. The resulting +values are then summed up by taking into account that each +band contains half of the samples. +D. Theoretical Results +The resulting curves are plotted in Figure 3. The results +strongly depend on the µ chosen for the structural part of +the signal. The value of µ in the center of the co-domain is +the ideal case because for a smaller or greater value of µ +clipping is introduced at smaller values of σ. The plot can +be divided into four areas along the abscissa denoted by (a)- +(d). In area (a) the noise is quantized to zero and thus has +no influence on the entropy. When σ gets bigger and only a +few samples are affected by noise, that is not quantized to +100 +102 +0 +2 +4 +6 +8 +noise standard deviation σ +entropy H in bit per sample +* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * +* +* +* +* +* +* +* +* +* * * +bC +bC +bC +bC +bC +bC +bC +bC +bC +bC +bC +bC +bC +bC +bC +bC +bC +bC +bC +bC +bC +bC +bC +bC +bC +bC +bC +bC +bC +bC +bC +bC +bC +bC +bC +bC +bC +bC +bC +bC +bC +bC +bC +| | | | | | +| +| +| +| | +| +| +| +| +| +| +| +| +| +| +| +| +| +| +| +| +| +| +| +| | | | +| +| +| +| +| +| +| +| +| +uT +uT +uT +uT +uT +uT +uT +uT +uT +uT +uT +uT +uT +uT +uT +uT +uT +uT +uT +uT +uT +uT +uT +uT +uT +uT +uT +uT +uT +uT +uT +uT +uT +uT +uT +uT +uT +uT +uT +uT +uT +uT +uT +direct +predictive +wavelet Haar +wavelet LeGall +* +bC +| +uT +(d) +(c) +(b) +(a) +Figure 3. +Theoretical results - entropy in bit per sample plotted over the +standard deviation σ of the noise on the abscissa in a lin-log plot +zero, the curves have a different slope as area (b) shows. The +reason for the steeper slope of the predictive and wavelet-based +methods is the combination of several samples. As soon as the +noise component in one sample is above quantization, several +coefficients are affected and cause an increase of the entropy. +For bigger values of σ more samples are affected by noise. +As long as the noise is small enough such that no clipping is +occurring, the curves run in parallel, as shown in area (c). The +offsets correspond to the derivation in the previous subsection. +Clipping begins to occur for larger values of the standard +deviation σ as shown in area (d). Direct entropy coding can +exploit the rising number of clipped values better resulting in +a lower entropy. The entropy of the predictive scheme and the +wavelet-based method drops slower due to the combination of +clipped samples with unclipped ones. The LeGall 5/3 wavelet +is even more sensitive to this than the Haar wavelet because +of the greater filter length. +III. SIMULATION RESULTS +In our simulation we used different images and added +Gaussian noise with increasing standard deviation σ before +coding them with different algorithms. To validate our model +we first used an image of size 512x512 pixels with a constant +code value of 128. +In order to prove that our model also fits for images we +used details from the Big Buck Bunny sequence. The reason +for the choice of artificial images is the difficulty of perfectly +denoising real images [7]. The chosen images are computer +generated and thus do not contain noise in the beginning. We +cut images of 512x512 pixels as shown in Figure 2 (a) and +(b). For illustration we added Gaussian noise with a standard +deviation of σ = 30 to the first detail shown in Figure 2 (a). +The second detail in Figure 2 (b) is shown in original, i.e., +without additive noise. In our simulation we used the green +color channel only. + +10−2 +100 +102 +0 +100 +200 +300 +noise standard deviation σ +file size in kbyte +* +****** ********* +* +* +*************************************************************** +bC +bC bC bC bC bC bC +bC +bC +bC +bC bC bC bC bC bC +bC +bC +bC +bC bC +bC +bC bC bC bC bC +bC bCbC bC bC bCbC bC bC bC bC bC bC bC bC bC bC bCbCbCbCbCbCbCbCbCbCbCbCbCbCbCbCbCbC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC +uT +uT uT uT uT uT uT +uT +uT +uT +uT uT uT uT uT uT +uT +uT +uT +uT uT +uT +uT uT uT uT uT +uT uTuT uT uT uTuT uT uT uT uT uT uT uT uT uT uT uTuTuTuTuTuTuTuTuTuTuTuTuTuTuTuTuTuT uT uT uT uT uT uT uT uT uT uT uT uT uT uT uT uT uT uT uT +BZ2 +llJP +JP2k +* +bC +uT +10−2 +100 +102 +0 +100 +200 +300 +noise standard deviation σ +file size in kbyte +* +****** ********* +* +* +*************************************************************** +bC +bC bC bC bC bC bC +bC +bC bC bC bC bC bC bC bC +bC +bC +bC bC bC +bC bC bC bC bC bC bC bCbC bC bC bCbC bC bC bC bC bC bC bC bC bC bC bCbCbCbCbCbCbCbCbCbCbCbCbCbCbCbCbCbC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC +uT +uT uT uT uT uT uT +uT +uT +uT uT uT uT uT uT uT +uT +uT +uT +uT uT +uT +uT uT uT uT uT uT uTuT uT uT uTuT uT uT uT uT uT uT uT uT uT uT uTuTuTuTuTuTuTuTuTuTuTuTuTuTuTuTuTuT uT uT uT uT uT uT uT uT uT uT uT uT uT uT uT uT uT uT uT +BZ2 +llJP +JP2k +* +bC +uT +10−2 +100 +102 +0 +100 +200 +300 +noise standard deviation σ +file size in kbyte +* +****** ********* * * *************************************************************** +bC +bC bC bC bC bC bC +bC +bC bC bC bC bC bC bC bC +bC +bC +bC bC bC +bC +bC bC bC bC bC +bC bCbC bC bC bCbC bC bC bC bC bC bC bC bC bC bC bCbCbCbCbCbCbCbCbCbCbCbCbCbCbCbCbCbC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC +uT +uT uT uT uT uT uT +uT +uT uT uT uT uT uT uT uT +uT +uT +uT +uT uT +uT +uT uT uT uT uT +uT uTuT uT uT uTuT uT uT uT uT uT uT uT uT uT uT uTuTuTuTuTuTuTuTuTuTuTuTuTuTuTuTuTuT uT uT uT uT uT uT uT uT uT uT uT uT uT uT uT uT uT uT uT +BZ2 +llJP +JP2k +* +bC +uT +(a) +(b) +(c) +Figure 4. +Simulation results show the file size in kbytes over the standard deviation σ of the added noise in lin-log plots. (a) shows the results for a constant +signal equal to 128. In (b) the results for the first test image (Figure 2 (a)) are shown. (c) shows the results for the second test image (Figure 2 (b)) +For direct entropy coding we used bzip2 [2] in version 1.0.4. +No additional parameters were given when calling the pro- +gram. For predictive coding we used lossless JPEG [3], [4]. +For wavelet-based coding we used JPEG 2000 [5], [6] with +4 decomposition steps. As long as the number of decom- +position steps is larger than 3 this parameter has not much +influence on the file size. +The results are shown in Figure 4 where the file size of +the compressed images is plotted in kbyte over the standard +deviation σ of the noise in lin-log plots. +In Figure 4 (a) the results are shown for the image with a +constant code value of 128 and additive noise. By comparing +Figure 4 (a) with the theoretical results in Figure 3 it can +be seen that our model matches with the results from the +simulation up to the offset of the lossless JPEG method for +small values of σ. The corresponding curves have the same +shape and the plot in Figure 4 (a) can also be divided into the +four areas indicated in Figure 3. +Quantitative statements cannot be given as the results +strongly depend on the structural information in the input +signal. +The offsets in Figure 4 (b) and (c) come from the structural +information. The three methods need a different amount of +bits to code the structural information. The two plots show +clearly that as long as the noise part is small enough advanced +methods as a predictive scheme and a wavelet decomposition +are advantageous because they are more capable to reduce the +redundancy in the structural information part of the signal. +Clipping in the structural information, e.g. due to wrong +exposure, often affects bigger areas of an image and thus is +beneficial for predictive and wavelet-based coding. The prob- +lem are isolated clipped values introduced by noise because +their combination with other unclipped samples lead to a +higher entropy. +The results in Figure 3 and Figure 4 (a) show that it is +advantageous to use direct entropy coding for noise. In [7] +several state of the art denoising algorithms are compared to +a theoretical bound. The result is that there is still room for +improvements. A separation of the noise from the structural +information leads to an increase of the encoder complexity. +The decoder has to decode both parts and add the noise to +the structural part. Compared with wavelet-based coding gains +can be achieved for small values of σ as shown in area (b) of +Figure 3. +Another result is that for very noisy images when clipping +is introduced by noise it is more advantageous to directly use +an entropy coder without any signal decomposition. These +results show that our model also fits for images with structural +information. +IV. CONCLUSION +In this paper we analyzed the impact of clipping to lossless +compression of noisy images. We derived an analytical de- +scription that models the behavior of different coding methods. +So the effects shown are general properties of the methods and +not of a special implementation. Simulation results support +our model. The results show that in general for noisy data it +is advantageous to code the noise and the signal separately. +Furthermore, the results show that for the case that clipping +is introduced by noise it is more advantageous to directly use +an entropy coder without advanced preprocessing steps. +ACKNOWLEDGMENT +We gratefully acknowledge that this work has been sup- +ported by the Deutsche Forschungsgemeinschaft (DFG) under +contract number KA 926/4-1. +REFERENCES +[1] A. Calderbank, I. Daubechies, W. Sweldens, and B.-L. Yeo, “Lossless +image compression using integer to integer wavelet transforms,” in Int. +Conf. on Image Processing (ICIP), vol. 1, pp. 596–599, oct. 1997. +[2] J. Seward, “bzip2 and libbzip2: A program and library for data compres- +sion,” http://www.bzip.org. +[3] ISO/IEC JTC1/SC29/ WG1 and ITU-T (JPEG), “ISO/IEC IS 10918- +1:1994 | ITU-T Rec. T.81 Information technology - Digital compression +and coding of continuous-tone still images: Requirements and guidelines,” +1994. +[4] G. Wallace, “The JPEG still picture compression standard,” IEEE Trans. +on Consumer Electronics, vol. 38, no. 1, pp. 18–34, feb. 1992. +[5] ISO/IEC JTC1/SC29/ WG1 and ITU-T SG-8 (JPEG), “ISO/IEC 15444- +1:2004 | ITU-T Rec. T.800 JPEG 2000 Image Coding System, Part 1: +Core Coding System,” 2000. +[6] C. Christopoulos, A. Skodras, and T. Ebrahimi, “The JPEG2000 still im- +age coding system: an overview,” IEEE Trans. on Consumer Electronics, +vol. 46, no. 4, pp. 1103–1127, nov. 2002. +[7] P. Chatterjee and P. Milanfar, “Fundamental limits of image denoising: +Are we there yet?” in IEEE Int. Conf. on Acoustics Speech and Signal +Processing (ICASSP), pp. 1358–1361, mar. 2010. + diff --git a/lNE3T4oBgHgl3EQfKAkA/content/tmp_files/load_file.txt b/lNE3T4oBgHgl3EQfKAkA/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fbece8a6e8f205d26e1b1bd4dcc9c3fb5796b3f4 --- /dev/null +++ b/lNE3T4oBgHgl3EQfKAkA/content/tmp_files/load_file.txt @@ -0,0 +1,476 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf,len=475 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content='04348v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content='IV] 11 Jan 2023 On the Influence of Clipping in Lossless Predictive and Wavelet Coding of Noisy Images Wolfgang Schnurrer, Jürgen Seiler, Michael Schöberl, and André Kaup Multimedia Communications and Signal Processing University of Erlangen-Nuremberg, Cauerstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' 7, 91058 Erlangen, Germany Email: {schnurrer, seiler, schoeberl, kaup}@lnt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content='de Abstract—Especially in lossless image coding the obtainable compression ratio strongly depends on the amount of noise included in the data as all noise has to be coded, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' Different approaches exist for lossless image coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' We analyze the compression performance of three kinds of approaches, namely direct entropy, predictive and wavelet-based coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' The results from our theoretical model are compared to simulated results from standard algorithms that base on the three approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' As long as no clipping occurs with increasing noise more bits are needed for lossless compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' We will show that for very noisy signals it is more advantageous to directly use an entropy coder without advanced preprocessing steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' INTRODUCTION Lossless compression is an important task in all areas where any modification of information is not allowed or at least not acceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' Examples are among many others measurements for quality assurance, archiving, surveillance, conservation of evidence material or medical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' Noise that is contained in the data has also to be coded in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' In the medical environment lossy compression is often not acceptable as the correct diagnosis cannot be guaranteed for the lossy coded images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' But medical images contain a lot of noise because on the one hand radiation has to be kept low to reduce the risks for the patients and on the other hand the acquisition time is kept short to avoid motion artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' Several different approaches exist for lossless coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' We observed that the performance of the different approaches varies significantly when the data contains different amounts of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' Prediction-based methods like lossless JPEG [3], [4] and wavelet-based methods like JPEG 2000 [5], [6] are advantageous to code the structural information but become less effective when the images contain a lot of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' We will provide a theoretical analysis on the behavior of the energy of the noise when a wavelet transform is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' We will compare this to direct entropy coding and a predictive coding scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' Figure 1 shows a block diagram of our signal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' Noise n [k] with a standard deviation σ is added to the signal s [k] that contains the structural information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' The signal f [k] results from quantizing the noisy signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' One of the methods, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=', direct, predictive and wavelet, is then applied in the gray box and the output is analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' In our study we assume that clipping occurs mainly when the additive noise leads to values that exceed the limits of the quantizer Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' PSfrag replacements σ f[k] n[k] s[k] Q H + noise direct wavelet predictive Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' Signal model 0 50 100 150 200 250 0 1000 2000 3000 4000 5000 code value frequency noisy original (a) ↑ / (b) ↓ (c) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' Two detail images from the Big Buck Bunny sequence (a), (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' (a) is shown with additive noise with σ = 30, (b) is shown original.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' (c) shows histograms of the image (a) with additive noise (solid red) and original of image (a), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=', without additive noise (dashed blue) In Section 2 we present the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' The description of our simulation and results are given in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' Section 4 will conclude this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' THEORETICAL ANALYSIS OF NOISE IMPACT We compare three different approaches for coding a signal without loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' The first method is called direct as the samples are directly entropy coded without any preprocessing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' The second method is called predictive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' Before entropy coding it is possible to subtract a prediction where the predictor for the current sample is computed from already decoded samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' The various predictors differ by the number and the weight of the incorporated samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' We analyze the prediction from one previous sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' The combination of more samples leads to a noise variance reduction due to averaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' But the overall noise variance of the predictor will stay greater than zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' The third class of methods in our analysis is based on the wavelet transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' Instead of subtracting a prediction, the samples are transformed and the coefficients from the sub-bands are then entropy coded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' In our analysis we compare two different wavelets, the Haar wavelet and the LeGall 5/3 wavelet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' We assume that the input signal f [k] consists of the structural information s [k] with additive noise n [k] after quantization as shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' For simplicity we show the analysis for the one dimensional case only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' At first we neglect the structural part and quantization step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' We analyze and compare the output of the different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' We then show how to calculate the entropy and finally add the structural information of the signal and the quantization step to our modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' Noise Variance for Different Coding Methods For our analysis we are mainly interested in the noise part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' We consider the structural signal s [k] = 0 in this subsection and assume Gaussian noise n [k] with zero mean and a variance σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' We analyze the influence of the different methods on the noise by comparing the noise variance at their output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' The direct method does not apply preprocessing and so the error distribution does not change and stays equal to E � n2 direct[k] � = σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' (1) The second method based on prediction uses one previous sample for the calculation of the predictor p[k] = f[k − 1] and will be called predictive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' The residuum of the predictor r[k] = f[k]− p [k] is then coded and the variance of the noise in the resulting sample doubles to E � n2 p[k] � = E � (n[k] − n[k − 1])2� = σ2 + σ2 = 2σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' (2) Wavelet transforms consist of a filter pair for the compu- tation of the high HP- and the low LP-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' For the Haar wavelet, the samples of the HPHaar- and the LPHaar-band are calculated to HPHaar [k] = f [2k] − f [2k − 1] and LPHaar [k] = � 1 2 (f [2k] − f [2k − 1]) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' The rounding operation is due to the integer wavelet transform [1] for lossless coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' The error variance for the coefficients of the HPHaar band calculates to E � n2 HPHaar[k] � = E � (n[2k] − n[2k − 1])2� = 2σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' (3) For the LPHaar-band the error variance is equal to E � n2 LPHaar[k] � = 1 4σ2 + 1 4σ2 = 1 2σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' (4) For the LeGall 5/3 wavelet the samples of the HPLeGall- and the LPLeGall- band are calculated to HPLeGall [k] = f [2k] − �1 2 (f [2k − 1] + f [2k + 1]) � (5) and LPLeGall[k]=f[2k−1]+ �1 4(HPLeGall[k]+HPLeGall[k−1]) � (6) as given in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' The error variance for the coefficients of the HPLeGall-band calculates to E � n2 HPLeGall[k] � = 1 4σ2 + σ2 + 1 4σ2 = 3 2σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' (7) For the LPLeGall-band the error variance needs to be calculated from the filter representation of the wavelet to preserve the signal independence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' The error variance is then equal to E � n2 LPLeGall[k] � = 12 + 22 + 62 + 22 + 12 64 σ2 = 46 64σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' (8) The combined error variance cannot be computed by aver- aging the two values E � n2 LP � and E � n2 HP � from the subbands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' The two subbands are calculated from the same signal values so the independence assumption does not hold anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' In the next subsection we will show how the values can be combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' From Noise Variance to Bits For the wavelet approach a combined variance cannot be computed by simply averaging the variances from the HP- and LP-bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' In order to get an estimation for the number of bits needed for coding the signal processed by the different methods with an optimum entropy coder we use the entropy H = − � i pilog2 (pi) [bit per sample] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' (9) The probabilities pi result from integrals over the probability density function with pi = ˆ i+ 1 2 i− 1 2 1 √ 2πσ2 e− 1 2( x σ) 2 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' (10) Without considering clipping on the upper and lower end of the co-domain i ∈ Z holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' For this case (9) can be solved analytically and be approximated by a formula that is dependent on the standard deviation σ of the noise Hdirect (σ) ≈ log2 � σ √ 2πe � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' (11) Now we can calculate the entropy for the different methods by evaluating (11) for the variances derived in the previous subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' For the predictive scheme the resulting entropy is given by Hpredictive (σ) ≈ H( √ 2σ) = 1 2 bit + Hdirect (σ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' (12) After the wavelet transform, the signal is represented by coefficients in the high and the low band in a ratio of 1:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' So the overall entropy can be computed by summing up the entropy of the high and the low band containing half the samples each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' For the Haar wavelet this results in HHaar (σ) ≈ 1 2 � H �� 1 2σ � + H �√ 2σ �� (13) = log2 � σ √ 2πe � = Hdirect (σ) and for the LeGall 5/3 wavelet in HLeGall (σ) ≈ 1 2 � H �� 3 2σ � + H �� 46 64σ �� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content='027 bit + Hdirect (σ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' (14) Even though the value range doubles for the HHaar-band that contains half of all coefficients, the entropy of the whole signal stays the same for the Haar wavelet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' Comparing the two wavelets, the entropy increases slightly for the LeGall 5/3 wavelet by an offset of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content='027 bit per sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' Comparing the entropy of the different methods we can conclude that without clipping all the methods need the same rate for coding the noise up to an offset that does not depend on σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' While this offset is very small for the wavelet-based methods for the predictive scheme that uses one previous sample the coding of the noise is more expensive by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content='5 bit per sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' The Influence of Clipping We now extend the modeling by the structural information and the quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' As we are interested in lossless coding we are limited to integer values in order to avoid rounding errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' To analyze the impact of clipping to zero mean noise we use a structural signal s [k] with a constant value in the center of the co-domain µ = � 28−1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' For this we extend (9) by a limitation of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' The probabilities pi, i ∈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content='.255 for a co- domain of 8 bit, of the signal result from integrating of the probability density function fN (n) of the noise over the bin size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' The calculation of the probabilities pi is given by pi = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 ´ −127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content='5 −∞ fN (ν) dν for i = 0 ´ ∞ 126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content='5 fN (ν) dν for i = 255 ´ i+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content='5 i−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content='5 fN (ν) dν for 0 < i < 254 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' (15) Values outside the co-domain are clipped to the minimum 0 and maximum 255 code value respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' This leads to peaks in the probability distribution of the signal at the borders of the co-domain as in Figure 2 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' In our analysis we assume statistically independent Gaus- sian noise, so the probability density function of the noise equals fN (n) = 1 √ 2πσ2 e− 1 2( x σ) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' As we assume a signal with a constant code value only, the probability distribution of the signal equals pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' The probability distribution of the input signal can now be combined according to the presented methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' Then the entropy can be calculated by inserting the values of the probabilities from 15 into the entropy formula (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' For the wavelet-based methods the entropy for the HP-band and for the LP-band have to be calculated separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' The resulting values are then summed up by taking into account that each band contains half of the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' Theoretical Results The resulting curves are plotted in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' The results strongly depend on the µ chosen for the structural part of the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' The value of µ in the center of the co-domain is the ideal case because for a smaller or greater value of µ clipping is introduced at smaller values of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' The plot can be divided into four areas along the abscissa denoted by (a)- (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' In area (a) the noise is quantized to zero and thus has no influence on the entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' When σ gets bigger and only a few samples are affected by noise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' that is not quantized to ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content='noise standard deviation σ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content='entropy H in bit per sample ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content='uT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content='uT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content='uT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content='direct ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content='predictive ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content='wavelet Haar ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content='wavelet LeGall bC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content='| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content='uT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content='(d) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content='(c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content='(b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content='(a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content='Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' Theoretical results - entropy in bit per sample plotted over the standard deviation σ of the noise on the abscissa in a lin-log plot zero, the curves have a different slope as area (b) shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' The reason for the steeper slope of the predictive and wavelet-based methods is the combination of several samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' As soon as the noise component in one sample is above quantization, several coefficients are affected and cause an increase of the entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' For bigger values of σ more samples are affected by noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' As long as the noise is small enough such that no clipping is occurring, the curves run in parallel, as shown in area (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' The offsets correspond to the derivation in the previous subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' Clipping begins to occur for larger values of the standard deviation σ as shown in area (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' Direct entropy coding can exploit the rising number of clipped values better resulting in a lower entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' The entropy of the predictive scheme and the wavelet-based method drops slower due to the combination of clipped samples with unclipped ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' The LeGall 5/3 wavelet is even more sensitive to this than the Haar wavelet because of the greater filter length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' SIMULATION RESULTS In our simulation we used different images and added Gaussian noise with increasing standard deviation σ before coding them with different algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' To validate our model we first used an image of size 512x512 pixels with a constant code value of 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' In order to prove that our model also fits for images we used details from the Big Buck Bunny sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' The reason for the choice of artificial images is the difficulty of perfectly denoising real images [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' The chosen images are computer generated and thus do not contain noise in the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' We cut images of 512x512 pixels as shown in Figure 2 (a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' For illustration we added Gaussian noise with a standard deviation of σ = 30 to the first detail shown in Figure 2 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' The second detail in Figure 2 (b) is shown in original, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=', without additive noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' In our simulation we used the green color channel only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content='10−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content='100 ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content='Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' Simulation results show the file size in kbytes over the standard deviation σ of the added noise in lin-log plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' (a) shows the results for a constant signal equal to 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' In (b) the results for the first test image (Figure 2 (a)) are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' (c) shows the results for the second test image (Figure 2 (b)) For direct entropy coding we used bzip2 [2] in version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' No additional parameters were given when calling the pro- gram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' For predictive coding we used lossless JPEG [3], [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' For wavelet-based coding we used JPEG 2000 [5], [6] with 4 decomposition steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' As long as the number of decom- position steps is larger than 3 this parameter has not much influence on the file size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' The results are shown in Figure 4 where the file size of the compressed images is plotted in kbyte over the standard deviation σ of the noise in lin-log plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' In Figure 4 (a) the results are shown for the image with a constant code value of 128 and additive noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' By comparing Figure 4 (a) with the theoretical results in Figure 3 it can be seen that our model matches with the results from the simulation up to the offset of the lossless JPEG method for small values of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' The corresponding curves have the same shape and the plot in Figure 4 (a) can also be divided into the four areas indicated in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' Quantitative statements cannot be given as the results strongly depend on the structural information in the input signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' The offsets in Figure 4 (b) and (c) come from the structural information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' The three methods need a different amount of bits to code the structural information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' The two plots show clearly that as long as the noise part is small enough advanced methods as a predictive scheme and a wavelet decomposition are advantageous because they are more capable to reduce the redundancy in the structural information part of the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' Clipping in the structural information, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' due to wrong exposure, often affects bigger areas of an image and thus is beneficial for predictive and wavelet-based coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' The prob- lem are isolated clipped values introduced by noise because their combination with other unclipped samples lead to a higher entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' The results in Figure 3 and Figure 4 (a) show that it is advantageous to use direct entropy coding for noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' In [7] several state of the art denoising algorithms are compared to a theoretical bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' The result is that there is still room for improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' A separation of the noise from the structural information leads to an increase of the encoder complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' The decoder has to decode both parts and add the noise to the structural part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' Compared with wavelet-based coding gains can be achieved for small values of σ as shown in area (b) of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' Another result is that for very noisy images when clipping is introduced by noise it is more advantageous to directly use an entropy coder without any signal decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' These results show that our model also fits for images with structural information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' CONCLUSION In this paper we analyzed the impact of clipping to lossless compression of noisy images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' We derived an analytical de- scription that models the behavior of different coding methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' So the effects shown are general properties of the methods and not of a special implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' Simulation results support our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' The results show that in general for noisy data it is advantageous to code the noise and the signal separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' Furthermore, the results show that for the case that clipping is introduced by noise it is more advantageous to directly use an entropy coder without advanced preprocessing steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' ACKNOWLEDGMENT We gratefully acknowledge that this work has been sup- ported by the Deutsche Forschungsgemeinschaft (DFG) under contract number KA 926/4-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' REFERENCES [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' Calderbank, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' Daubechies, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' Sweldens, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' Yeo, “Lossless image compression using integer to integer wavelet transforms,” in Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' on Image Processing (ICIP), vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' 596–599, oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' [2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' Seward, “bzip2 and libbzip2: A program and library for data compres- sion,” http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content='bzip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' [3] ISO/IEC JTC1/SC29/ WG1 and ITU-T (JPEG), “ISO/IEC IS 10918- 1:1994 | ITU-T Rec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content='81 Information technology - Digital compression and coding of continuous-tone still images: Requirements and guidelines,” 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' [4] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' Wallace, “The JPEG still picture compression standard,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' on Consumer Electronics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' 38, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' 18–34, feb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' [5] ISO/IEC JTC1/SC29/ WG1 and ITU-T SG-8 (JPEG), “ISO/IEC 15444- 1:2004 | ITU-T Rec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content='800 JPEG 2000 Image Coding System, Part 1: Core Coding System,” 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' [6] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' Christopoulos, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' Skodras, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' Ebrahimi, “The JPEG2000 still im- age coding system: an overview,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' on Consumer Electronics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' 46, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' 1103–1127, nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' [7] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' Chatterjee and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' Milanfar, “Fundamental limits of image denoising: Are we there yet?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' in IEEE Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' on Acoustics Speech and Signal Processing (ICASSP), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' 1358–1361, mar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} +page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfKAkA/content/2301.04348v1.pdf'} diff --git a/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf b/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..00f591e2038fe7b1b188b497f845e572a5fa7a57 --- /dev/null +++ b/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf 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+size 107746 diff --git a/nNE4T4oBgHgl3EQfuQ2l/content/tmp_files/2301.05231v1.pdf.txt b/nNE4T4oBgHgl3EQfuQ2l/content/tmp_files/2301.05231v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..4a26bce35c80fed41da9075a05d57f3633f304dd --- /dev/null +++ b/nNE4T4oBgHgl3EQfuQ2l/content/tmp_files/2301.05231v1.pdf.txt @@ -0,0 +1,363 @@ +Extended Abstract Track +Extended Abstract Track 2022 +NeurIPS Workshop on Symmetry and Geometry in Neural Representations +Equivariant Representations for Non-Free Group Actions +Luis Armando P´erez Rey∗1,2,3 +l.a.perez.rey@tue.nl +Giovanni Luca Marchetti∗4 +glma@kth.se +Danica Kragic4 +dani@kth.se +Dmitri Jarnikov1,3 +d.s.jarnikov@tue.nl +Mike Holenderski1 +m.holenderski@tue.nl +1. Eindhoven University of Technology, Eindhoven, The Netherlands +2. Eindhoven Artificial Intelligence Systems Institute (EAISI), Eindhoven, The Netherlands +3. Prosus, Amsterdam, The Netherlands +4. Royal Institute of Technology (KTH), Stockholm, Sweden +Editors: Sophia Sanborn, Christian Shewmake, Simone Azeglio, Arianna Di Bernardo, Nina Miolane +Abstract +We introduce a method for learning representations that are equivariant with respect to +general group actions over data. Differently from existing equivariant representation learn- +ers, our method is suitable for actions that are not free i.e., that stabilize data via nontrivial +symmetries. Our method is grounded in the orbit-stabilizer theorem from group theory, +which guarantees that an ideal learner infers an isomorphic representation. Finally, we +provide an empirical investigation on image datasets with rotational symmetries and show +that taking stabilizers into account improves the quality of the representations. +Keywords: Representation Learning, Symmetries, Group Theory +1. Introduction +Figure 1 +The problem of incorporating symmetries into representations defines a +fundamental challenge and has been considered in a number of recent +works (Quessard et al., 2020; Higgins et al., 2022; Cohen and Welling, +2014; Tonnaer et al., 2022; Ahuja et al., 2021). The overall aim is to +design representations which preserve symmetries – a property known +as equivariance. +This is because preservation of symmetries leads to +the extraction of geometric and semantic structure in data, which can +be exploited for reasoning, efficiency and generalization (Bengio et al., +2013). As an example, the challenge of disentangling semantic factors +of variations has been rephrased in terms of equivariant representations +(Higgins et al., 2018; Caselles-Dupr´e et al., 2019). +The majority of literature relies on the assumption that the group +of symmetries acts freely on data (Marchetti et al., 2022) i.e., that no +datapoint is stabilized by (nontrivial) symmetries. This avoids the need +to model stabilizers, which are unknown subgroups of the symmetry +group considered. However, non-free group actions arise in several prac- +tical scenarios. This happens for example when considering images of +∗ Equal contribution +© 2022 L.A. P´erez Rey, G.L. Marchetti, D. Kragic, D. Jarnikov & M. Holenderski. +arXiv:2301.05231v1 [cs.LG] 12 Jan 2023 + +2 +=T +5Extended Abstract Track +P´erez Rey Marchetti Kragic Jarnikov Holenderski +objects acted upon by the rotation group via change of orientation. Such objects might be +symmetrical, resulting in rotations leaving the image (almost) identical and consequently +ambiguous in its orientation (see Figure 1). +In this work we propose a method for learning equivariant representation for general +and potentially non-free group actions. Based on the orbit-stabilizer theorem from group +theory, we design a model that outputs cosets of the stabilizer subgroup. The representation +learner optimizes an equivariance loss based on supervision from symmetries alone. The +above-mentioned theoretical results guarantee that a learner infers representations that are +isomorphic to the original dataset. +2. Group Theory Background +Let G be the group of symmetries with multiplication denoted by (g, h) → gh and identity +denoted by 1 ∈ G. Suppose that G acts on a set X via (g, x) → g · x. The action defines +a set of orbits X/G given by the equivalence classes of the relation x ∼ y iff y = g · x for +some g ∈ G. For each x ∈ X, the stabilizer subgroup is defined as Gx = {g ∈ G | g · x = x}. +Stabilizers are conjugate as x varies in its orbit, and by abuse of notation we refer to the +conjugacy class GO for O ∈ X/G. The action is said to be free if GO = {1} for every O. +Recall that a map ϕ : X → Z between sets acted upon by G is said to be equivariant +if ϕ(g · x) = g · ϕ(x) for every x ∈ X and g ∈ G. An equivariant bijection is referred to as +isomorphism. The following is the fundamental result on group actions (Rotman, 2012). +Theorem 1 (Orbit-Stabilizer) +Each orbit O is isomorphic to the set of (left) cosets +G/GO = {gGO | g ∈ G}. In other words, there is an isomorphism: +X ≃ +� +O∈X/G +G/GO +⊆ 2G × X/G +(1) +where 2G denotes the power-set of G on which G acts by left multiplication g ·A = {ga | a ∈ +A}. Moreover, any equivariant map ϕ : X → � +O∈X/G G/GO which induces a bijection on +orbits is an isomorphism. +3. Equivariant Representation Learning +Our goal is to design an equivariant representation learner based on Theorem 1. We aim to +train a model ϕ : X → Z with a latent space Z on a loss encouraging equivariance. While +we assume that G is known a priori, its action on X is not and has to be conveyed through +data. The ideal choice for Z is given by � +O∈X/G G/GO since the latter is the isomorphic +to X (Theorem 1). In other words, ϕ ideally outputs cosets of stabilizers of the inputs. +However, the stabilizers are unknown a priori since they depend on the group action. In +order to circumvent the modeling of stabilizers and their cosets, we appeal to the following +simple result (a proof is provided in the Appendix): +Proposition 2 +Let ϕ : X → 2G be an equivariant map. Then for each x ∈ X from an +orbit O, ϕ(x) contains a coset of (a conjugate of) GO. +2 + +Extended Abstract Track +Non-Free Group Actions +Proposition 2 enables ϕ to output subsets of G instead of cosets of stabilizers. As long as +those subsets are minimal w.r.t. to inclusion, they will coincide with the desired cosets. +Based on this, we define the latent space as Z = ZG × ZO and implement the map ϕ as a +pair of neural networks ϕG : X → ZG, ϕO : X → ZO. The component ZG represents cosets +of stabilizers while ZO represents orbits. Since the output space of a neural network is a +finite-dimensional vector space, we assume that G is a linear Lie group, i.e. G is a manifold +of matrices, and that the stabilizers of the action are finite. The model ϕG first outputs N +elements (ϕ1 +G(x), · · · , ϕN +G(x)) = ϕG(x) in the matrix Lie algebra g that are converted to G +by the exponential map exp : g → G. The hyper-parameter N ideally should be chosen +to be larger than the cardinality of the stabilizers. On the other hand, the output of ϕO +consists of a vector of arbitrary dimensionality. The only requirement is that the output +space of ϕO should have enough capacity to contain X/G. +Our dataset D consists of samples from the (unknown) group action, meaning that +datapoints are triplets (x, g, y) ∈ X × G × X with y = g · x. Given a datapoint (x, g, y) ∈ D +the learner ϕG optimizes the equivariance loss over its parameters: +LG(x, g, y) = d(g · ϕG(x), ϕG(y)) +(2) +where d is a (semi) metric for sets. We opt for the (asymmetric) Chamfer distance d(A, B) = +1 +|A| +� +a∈A minb∈B dG(a, b) because of its differentiability properties. Here dG is a metric on +G and is typically set as the (squared) Euclidean one for G = Rn and as the (squared) +Frobenius one for G = SO(n). As previously discussed we wish ϕG(x), when seen as a +set, to be minimal in cardinality. To this end we add the following regularization term +measuring the discrete entropy: +�LG(x) = λ +N2 +� +1≤i,j≤N +dG(ϕi +G(x), ϕj +G(x)) +(3) +where λ is a small weight set canonically to 0.001. On the other hand, since orbits are +invariant to the group action ϕO optimizes a contrastive loss. +We opt for the popular +InfoNCE loss from the literature (Chen et al., 2020): +LO(x, y) = dO(ϕO(x), ϕO(y)) + log Ex′ +� +e−dO(ϕO(x′), ϕO(x))� +(4) +where x′ is marginalized from D. As customary for the InfoNCE loss, we normalize the +output of ϕO and set dO(a, b) = − cos(∠ab) = −a·b. The second summand of LO encourages +injectivity of ϕO as and prevents orbits from collapsing in the representation. +The Orbit-Stabilizer Theorem guarantees that an ideal learner achieves isomorphic rep- +resentations in the following sense. If the LG(x, g, y) and the first summand of LO(x, y) +vanish for every (x, g, y) then ϕ is equivariant. If moreover the regularizations ( �LG and +the second summand of LO) are at a minimum then ϕG(x) coincides with a coset of GO +for every x ∈ O (Proposition 2) and ϕO is injective. Theorem 1 implies then that the +representation is isomorphic (on its image) as desired. +4. Experiments +We test an implementation of the neural networks ϕG and ϕO on the following four datasets +consisting of 64 × 64 images subject to non-free group actions: +3 + +Extended Abstract Track +P´erez Rey Marchetti Kragic Jarnikov Holenderski +• Rotating Arrows: images of radial configurations of ν ∈ {4, 5, 6} arrows rotated +by G = SO(2). The number of arrows ν determines the orbit with stabilizer the cyclic +group Cν ⊆ G of order ν. +• Double Arrows: images of two radial configurations of 2 and 3 arrows respectively +rotated by G = SO(2)×SO(2). The action produces a single orbit, i.e. it is transitive, +and the stabilizer is a product of cyclic groups C2 × C3. +• Chair: images of a monochromatic chair from ModelNet40 (Wu et al., 2015) rotated +by G = SO(2) along a vertical axis with a single orbit with stabilizer the cyclic group +C4 ⊆ G of order 4. +• Tetrahedron: images of a monochromatic tetrahedron (Murphy et al., 2021) ro- +tated by G = SO(3) with a single orbit and stabilizer the alternating group A4 of +order 12. +We compare our model with the baseline where ϕG produces a single output i.e., N = 1. +The latent space is thus Z = G × ZO, on which G acts freely. This is similar to what +has been proposed in previous work (Caselles-Dupr´e et al., 2019; Marchetti et al., 2022; +Tonnaer et al., 2022). The models are compared based on two evaluation metrics. First, +the equivariance loss (Equation 3) on a test set. Second, the reconstruction loss (pixel-wise +cross-entropy) on a test set of a decoder ψ : Z → X trained jointly with ϕ. Quantitative +results are presented in Table 1 while qualitative visualizations are shown in Figure 2. As +can be seen, our model correctly infers the stabilizers (the cyclic subgroups of SO(2) and the +alternating subgroup of SO(3)) by overlapping components of ϕG(x) and distributing them +geometrically. Moreover, our model achieves significantly lower scores than the baseline. +The latter is not able to capture the stabilizers in its latent space, leading to representations +of poor quality and loss of information. +Table 1: Mean and std over 3 runs of the evaluation metrics for our model and the baseline. +Dataset +Model +N +Equivariance +Reconstruction +Rotating Arrows +Baseline +1 +1.985±0.027 +0.551±0.051 +Ours +10 +0.011±0.002 +0.372±0.040 +Double Arrows +Baseline +1 +4.016±0.027 +0.219±0.002 +Ours +6 +0.009±0.006 +0.152±0.011 +Chair +Baseline +1 +1.944±0.045 +0.603±0.037 +Ours +5 +0.098±0.061 +0.424±0.022 +Tetrahedron +Baseline +1 +6.025±0.063 +0.470±0.045 +Ours +20 +0.032±0.007 +0.286±0.029 +5. Conclusions and Future Work +In this work we introduced a method for learning equivariant representations for general and +potentially non-free group actions. We discussed the theoretical foundations and empirically +investigated the method on images with rotational symmetries. +4 + +Extended Abstract Track +Non-Free Group Actions +x ∈ X +ϕG(x) ⊆ G +Figure 2: Visualization of the data x for the four datasets and the predicted stabilizer +ϕG(x). For the double arrows, the torus G = SO(2) × SO(2) is visualized as an identified +square. For the tetrahedron, G is visualized as a projective space RP3 ≃ SO(3). +Our model relies on the assumptions that the stabilizers are finite. However, non-discrete +stabilizer subgroups sometimes occur, for example in the case of symmetrical objects such +as a cone or a cylinder. An interesting future direction is designing an equivariant repre- +sentation learner suitable for group actions with non-discrete stabilizers and the evaluation +of our method on more complex datasets. +6. Acknowledgements +The second and third named authors thank the Swedish Research Council, the Knut and +Alice Wallenberg Foundation and the European Research Council (ERC-BIRD-884807) +for their support. This work has also received funding from the NWO-TTW Programme +“Efficient Deep Learning” (EDL) P16-25. +5 + +Extended Abstract Track +P´erez Rey Marchetti Kragic Jarnikov Holenderski +References +Kartik Ahuja, Jason Hartford, and Yoshua Bengio. Properties from Mechanisms: An Equiv- +ariance Perspective on Identifiable Representation Learning. 2021. +Yoshua Bengio, Aaron Courville, and Pascal Vincent. Representation learning: A review +and new perspectives. IEEE Transactions on Pattern Analysis and Machine Intelligence, +35(8):1798–1828, 2013. ISSN 01628828. doi: 10.1109/TPAMI.2013.50. +Hugo Caselles-Dupr´e, Michael Garcia-Ortiz, and David Filliat. Symmetry-Based Disen- +tangled Representation Learning Requires Interaction with Environments. Advances in +Neural Information Processing Systems, 32(NeurIPS):1–10, 2019. ISSN 10495258. +Ting Chen, Simon Kornblith, Mohammad Norouzi, and Geoffrey Hinton. A simple frame- +work for contrastive learning of visual representations. In International conference on +machine learning, pages 1597–1607. PMLR, 2020. +Taco Cohen and Max Welling. Learning the Irreducible Representations of Commutative +Lie Groups. International Conference on Machine Learning, 5:3757–3770, 2014. +Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep Residual Learning for +Image Recognition. In 2016 IEEE Conference on Computer Vision and Pattern Recog- +nition (CVPR), volume 37, pages 770–778. IEEE, 6 2016. +ISBN 978-1-4673-8851-1. +doi: 10.1109/CVPR.2016.90. URL https://onlinelibrary.wiley.com/doi/10.1002/ +chin.200650130http://ieeexplore.ieee.org/document/7780459/. +Irina Higgins, David Amos, David Pfau, Sebastien Racaniere, Loic Matthey, Danilo +Rezende, and Alexander Lerchner. +Towards a Definition of Disentangled Representa- +tions. pages 1–29, 2018. URL http://arxiv.org/abs/1812.02230. +Irina Higgins, S´ebastien Racani`ere, and Danilo Rezende. +Symmetry-Based Represen- +tations for Artificial and Biological General Intelligence. +Frontiers in Computational +Neuroscience, 16, 4 2022. +ISSN 1662-5188. +doi: 10.3389/fncom.2022.836498. +URL +https://www.frontiersin.org/articles/10.3389/fncom.2022.836498/full. +Ilya Loshchilov and Frank Hutter. Decoupled Weight Decay Regularization. In International +Conference on Learning Representations, 2 2019. +Giovanni Luca Marchetti, Gustaf Tegn´er, Anastasiia Varava, and Danica Kragic. Equivari- +ant Representation Learning via Class-Pose Decomposition. 7 2022. doi: 10.48550/arxiv. +2207.03116. URL https://arxiv.org/abs/2207.03116v2. +Kieran Murphy, Carlos Esteves, Varun Jampani, Srikumar Ramalingam, and Ameesh Maka- +dia. +Implicit representation of probability distributions on the rotation manifold. +In +International Conference on Machine Learning, 2021. +Robin Quessard, Thomas D Barrett, and William R Clements. +Learning Disentangled +Representations and Group Structure of Dynamical Environments. In 34th Conference +on Neural Information Processing Systems, 2020. +6 + +Extended Abstract Track +Non-Free Group Actions +Joseph J Rotman. An introduction to the theory of groups, volume 148. Springer Science +& Business Media, 2012. +Loek Tonnaer, Luis Armando Perez Rey, Vlado Menkovski, Mike Holenderski, and Jim +Portegies. Quantifying and Learning Linear Symmetry-Based Disentanglement. In Pro- +ceedings of the 39th International Conference on Machine Learning, pages 21584–21608. +PMLR, 6 2022. URL https://proceedings.mlr.press/v162/tonnaer22a.html. +Zhirong Wu, Shuran Song, Aditya Khosla, Fisher Yu, Linguang Zhang, Xiaoou Tang, and +Jianxiong Xiao. 3d shapenets: A deep representation for volumetric shapes. In Pro- +ceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), +June 2015. +7 + +Extended Abstract Track +P´erez Rey Marchetti Kragic Jarnikov Holenderski +7. Appendix +7.1. Proofs of Theoretical Results +Proposition 3 Let ϕ : +X → 2G be an equivariant map. Then for each x ∈ X from an +orbit O, ϕ(x) contains a coset of (a conjugate of) GO. +Proof +Pick x ∈ X. Then for every g ∈ Gx it holds that ϕ(x) = ϕ(g · x) = g · ϕ(x). In +other words Gxh = hh−1Gxh ⊆ ϕ(x) for each h ∈ ϕ(x). Since h−1Gxh is conjugate to Gx +the thesis follows. +7.2. Training Details +We implement the neural networks ϕG and ϕO with a backbone ResNet18 (He et al., 2016). +For a datapoint x ∈ X, the network implements multiple heads to produce embeddings +� +ϕ1 +G(x), · · · , ϕN +G(x) +� +with ϕi +G(x) ∈ G. The output dimension of ϕO is set to 3. We train the +model for 50 epochs (except for the chair dataset which requires a longer training of 100 +epochs) using the AdamW optimizer (Loshchilov and Hutter, 2019) with a learning rate of +10−4 and batches of 16 triplets (x, g, y) ∈ D. +The rotating arrows and chair dataset consists of 5000 datapoints per orbit while the +double arrows and the tetrahedron datasets consist of 20000 datapoints. For all datasets +the test set consists of a random 10% split. +7.3. Reconstructions +We present in Figure 3 some examples of images reconstructed by a decoder ψ trained +jointly with the encoder ϕ. The baseline model is not capable of clearly reconstructing the +images compared to our model. +As seen from the quantitative results in Table 1, the baseline model is not capable of +optimizing the equivariance loss from Equation 2. This provides a hint that the encoder ϕ +might not be converging to a stable representation. Consequently, the decoder is incapable +of consistently reconstructing the data which results in a higher reconstruction loss. +Baseline +Ours +Figure 3: Reconstruction examples produced by a decoder trained on the latent variables +produced by our model and the baseline. +8 + +**大* \ No newline at end of file diff --git a/nNE4T4oBgHgl3EQfuQ2l/content/tmp_files/load_file.txt b/nNE4T4oBgHgl3EQfuQ2l/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..410fa2f6d81f5d6cd6c0d686e65bb8d2c20c50a1 --- /dev/null +++ b/nNE4T4oBgHgl3EQfuQ2l/content/tmp_files/load_file.txt @@ -0,0 +1,304 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf,len=303 +page_content='Extended Abstract Track Extended Abstract Track 2022 NeurIPS Workshop on Symmetry and Geometry in Neural Representations Equivariant Representations for Non-Free Group Actions Luis Armando P´erez Rey∗1,2,3 l.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='jarnikov@tue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='nl Mike Holenderski1 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='holenderski@tue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='nl 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Eindhoven University of Technology, Eindhoven, The Netherlands 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Eindhoven Artificial Intelligence Systems Institute (EAISI), Eindhoven, The Netherlands 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Prosus, Amsterdam, The Netherlands 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Royal Institute of Technology (KTH), Stockholm, Sweden Editors: Sophia Sanborn, Christian Shewmake, Simone Azeglio, Arianna Di Bernardo, Nina Miolane Abstract We introduce a method for learning representations that are equivariant with respect to general group actions over data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Differently from existing equivariant representation learn- ers, our method is suitable for actions that are not free i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=', that stabilize data via nontrivial symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Our method is grounded in the orbit-stabilizer theorem from group theory, which guarantees that an ideal learner infers an isomorphic representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Finally, we provide an empirical investigation on image datasets with rotational symmetries and show that taking stabilizers into account improves the quality of the representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Keywords: Representation Learning, Symmetries, Group Theory 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Introduction Figure 1 The problem of incorporating symmetries into representations defines a fundamental challenge and has been considered in a number of recent works (Quessard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Higgins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Cohen and Welling, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Tonnaer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Ahuja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' The overall aim is to design representations which preserve symmetries – a property known as equivariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' This is because preservation of symmetries leads to the extraction of geometric and semantic structure in data, which can be exploited for reasoning, efficiency and generalization (Bengio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' As an example, the challenge of disentangling semantic factors of variations has been rephrased in terms of equivariant representations (Higgins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Caselles-Dupr´e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' The majority of literature relies on the assumption that the group of symmetries acts freely on data (Marchetti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=', 2022) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=', that no datapoint is stabilized by (nontrivial) symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' This avoids the need to model stabilizers, which are unknown subgroups of the symmetry group considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' However, non-free group actions arise in several prac- tical scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' This happens for example when considering images of ∗ Equal contribution © 2022 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' P´erez Rey, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Marchetti, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Kragic, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Jarnikov & M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Holenderski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='05231v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='LG] 12 Jan 2023 2 =T 5Extended Abstract Track P´erez Rey Marchetti Kragic Jarnikov Holenderski objects acted upon by the rotation group via change of orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Such objects might be symmetrical, resulting in rotations leaving the image (almost) identical and consequently ambiguous in its orientation (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' In this work we propose a method for learning equivariant representation for general and potentially non-free group actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Based on the orbit-stabilizer theorem from group theory, we design a model that outputs cosets of the stabilizer subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' The representation learner optimizes an equivariance loss based on supervision from symmetries alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' The above-mentioned theoretical results guarantee that a learner infers representations that are isomorphic to the original dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Group Theory Background Let G be the group of symmetries with multiplication denoted by (g, h) → gh and identity denoted by 1 ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Suppose that G acts on a set X via (g, x) → g · x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' The action defines a set of orbits X/G given by the equivalence classes of the relation x ∼ y iff y = g · x for some g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' For each x ∈ X, the stabilizer subgroup is defined as Gx = {g ∈ G | g · x = x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Stabilizers are conjugate as x varies in its orbit, and by abuse of notation we refer to the conjugacy class GO for O ∈ X/G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' The action is said to be free if GO = {1} for every O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Recall that a map ϕ : X → Z between sets acted upon by G is said to be equivariant if ϕ(g · x) = g · ϕ(x) for every x ∈ X and g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' An equivariant bijection is referred to as isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' The following is the fundamental result on group actions (Rotman, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Theorem 1 (Orbit-Stabilizer) Each orbit O is isomorphic to the set of (left) cosets G/GO = {gGO | g ∈ G}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' In other words, there is an isomorphism: X ≃ � O∈X/G G/GO ⊆ 2G × X/G (1) where 2G denotes the power-set of G on which G acts by left multiplication g ·A = {ga | a ∈ A}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Moreover, any equivariant map ϕ : X → � O∈X/G G/GO which induces a bijection on orbits is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Equivariant Representation Learning Our goal is to design an equivariant representation learner based on Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' We aim to train a model ϕ : X → Z with a latent space Z on a loss encouraging equivariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' While we assume that G is known a priori, its action on X is not and has to be conveyed through data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' The ideal choice for Z is given by � O∈X/G G/GO since the latter is the isomorphic to X (Theorem 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' In other words, ϕ ideally outputs cosets of stabilizers of the inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' However, the stabilizers are unknown a priori since they depend on the group action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' In order to circumvent the modeling of stabilizers and their cosets, we appeal to the following simple result (a proof is provided in the Appendix): Proposition 2 Let ϕ : X → 2G be an equivariant map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Then for each x ∈ X from an orbit O, ϕ(x) contains a coset of (a conjugate of) GO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' 2 Extended Abstract Track Non-Free Group Actions Proposition 2 enables ϕ to output subsets of G instead of cosets of stabilizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' As long as those subsets are minimal w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' to inclusion, they will coincide with the desired cosets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Based on this, we define the latent space as Z = ZG × ZO and implement the map ϕ as a pair of neural networks ϕG : X → ZG, ϕO : X → ZO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' The component ZG represents cosets of stabilizers while ZO represents orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Since the output space of a neural network is a finite-dimensional vector space, we assume that G is a linear Lie group, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' G is a manifold of matrices, and that the stabilizers of the action are finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' The model ϕG first outputs N elements (ϕ1 G(x), · · · , ϕN G(x)) = ϕG(x) in the matrix Lie algebra g that are converted to G by the exponential map exp : g → G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' The hyper-parameter N ideally should be chosen to be larger than the cardinality of the stabilizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' On the other hand, the output of ϕO consists of a vector of arbitrary dimensionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' The only requirement is that the output space of ϕO should have enough capacity to contain X/G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Our dataset D consists of samples from the (unknown) group action, meaning that datapoints are triplets (x, g, y) ∈ X × G × X with y = g · x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Given a datapoint (x, g, y) ∈ D the learner ϕG optimizes the equivariance loss over its parameters: LG(x, g, y) = d(g · ϕG(x), ϕG(y)) (2) where d is a (semi) metric for sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' We opt for the (asymmetric) Chamfer distance d(A, B) = 1 |A| � a∈A minb∈B dG(a, b) because of its differentiability properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Here dG is a metric on G and is typically set as the (squared) Euclidean one for G = Rn and as the (squared) Frobenius one for G = SO(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' As previously discussed we wish ϕG(x), when seen as a set, to be minimal in cardinality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' To this end we add the following regularization term measuring the discrete entropy: �LG(x) = λ N2 � 1≤i,j≤N dG(ϕi G(x), ϕj G(x)) (3) where λ is a small weight set canonically to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' On the other hand, since orbits are invariant to the group action ϕO optimizes a contrastive loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' We opt for the popular InfoNCE loss from the literature (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=', 2020): LO(x, y) = dO(ϕO(x), ϕO(y)) + log Ex′ � e−dO(ϕO(x′), ϕO(x))� (4) where x′ is marginalized from D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' As customary for the InfoNCE loss, we normalize the output of ϕO and set dO(a, b) = − cos(∠ab) = −a·b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' The second summand of LO encourages injectivity of ϕO as and prevents orbits from collapsing in the representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' The Orbit-Stabilizer Theorem guarantees that an ideal learner achieves isomorphic rep- resentations in the following sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' If the LG(x, g, y) and the first summand of LO(x, y) vanish for every (x, g, y) then ϕ is equivariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' If moreover the regularizations ( �LG and the second summand of LO) are at a minimum then ϕG(x) coincides with a coset of GO for every x ∈ O (Proposition 2) and ϕO is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Theorem 1 implies then that the representation is isomorphic (on its image) as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Experiments We test an implementation of the neural networks ϕG and ϕO on the following four datasets consisting of 64 × 64 images subject to non-free group actions: 3 Extended Abstract Track P´erez Rey Marchetti Kragic Jarnikov Holenderski Rotating Arrows: images of radial configurations of ν ∈ {4, 5, 6} arrows rotated by G = SO(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' The number of arrows ν determines the orbit with stabilizer the cyclic group Cν ⊆ G of order ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Double Arrows: images of two radial configurations of 2 and 3 arrows respectively rotated by G = SO(2)×SO(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' The action produces a single orbit, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' it is transitive, and the stabilizer is a product of cyclic groups C2 × C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Chair: images of a monochromatic chair from ModelNet40 (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=', 2015) rotated by G = SO(2) along a vertical axis with a single orbit with stabilizer the cyclic group C4 ⊆ G of order 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Tetrahedron: images of a monochromatic tetrahedron (Murphy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=', 2021) ro- tated by G = SO(3) with a single orbit and stabilizer the alternating group A4 of order 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' We compare our model with the baseline where ϕG produces a single output i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=', N = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' The latent space is thus Z = G × ZO, on which G acts freely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' This is similar to what has been proposed in previous work (Caselles-Dupr´e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Marchetti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Tonnaer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' The models are compared based on two evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' First, the equivariance loss (Equation 3) on a test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Second, the reconstruction loss (pixel-wise cross-entropy) on a test set of a decoder ψ : Z → X trained jointly with ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Quantitative results are presented in Table 1 while qualitative visualizations are shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' As can be seen, our model correctly infers the stabilizers (the cyclic subgroups of SO(2) and the alternating subgroup of SO(3)) by overlapping components of ϕG(x) and distributing them geometrically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Moreover, our model achieves significantly lower scores than the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' The latter is not able to capture the stabilizers in its latent space, leading to representations of poor quality and loss of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Table 1: Mean and std over 3 runs of the evaluation metrics for our model and the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Dataset Model N Equivariance Reconstruction Rotating Arrows Baseline 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='985±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='027 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='551±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='051 Ours 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='011±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='372±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='040 Double Arrows Baseline 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='016±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='027 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='219±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='002 Ours 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='009±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='152±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='011 Chair Baseline 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='944±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='603±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='037 Ours 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='098±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='061 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='424±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='022 Tetrahedron Baseline 1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='025±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='063 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='470±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='045 Ours 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='032±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='286±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='029 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Conclusions and Future Work In this work we introduced a method for learning equivariant representations for general and potentially non-free group actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' We discussed the theoretical foundations and empirically investigated the method on images with rotational symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' 4 Extended Abstract Track Non-Free Group Actions x ∈ X ϕG(x) ⊆ G Figure 2: Visualization of the data x for the four datasets and the predicted stabilizer ϕG(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' For the double arrows, the torus G = SO(2) × SO(2) is visualized as an identified square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' For the tetrahedron, G is visualized as a projective space RP3 ≃ SO(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Our model relies on the assumptions that the stabilizers are finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' However, non-discrete stabilizer subgroups sometimes occur, for example in the case of symmetrical objects such as a cone or a cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' An interesting future direction is designing an equivariant repre- sentation learner suitable for group actions with non-discrete stabilizers and the evaluation of our method on more complex datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Acknowledgements The second and third named authors thank the Swedish Research Council, the Knut and Alice Wallenberg Foundation and the European Research Council (ERC-BIRD-884807) for their support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' This work has also received funding from the NWO-TTW Programme “Efficient Deep Learning” (EDL) P16-25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' 5 Extended Abstract Track P´erez Rey Marchetti Kragic Jarnikov Holenderski References Kartik Ahuja, Jason Hartford, and Yoshua Bengio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Properties from Mechanisms: An Equiv- ariance Perspective on Identifiable Representation Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Yoshua Bengio, Aaron Courville, and Pascal Vincent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Representation learning: A review and new perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' IEEE Transactions on Pattern Analysis and Machine Intelligence, 35(8):1798–1828, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' ISSN 01628828.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='1109/TPAMI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Hugo Caselles-Dupr´e, Michael Garcia-Ortiz, and David Filliat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Symmetry-Based Disen- tangled Representation Learning Requires Interaction with Environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 32(NeurIPS):1–10, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' ISSN 10495258.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Ting Chen, Simon Kornblith, Mohammad Norouzi, and Geoffrey Hinton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' A simple frame- work for contrastive learning of visual representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' In International conference on machine learning, pages 1597–1607.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' PMLR, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Taco Cohen and Max Welling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Learning the Irreducible Representations of Commutative Lie Groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' International Conference on Machine Learning, 5:3757–3770, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Deep Residual Learning for Image Recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' In 2016 IEEE Conference on Computer Vision and Pattern Recog- nition (CVPR), volume 37, pages 770–778.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' IEEE, 6 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' ISBN 978-1-4673-8851-1.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='1002/ chin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='200650130http://ieeexplore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='ieee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='org/document/7780459/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Irina Higgins, David Amos, David Pfau, Sebastien Racaniere, Loic Matthey, Danilo Rezende, and Alexander Lerchner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Towards a Definition of Disentangled Representa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' pages 1–29, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' URL http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='org/abs/1812.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='02230.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Irina Higgins, S´ebastien Racani`ere, and Danilo Rezende.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Symmetry-Based Represen- tations for Artificial and Biological General Intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Frontiers in Computational Neuroscience, 16, 4 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' ISSN 1662-5188.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='3389/fncom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='836498.' metadata={'source': 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Loshchilov and Frank Hutter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Decoupled Weight Decay Regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' In International Conference on Learning Representations, 2 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Giovanni Luca Marchetti, Gustaf Tegn´er, Anastasiia Varava, and Danica Kragic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Equivari- ant Representation Learning via Class-Pose Decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' 7 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='48550/arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' 2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='03116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' URL https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='org/abs/2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='03116v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Kieran Murphy, Carlos Esteves, Varun Jampani, Srikumar Ramalingam, and Ameesh Maka- dia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Implicit representation of probability distributions on the rotation manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' In International Conference on Machine Learning, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Robin Quessard, Thomas D Barrett, and William R Clements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Learning Disentangled Representations and Group Structure of Dynamical Environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' In 34th Conference on Neural Information Processing Systems, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' 6 Extended Abstract Track Non-Free Group Actions Joseph J Rotman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' An introduction to the theory of groups, volume 148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Springer Science & Business Media, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Loek Tonnaer, Luis Armando Perez Rey, Vlado Menkovski, Mike Holenderski, and Jim Portegies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Quantifying and Learning Linear Symmetry-Based Disentanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' In Pro- ceedings of the 39th International Conference on Machine Learning, pages 21584–21608.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' PMLR, 6 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' URL https://proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='mlr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='press/v162/tonnaer22a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Zhirong Wu, Shuran Song, Aditya Khosla, Fisher Yu, Linguang Zhang, Xiaoou Tang, and Jianxiong Xiao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' 3d shapenets: A deep representation for volumetric shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' In Pro- ceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), June 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' 7 Extended Abstract Track P´erez Rey Marchetti Kragic Jarnikov Holenderski 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Appendix 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Proofs of Theoretical Results Proposition 3 Let ϕ : X → 2G be an equivariant map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Then for each x ∈ X from an orbit O, ϕ(x) contains a coset of (a conjugate of) GO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Proof Pick x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Then for every g ∈ Gx it holds that ϕ(x) = ϕ(g · x) = g · ϕ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' In other words Gxh = hh−1Gxh ⊆ ϕ(x) for each h ∈ ϕ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Since h−1Gxh is conjugate to Gx the thesis follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Training Details We implement the neural networks ϕG and ϕO with a backbone ResNet18 (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' For a datapoint x ∈ X, the network implements multiple heads to produce embeddings � ϕ1 G(x), · · · , ϕN G(x) � with ϕi G(x) ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' The output dimension of ϕO is set to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' We train the model for 50 epochs (except for the chair dataset which requires a longer training of 100 epochs) using the AdamW optimizer (Loshchilov and Hutter, 2019) with a learning rate of 10−4 and batches of 16 triplets (x, g, y) ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' The rotating arrows and chair dataset consists of 5000 datapoints per orbit while the double arrows and the tetrahedron datasets consist of 20000 datapoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' For all datasets the test set consists of a random 10% split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Reconstructions We present in Figure 3 some examples of images reconstructed by a decoder ψ trained jointly with the encoder ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' The baseline model is not capable of clearly reconstructing the images compared to our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' As seen from the quantitative results in Table 1, the baseline model is not capable of optimizing the equivariance loss from Equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' This provides a hint that the encoder ϕ might not be converging to a stable representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Consequently, the decoder is incapable of consistently reconstructing the data which results in a higher reconstruction loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' Baseline Ours Figure 3: Reconstruction examples produced by a decoder trained on the latent variables produced by our model and the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} +page_content=' 8 **大*' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE4T4oBgHgl3EQfuQ2l/content/2301.05231v1.pdf'} diff --git a/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf b/o9FQT4oBgHgl3EQfrjZh/content/2301.13384v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..b36b5197366febd5f13c59a65cb4f9e2097fc386 --- 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Jan 2023 +INTERIOR W 2,p ESTIMATE FOR SMALL PERTURBATIONS TO THE +COMPLEX MONGE-AMPERE EQUATION +JINGRUI CHENG, YULUN XU +Abstract. Let w0 be a bounded, C3, strictly plurisubharmonic function defined on +B1 ⊂ Cn. Then w0 has a neighborhood in L∞(B1) with the following property: for +any continuous, plurisubharmonic function u in this neighborhood solving 1 − ε ≤ +MA(u) ≤ 1 + ε, one has u ∈ W 2,p(B 1 +2 ), as long as ε > 0 is small enough depending +only on n and p. This partially generalizes Caffarelli’s interior W 2,p estimates for real +Monge-Ampere to the complex version. +1. Introduction +Monge-Ampere equations are second-order partial differential equations whose lead- +ing term is the determinant of the Hessian of a real unknown function. The Hessian +is required to be positive or at least nonnegative, so that the equations are elliptic or +degenerate elliptic. Monge-Ampere equations can be divided into real or complex, de- +pending on whether one is considering real Hessian or complex Hessian. In the real case, +the Hessian is uij, so that the positivity of the Hessian is a convexity condition. In the +complex case, the Hessian is ui¯j, and its positivity is a plurisubharmonicity condition. +For both real and complex Monge-Ampere, the existence and regularity theory with +smooth data has been well established. +In the real case, it is proved by Caffarelli- +Nirenberg-Spruck [4] on smooth strictly convex domains on Rn. In the complex case, +the foundations of an existence and regularity theory were laid out by Yau [18] in the +setting of a compact K¨ahler manifold, and by Caffarelli-Kohn-Nirenberg-Spruck [5], in +the setting of a smooth pseudo-convex domain. +Another important aspect about the Monge-Ampere equations is their apriori esti- +mates, starting with interior ones. In general, the results known for the real case is much +stronger than the complex case, due to the fact that the solution being convex gives +much more stringent constraint than being plurisubharmonic. For example, the interior +gradient estimate for real Monge-Ampere equation is more or less a trivial matter (if we +know the solution is bounded), since the underlying solution considered is convex. This +is not the case for complex Monge-Ampere, and the boundedness of plurisubharmonic +function only gives the gradient being in L2. +Arguably the most important estimate of Monge-Ampere is to get second derivative +estimates. If we get such estimates in L∞, then the equation becomes unformly elliptic +and the standard theory can apply. On the other hand, the bad news is that for both +real and complex Monge-Ampere equations, there are no purely interior C2 estimates. +Indeed, having a convex solution to det uij = 1 in a domain doesn’t imply u ∈ C2 in the +interior, due to a counterexample by Pogorelov [14] (a counterexample for the complex +version is given by He in [9]). In general, one needs to impose some boundary conditions +Date: Nov 2022. +1 + +2 +JINGRUI CHENG, YULUN XU +(say, u = 0 on the boundary), in order to conclude that D2u is bounded in the interior +(Pogorelov’s estimate [13]). Based on that, Caffarelli’s proved the following interior W 2,p +estimate when the right hand side is a small perturbation of a constant: +Theorem 1.1. Let Ω ⊂ Rn be a convex domain such that B1 ⊂ Ω ⊂ Bn and u is a weak +solution to det uij = f with |f − 1| < ε and u = 0 on ∂Ω. Then for any 1 < p < ∞, if +ε > 0 is small enough depending only on p and n, then ||u||W 2,p(B 1 +2 ) can be bounded by +a constant depending only on p and n. +In the above theorem, the weak solution is defined using the measure of the im- +age of the gradient mapping. +One could also replace the boundary condition by a +strict convexity assumption, meaning that the supporting plane of a convex function +touches the function only at one point. That is, we have: +Theorem 1.2. Let Ω ⊂ Rn be a bounded convex domain, and u is a weak solution to +det uij = f with |f − 1| < ε which is strictly convex. Then for any compact subdomain +Ω′, one has ||u||W 2,p(Ω′) ≤ C, where C depends on p, n, dist(Ω′, ∂Ω), the modulus of +strict convexity of u, as long as ε is small enough depending only on p and n. +Theorem 1.2 actually follows from Theorem 1.1. Indeed, for any x0 ∈ Ω′, we can take +lx0 to be a linear function touching u from below, then Sc := {u(x) ≤ lx0(x) + c} will be +contained in Ω for c small enough, due to the strict convexity. Then one can normalize +Sc to be in the situation of Theorem 1.1. +The goal of this paper is to generalize (partially) Theorem 1.1 and Theorem 1.2 to +the complex Monge-Ampere equations. More precisely, we show that +Theorem 1.3. Let Ω ⊂ Cn be a bounded domain with B1−γ0 ⊂ Ω ⊂ B1+γ0 for some +γ0 > 0. Let u ∈ C2(Ω) ∩ PSH(Ω) ∩ C(¯Ω) be such that 1 − ε ≤ det ui¯j ≤ 1 + ε in Ω and +u = 0 on ∂Ω. Given 1 < p < ∞, if γ0 is small enough depending only on n, and ε small +enough depending only on n and p, then +||u||W 2,p(B 1 +2 ) ≤ C, +|| +� +i +1 +ui¯i +||Lp(B 1 +2 ) ≤ C, +where the constant C depends only on n and p. +This theorem should be understood as the analogue of Theorem 1.1. More generally, +we have: +Theorem 1.4. Let w0 be a smooth function in the unit ball such that for some C0 > 1: +1 +C0 +I ≤ (w0)zi¯zj ≤ C0I, |D3w0| ≤ C0 in B1. +Then there exists δ0 > 0 small enough, depending only on C0 and n, such that for all +u ∈ PSH(B1) ∩ C(B1) with |u − w0| ≤ δ0 on B1, solving 1 − ε ≤ MA(u) ≤ 1 + ε, we +have u ∈ W 2,p(B 1 +2 ) and � +i +1 +ui¯i ∈ Lp(B 1 +2 ), as long as ε is small enough depending only +on n and p. +In the above, MA(u) is the complex Monge-Ampere operator defined for continuous +plurisubharmonic functions, in the Bedford-Taylor sense (see [1]), so that MA(u) = +det ui¯j when u ∈ C2. + +INTERIOR W 2,p ESTIMATE FOR SMALL PERTURBATIONS TO THE COMPLEX MONGE-AMPERE EQUATION3 +If one compares Theorem 1.3 to Theorem 1.1, or Theorem 1.4 to Theorem 1.2, the +biggest difference is that we have to assume our solution is close to a smooth plurisub- +harmonic function. The reason we have to make this assumption is related to whether +one has Pogorelov type estimates for complex Monge-Ampere equations, which has been +open until now. Note that the interior C2 estimates for the complex Monge-Ampere +equation with zero boundary values were studied by F. Schulz in [16], using the integral +approach of N. M. Ivochikina [10] for real Monge-Ampere equations. However, the proof +in [16] is not complete, which was first pointed out by Blocki [2]. We will comment more +on the technical aspect later, but for now, let us present some direct consequences of our +main theorem, which seems new and interesting. +First we observe that Theorem 1.4 would give us the following result in the manifold +setting: +Corollary 1.1. Let (M, ω0) be a compact K¨ahler manifold. Let ϕ ∈ PSH(M, ω0)∩C(M) +be the solution to: +(ω0 + +√ +−1∂ ¯∂ϕ)n = fωn +0 , ω0 + +√ +−1∂ ¯∂ϕ > 0, +where |f −1| < ε and +� +M fωn +0 = +� +M ωn +0 . Let 1 < p < ∞, then we have that ϕ ∈ W 2,p(M) +as long as ε is small enough depending only on p, n and the background metric ω0. +Similar results would also hold for the setting of bounded domains. In other words, +we have: +Corollary 1.2. Let Ω ⊂ Cn be a bounded domain. Let u0 ∈ C3(Ω) ∩ C(¯Ω) ∩ PSH(Ω) +be the solution to det(u0)i¯j = f0 > 0 in Ω and u0|∂Ω = ϕ0. Let u ∈ C(¯Ω) ∩ PSH(Ω) be +the solution to MA(u) = f and u|∂Ω = ϕ such that |f − f0| < ε, |ϕ − ϕ0| < ε. Let Ω′ +be a compact subdomain of Ω, then we have u ∈ W 2,p(Ω′), as long as ε is small enough +depending only on Ω′, Ω, the C3 bound and complex Hessian lower bound of u0 in a +neighborhood of Ω′, p and n. +One more application of Theorem 1.4 is the following Liouville theorem for entire +scalar flat metric on Cn, which is a generalization of a result by Yu Wang [17]: +Corollary 1.3. Let u be a C2 plurisubharmonic function on Cn. Denote ωu = √−1∂ ¯∂u +and assume that ωu is scalar flat, namely +n +� +i,j=1 +ui¯j∂i¯j +� +log det ua¯b +� += 0. +Then there exists εn > 0 small enough depending only on n, such that we can deduce u +is quadratic, provided that: +(1) limr→∞ +supBr |u(z)−|z|2| +r2 +≤ εn, +(2) 1 − εn ≤ det ua¯b ≤ 1 + εn. +The result by Yu Wang [17] is a special case of the above Corollary with det ua¯b = 1. +Finally we observe the following C2,α estimate for complex Monge-Ampere. For K > +0, 0 < α < 1 and C1 > 1, we define the following class of functions: +F(K, α, C1) = {f is defined on B1 : ||f||α,B1 ≤ K, +1 +C1 +≤ f ≤ C1 on B1}. + +4 +JINGRUI CHENG, YULUN XU +Corollary 1.4. Let w0 be as in Theorem 1.4. Then there exists δ0 > 0, depending only +on C0, n, K, α, C1, such that for any u ∈ PSH(B1) ∩ C(B1) with |u − w0| ≤ δ0 on B1 +and solving MA(u) = f for some f ∈ F(K, α, C1), we have u ∈ C2,α(B 1 +2). +Next we would like to explain the ideas of proof for Theorem 1.3. The heart of the +idea is from Caffarelli’s paper [3] which we explain first. Since the solution u is strictly +convex, we may consider sections of u of the form Sc(x0) := {(u − l)(x) ≤ u(x0) + c} +which is strictly contained in Ω, where l(x) is the supporting linear function of u at x0 +and c > 0 is called the “height” of the section. Now we solve det wij = 1 on this open +set, equaling u on the boundary. From Pogorelov estimate, we know that w is smooth +in the interior. By doing Taylor expansion for w, we find that the sections of w will be +close to ellipsoids. On the other hand, since f is close to 1, we also have u is very close +to w by maximum principle, hence the sections of u are close to ellipsoids as well. If the +shape of the ellipsoids are comparable to a ball for heights going to 0, then the second +derivatives are under control at that point. The whole point of W 2,p estimate is then to +estimate the measure of the set where the shape of such ellipsoids loses control. (which +is reflected by the opening of the paraboloid touching u from below) For this purpose, +we will need a version of Vitali’s covering lemma, but adapted to sections. To establish +the covering lemma for sections, a crucial property we need is the following engulfing +property: +If Sµ1(x1) ∩ Sµ2(x2) ̸= ∅, with µ1 ≤ µ2, then Sµ1(x1) ⊂ S10µ2(x2). +This property would be a result of compactness. +Indeed, if u were the standard +solution 1 +2|x|2, we would have Sµ1(x1) = B√µ1(x1), Sµ2(x2) = B√µ2(x2) and the engulfing +property indeed holds. We can still expect this property if u is close to a quadratic +polynomial. +We follow similar lines of argument in the proof of Theorem 1.3. The first hurdle +we face is to take sections with u. Unlike the convex function, given x0 ∈ Ω, it is not +clear whether one can find a pluriharmonic function h, for which {u − h < u(x0) + c} +is compactly contained in Ω for c > 0 small. Even though this is not clear in general, +we show that, however, it is indeed possible if u is close to a smooth plurisubharmonic +function whose complex Hessian has a lower bound. +Next we need to solve the Dirichlet problem det wi¯j = 1, w = u on the boundary of a +small section of u, similar to what we did in the real case. The problem we are facing +now, is that we do not know if w is smooth, since Pogorelov’s estimate is not known +for the complex Monge-Ampere equations. However, if u is close to a smooth, strictly +plurisubharmonic function, then the section defined by u will be close to an ellipsoid from +the very beginning. This will allow us to use Savin’s perturbation result to conclude that +w is indeed smooth in the interior. In this paper, we use an induction process to construct +sections Sµ(x0) for µ > 0 and small, which takes the form {(u − hµ,x0)(z) ≤ u(x0) + µ}, +where hµ,x0(z) is pluriharmonic. Moreover, we also show that Sµ(x0) remains close to +an ellipsoid in the induction process. +A drawback with our construction is that it is highly non-canonical, since it relies on +solving Dirichlet problem on a sequence of smaller and smaller sections “centered at” +x0. This construction of Sµ(x0) does not commute with linear transformations we use +to normalize the ellipsoids. We explain this matter in greater detail in Subsection 4.2, +under Proposition 4.16. +The fact that Sµ(x0) is non-canonical makes it apparently very hard to relate Sµ(x0) +to Sµ(x1), even if x0 and x1 are very close. This would make it seeming impossible to + +INTERIOR W 2,p ESTIMATE FOR SMALL PERTURBATIONS TO THE COMPLEX MONGE-AMPERE EQUATION5 +prove the engulfing property for the Sµ(x0) we constructed. (let us also comment that +the hµ,x0(z) above is unbounded in general as µ → 0, which is totally different from the +real case.) However, one surprising thing we observed is that, if a section in the form +{u − h ≤ u(x0) + µ} happens to be close to an ellipsoid, then the shape of this ellipsoid +is “unique” in a quatitative sense. This key observation allows us to show the engulfing +property. +The above discussion shows the importance to understand whether we have Pogorelov +estimates for complex Monge-Ampere equations. +In particular, we are motivated to +make the following definition: +Definition 1.5. Let Ω ⊂ Cn be a bounded domain and Ω′ ⊂ Ω be compactly contained +in Ω. We say that (Ω, Ω′) has the Pogorelov property if: +(1) There exists u ∈ PSH(Ω) ∩ C(¯Ω), solving MA(u) = 1 in Ω in the sense of +Bedford-Taylor, and u = 0 on ∂Ω, +(2) The solution u is C2 (hence C∞) on Ω′. +If one carefully checks the argument of the present paper, what we really proved is +the following result: +Theorem 1.5. Let u ∈ PSH(B1) ∩ C(B1) solve 1 − ε ≤ MA(u) ≤ 1 + ε in B1. Assume +that there exists finitely many Pogorelov pairs (Ωi, Ω′ +i), 1 ≤ i ≤ N, such that +(1) Each Ωi is of the form {u − hi < ci} for some pluriharmonic function hi and +ci ∈ R. +(2) ¯B 1 +2 ⊂ ∪N +i=1Ω′ +i. +Then we have u ∈ W 2,p(B 1 +2), as long as ε is small enough, depending only on n, p, the +lower and upper Hessian bound and C3 bound for ui on Ω′ +i. Here ui is the solution to +MA(ui) = 1 on Ωi and ui = 0 on ∂Ωi. +For now, it seems mysterious to characterize when the assumptions (1) and (2) above +hold. We are only able to verify such assumptions when u is close to a smooth function +for the moment. For example, in the setting of Theorem 1.3, Ω = {u < 0}, and (Ω, B0.6) +has the Pogorelov property, as long as Ω is close to a unit ball, thanks to Savin’s C2,α +estimates for small perturbations. +To conclude the Introduction, we will explain the organization of the rest of the paper. +In Section 2, we include some definitions, notations and some preliminary results we +will use again and again in this paper. +In Section 3, we show how to reduce Theorem 1.4 to Theorem 1.3. The later is a +special case of the former by taking w0 = |z|2 − 1. Section 4-6 below are devoted to the +proof of Theorem 1.3. +In Section 4, we construct sections Sµ(x0) for u which are of the form {(u−hµ,x0)(z) ≤ +u(x0) + µ}, where hµ,x0 is pluriharmonic and Sµ(x0) is close to an ellipsoid. The second +half of this section focuses on the engulfing property of sections. +In Section 5, we prove some measure-theoretic lemmas which will be needed to estimate +the “bad” set where the second derivative loses control. These lemmas are all standard +results for balls, but we have to adapt them to Sµ(x0) we constructed in Section 4. The +engulfing property of Sµ(x0) is crucially used in establishing these lemmas. + +6 +JINGRUI CHENG, YULUN XU +In Section 6, we verify that the “bad sets” fits in the assumptions of the measure +theoretic lemmas in Section 5, and obtain the power decay of the measure of the bad +set. Contrary to the real case, we first obtain control for the mixed Hessian ui¯j, then +the full W 2,p estimate follows from the classical Lp estimate for Laplacian. +In Section 7, we discuss some implications of Theorem 1.4. In particular, we give +detailed proofs for Corollary 1.1, 1.2 ,1.3 and 1.4. +2. preliminaries +The key result we will need again and again is the following lemma: +Lemma 2.1. Let u be a viscosity solution to det(ui¯j) = 1 in B1. Suppose that ||u − +w||L∞ ≤ δ, where w is a smooth solution to det(wi¯j) = 1 in B1. If δ is small enough +depending only on the smoothness of w and n, we have ||u||C4(B0.99) ≤ C, where C has +the same dependence as δ. +The small perturbation theorem of Savin is for more general fully nonlinear elliptic +equations, and applies to equations of the form F(D2u, x) = 0, where F(r, x) : S ×B1 → +R is C2 in the x variable, elliptic in the r variable, and uniformly elliptic only for r in +a neighborhood of 0, with F(0, x) = 0. Then for any solution to F(D2u, x) = 0 with +||u||L∞ small enough, we would get C2,α estimate in the interior. +Lemma 2.1 follows from the general perturbation theorem of Savin in [15] by writing +the complex Monge-Ampere operator in the real form. The details can be found in Yu +Wang [17]. Also once we get C2,α estimate the above equation, it is straightforward to +apply standard elliptic estimates to improve the estimate to C4. In this paper, we will +mostly apply Lemma 2.1 with w = |z|2 − 1. +Another thing we will need is the following interpolation lemma: +We need the following interpolation estimate. +Lemma 2.2. Let u be a C4 function in Br0(0) with |D4u|L∞(Br0) ≤ C, and |u|L∞(Br0) ≤ +µ, then for any 0 < λ < r0, one has +|u(0)| ≤ µ, |Du(0)| ≤ C(λ3 + µ +λ), +|D2u(0)| ≤ C(λ2 + µ +λ2 ), |D3u(0)| ≤ C(λ + µ +λ3 ). +Proof. From the Taylor expansion, we find that, for x ∈ Br0, the following estimate +holds: +|u(x) − +� +|α|≤3 +Dαu(0) +α! +xα| ≤ CnC|x|4. +If we restrict to |x| ≤ λ, we find that +| +� +|α|≤3 +Dαu(0) +α! +xα| ≤ CnCλ4 + µ. +This is equivalent to: +sup +|y|≤1 +| +� +|α|≤3 +Dαu(0) +α! +λ|α|yα| ≤ CnCλ4 + µ. + +INTERIOR W 2,p ESTIMATE FOR SMALL PERTURBATIONS TO THE COMPLEX MONGE-AMPERE EQUATION7 +Hence we would have: +|Dmu(0)|λm ≤ C′ +nCλ4 + µ, m = 1, 2, 3. +□ +In the present paper, we will frequently use differentiation with respect to complex +variables. Following the usual conventions, we define: +(2.1) +∂zi = 1 +2(∂xi − +√ +−1∂yi), ∂¯zi = 1 +2(∂xi + +√ +−1∂yi), zi = xi + +√ +−1yi. +So that we find the Laplacian operator can be written as: +∆ = 4 +n +� +i=1 +∂zi¯zi. +A notion we will encounter again and again is pluriharmonic function, which we explain +below. +Definition 2.3. Let Ω ⊂ Cn be a domain. Let h ∈ C2(Ω). We say that h is plurihar- +monic if hzi¯zj(z) = 0 for all 1 ≤ i, j ≤ n. +Note that h being pluriharmonic will imply h being harmonic, but not the other +way. One can also see that if h is the real part of a holomorphic function, then h is +pluriharmonic. +Another definition we need is: +Lemma 2.4. Let T : Cn → Cn be a C-linear transformation, we define ||T|| to be the +operator norm of T, namely: +||T|| = sup +|z|≤1 +|T(z)|. +In the proof of engulfing property of sections, we will need to frequently consider +dilation maps. Hence we introduce the following definition to make the notations simpler. +Definition 2.5. Let E ⊂ Cn be a set and x0 ∈ E. We will sometimes denote E to be +E(x0) to indicate it is a “pointed set”. Let c > 0, we define: +cE(x0) = {x0 + c(y − x0) : y ∈ E(x0)}. +Namely cE(x0) is the image of the dilation map centered at x0 by a factor c. +3. Reduction of Theorem 1.4 to Theorem 1.3 +In this section, we will show how to use Theorem 1.3 to deduce Theorem 1.4. +To see the implication in an intuitive way, we can take any point x0 ∈ B 1 +2. After +subtracting a pluriharmonic function hx0, we will see that {w − hx0(z) ≤ w0(x0) + µ} +will be close to an ellipsoid (centered at x0) when µ is small enough. The same would be +true for {u − hx0(z) ≤ µ}. After normalization the ellipsoid to a unit ball, we are in the +situation of Theorem 1.3 and we get u ∈ W 2,p in a neighborhood of x0. We will make +this idea precise in the rest of this section. + +8 +JINGRUI CHENG, YULUN XU +Let w0 be as in Theorem 1.4. Take any point x0 ∈ B 1 +2, we can write down the Taylor +expansion of w0 at x0: +w0 = w0(x0) + Re( +� +i +lx0,i(z − x0)i) + +� +i,j +ax0,i¯j(z − x0)i(z − x0)j ++ Re( +� +i,j +bx0,ij(z − x0)i(z − x0)j) + O(|z − x0|3). +(3.1) +Define hx0(z) = Re(� +i lx0,i(z − x0)i) + Re(� +i,j bx0,ij(z − x0)i(z − x0)j), First we want +to show that if we have another function u0, such that |u0 − w0| ≤ δ, then the section +{z : u0 − hx0(z) ≤ u0(x0) + µ} will be close to an ellipsoid if µ is small, but much larger +than δ. More precisely +Lemma 3.1. Let w0 be as stated in Theorem 1.4. Namely we assume that w0 ∈ C3(B1), +and +1 +C0I ≤ (w0)i¯j ≤ C0I, |D3w0| ≤ C0 on B0.99. Let δ ≥ 0 and u0 is a function on B1 +with |u0 − w0| ≤ δ on B0.95. Then there exists C1 > 0 large enough and µ0 > 0 small +enough depending only on C0, such that for all µ with 4C1δ ≤ µ ≤ µ0, we have: +(1 − C1γ)Eµ(x0) ⊂ {z ∈ B +1 +2C2 +0 +(x0) : (u0 − hx0)(z) ≤ u0(x0) + µ} ⊂ (1 + C1γ)Eµ(x0). +Moreover, (u0 − hx0)(z) = u0(x0) + µ on ∂{z ∈ B +1 +2C2 +0 +(x0) : (u0 − hx0)(z) ≤ u0(x0) + µ}. +Here γ = δ +µ + µ +1 +2 and Eµ(x0) = {z ∈ Cn : �n +i,j=1 ax0,ij(z − x0)i(z − x0)j ≤ µ}. +Proof. Using (3.1), we see that on B1: +(3.2) − C0|z − x0|3 ≤ w0 − w0(x0) − hx0(z) − +� +i,j +ax0,ij(z − x0)i(z − x0)j ≤ C0|z − x0|3. +Since |u0 − w0| ≤ 2δ0 on B0.95, we see that for any x0 ∈ B0.95: +−2δ − C0|z − x0|3 ≤ u0 − u0(x0) − hx0(z) − +� +i,j +ax0,ij(z − x0)i(z − x0)j +≤ C0|z − x0|3 + 2δ. +Let z ∈ B +1 +2C2 +0 +(x0) and (u0 − hx0)(z) ≤ u0(x0) + µ, we get +−2δ − C0|z − x0|3 ≤ µ − +� +i,j +ax0,ij(z − x0)i(z − x0)j. +Since ax0,ij ≥ +1 +C0 I, we get +1 +C0 +|z − x0|2 ≤ +� +i,j +ax0,ij(z − x0)i(z − x0)j ≤ µ + 2δ + C0 · +1 +2C2 +0 +|z − x0|2. +So that +|z − x0| ≤ +� +2C0(µ + 2δ) ≤ +� +3C0µ. +So that +� +i,j +ax0,ij(z − x0)i(z − x0)j ≤ u0 − u0(x0) − hx0(z) + 2δ + C0|z − x0|3 +≤ µ + 2δ + C0(3C0µ) +3 +2 ≤ (1 + C1γ)µ. + +INTERIOR W 2,p ESTIMATE FOR SMALL PERTURBATIONS TO THE COMPLEX MONGE-AMPERE EQUATION9 +This proves the inclusion +{z ∈ B +1 +2C2 +0 +(x0) : (u0 − hx0)(z) ≤ u0(x0) + µ} ⊂ (1 + C1γ)Eµ(x0). +Now we prove the other inclusion. Let z ∈ (1 − C1γ)Eµ(x0), which implies +� +i,j +ax0,ij(z − x0)i(z − x0)j ≤ (1 − C1γ)2µ ≤ (1 − C1γ)µ. +So that +|z − x0|2 ≤ C0 +� +i,j +ax0,ij(z − x0)i(z − x0)j ≤ C0(1 − C1γ)µ < +1 +2C2 +0 +, +as long as µ ≤ µ0 with µ0 small enough. Moreover +u0 − u0(x0) − hx0(z) ≤ +� +i,j +ax0,ij(z − x0)i(z − x0)j + C0|z − x0|3 + 2δ +≤ (1 − C1γ)µ + C0(3C0µ) +3 +2 + 2δ ≤ µ. +The last inequality would hold if we take C1 to be large enough depending on C0. This +proves the inclusion: +(1 − C1γ)Eµ(x0) ⊂ {z ∈ B +1 +2C2 +0 +(x0) : (u0 − hx0)(z) ≤ u0(x0) + µ}. +□ +Now we are ready to verify the implication from Theorem 1.3 to 1.4. +Let u and w0 be as stated in Theorem 1.4. Let µ > 0 and x0 ∈ B0.8. Let Tµ,x0 be a +C-affine transformation such that Tµ,x0(B√µ(0)) = Eµ(x0). Define +(3.3) +uµ,x0(ζ) = +1 +µ| det Tµ,x0| +2 +n +(u − hx0 − µ)(Tµ,x0(√µζ)). +Since Eµ(x0) is defined in terms of ax0,ij, with +1 +C0 ≤ ax0,ij ≤ C0I, it is easy to see that: +||Tµ,x0|| ≤ C2, ||T −1 +µ,x0|| ≤ C2, +1 +C2 +≤ | det Tµ,x0|2 ≤ C2. +Here C2 is a large enough constant depending only on C0 and n. Define Ωµ = T −1 +µ,x0({z ∈ +B +1 +2C2 +0 +: (u − hx0)(z) ≤ u(x0) + µ}). Then by straightforward calculation and Lemma 3.1, +we can see the following: +Lemma 3.2. There is µ0 > 0 small enough depending only on C0 such that for all +4C1δ0 ≤ µ ≤ µ0 (with C1 > 0 being the constant given by Lemma 3.1), we have +(1) B1−C1γ ⊂ Ωµ ⊂ B1+C1γ, with γ = δ0 +µ + µ +1 +2 . +(2) det(uµ,x0)ζi ¯ζj = f(Tµ,x0(√µζ)) in Ωµ, uµ,x0 = 0 on ∂Ωµ. +The renormalized function uµ,x0 fits in the assumptions for Theorem 1.3 after suitably +choosing the parameters, and Theorem 1.4 follows as a direct consequence: +Corollary 3.3. Theorem 1.4 holds, if we assume Theorem 1.3 and u ∈ C2(B1). + +10 +JINGRUI CHENG, YULUN XU +Proof. We wish to apply Theorem 1.3 to each uµ,x0. In order to do so, we just need: +C1γ = C1(δ0 +µ + µ +1 +2 ) ≤ γ0(n), |f(Tµ,x0(√µζ)) − 1| ≤ ε(n, p). +Here γ0(n) and ε(n, p) are the constants given by Theorem 1.3. +So we could just take µ so that 2C1µ +1 +2 ≤ 1 +2γ0(n) and also µ ≤ µ0 (given by Lemma +3.2). With this µ, we can take δ0 so that C1 δ0 +µ ≤ 1 +2γ0(n) and also that 4C1δ0 ≤ µ. We +fix this choice from now on. +Since we assumed that Theorem 1.3 holds, we conclude that: +||uµ,x0||W 2,p(B 1 +2) ≤ C, +where C is a constant depending only on n and p. Then using (3.3) we may go back to +u and obtain that +||u||W 2,p(E 1 +2 µ(x0)) ≤ C′. +Here C′ depends on C0, p and n. Note that µ is already chosen which depends only on +C0 and n, and hx0 is defined using w0, which can be bounded in terms of C0 and n as +well. +Note that E 1 +2µ(x0) contains Br0(x0) for some r0 > 0 small enough (depending only on +C0) for any x0 ∈ B0.8. The result of Theorem 1.4 would follow right away. +□ +Next we can use an approximation argument to remove the assumption that u ∈ +C2(B1). +Proof. (of Theorem 1.4, without assuming u ∈ C2(B1)) +First, we can find fk ∈ C∞(B0.9) such that fk → f in L2(B0.9) and |fk − 1| ≤ ε (since +|f − 1| ≤ ε, one can see that the standard smoothing by convolution will preserve this +property). We can also find a sequence of gk ∈ C∞(∂B0.9), such that gk → u uniformly +on ∂B0.9 (since u is assumed to be continuous). +Let vk be the solution to the Dirichlet problem: +det(vk)i¯j = fk +in B0.9, vk = gk on ∂B0.9. +From Caffarelli-Kohn-Nirenberg-Spruck [5], we know that vk ∈ C∞( ¯B0.9). Also from the +following Lemma 3.4, we know that vk → u uniformly on ¯B0.9. Hence for large enough +k, vk will fullfil the assumption of Theorem 1.4, and each vk is smooth. Hence we may +use Corollary 3.3 to conclude that +||vk||W 2,p(B 1 +2 ) ≤ C, +where C depends only on C0, n and p. In particular, C is uniform in k. Passing to the +limit, we see that u ∈ W 2,p(B 1 +2), with the same bound C. +□ +In the above, we used the following stability estimate from Dinew-Kolodziej [7] to +deduce the uniform convergence of the approximation sequence: +Lemma 3.4. ([7]) Let ωE be the Euclidean K¨ahler form on Cn. Let q > 1. Consider +u, v ∈ PSH(Ω) ∩ C(¯Ω). Assume that +MA(u) = f, MA(v) = g. + +INTERIOR W 2,p ESTIMATE FOR SMALL PERTURBATIONS TO THE COMPLEX MONGE-AMPERE EQUATION +11 +for some f, g ∈ Lq(Ω, dV ). Then +sup +Ω +(v − u) ≤ sup +∂Ω +(v − u) + c(q, n, diam(Ω))||f − g|| +1 +n +Lq(Ω). +4. construction of sections +From now on we will focus on the proof of Theorem 1.3. Our first step is to construct +sections of u which are close to ellipsoids via an induction process. Next, we prove some +fine properties of the sections which ensures that they are good differentiation basis. +4.1. Inductive construction of sections. Let us summarize our construction into the +following proposition: +Proposition 4.1. Let Ω and u be as stated in Theorem 1.3, with γ0 small enough +depending only on n. Let 0 < σ < 1 be given. Then there exists ε > 0 depending only on +σ and n, such that if |f − 1| ≤ ε, the following hold: +(1) There exists µ0 > 0 small enough depending only on n and σ, such that for +all x0 ∈ B0,8 and all µ ≤ µ0, there exists a degree 2 pluriharmonic polynomial +hµ,x0(z) with hµ,x0(x0) = 0, such that +(1 − 0.1σ)Eµ(x0) ⊂ Sµ(x0) := {z ∈ Ω : (u − hµ,x0)(z) ≤ u(x0) + µ} +⊂ (1 + 0.1σ)Eµ(x0). +In the above, Eµ(x0) = {z ∈ Cn : �n +i,j=1 aµ,x0,ij(z − x0)i(z − x0)j ≤ µ}, with +aµ,x0,ij being positive Hermitian and det aµ,x0,ij = 1. +(2) There is a function c(σ) : σ ∈ (0, 1) → R>0, such that for any x0 ∈ B0.8 and any +0 < µ1 ≤ µ2 ≤ +µ0 +1+c(σ), one has Sµ1(x0) ⊂ S(1+c(σ))µ2(x0). Moreover, 0 < c(σ) ≤ +C2,nσ +1 +2 for some dimensional constant C2,n. +(3) There is a dimensional constant C3,n > 0 such that for all 0 < µ ≤ µ0 and any +x0 ∈ B0.8, there exists a C-linear transformation Tµ,x0, such that | det Tµ,x0| = 1, +Tµ,x0(B√µ(0)) = Eµ(x0), Tµ0,x0 = id. Moreover, for any 0 < µ1 < µ2 ≤ µ0 and +any x0 ∈ B0.8: +||Tµ1,x0 ◦ T −1 +µ2,x0|| ≤ C3,n(µ2 +µ1 +) +C3,nσ +1 +2 +− log(0.1σ) , ||Tµ2,x0 ◦ T −1 +µ1,x0|| ≤ C3,n(µ2 +µ1 +) +C3,nσ +1 +2 +− log(0.1σ) . +Remark 4.2. We will make a choice of σ later on, depending on the value of p in the +W 2,p estimate. (The larger p is, the smaller σ needs to be.) So that the choice of ε +eventually depends only on p and n. +We fix some 0 < σ < 1, and describe the construction of Sµ(x0). +First we solve the following Dirichlet problem on Ω. +det((v0)i¯j) = 1 in Ω +v0 = 0 on ∂Ω. +(4.1) +To start the process, we need that v0 is smooth in the interior. This is guaranteed by +the fact that Ω is close to B1. More precisely, we have: +Lemma 4.3. Let Ω ⊂ Cn be a bounded domain and B1−γ(0) ⊂ Ω ⊂ B1+γ(0) for some +0 < γ < 1. Let v0 be the solution to the Dirichlet problem in (4.1), then +|z|2 − 1 − 3γ ≤ v0 ≤ |z|2 − 1 + 3γ. + +12 +JINGRUI CHENG, YULUN XU +Moreover, there exists γn > 0 small enough, such that if γ ≤ γn, we have v0 ∈ C4( ¯B0.9) +with ||v0 − (|z|2 − 1)||C4,B0.9 ≤ C. Here C depends only on n. +Proof. From the assumption, we see that 1 − γ ≤ |z| ≤ 1 + γ on ∂Ω, hence +|z|2 − 1 − 3γ ≤ v0 ≤ |z|2 − 1 + 3γ, +on ∂Ω. +Note that both |z|2 − 1 − 3γ and |z|2 − 1 + 3γ satisfy det ui¯j = 1. Hence from maximum +principle, we see that +|z|2 − 1 − 3γ ≤ v0 ≤ |z|2 − 1 + 3γ in Ω. +If γ is small enough, then we may use Savin’s estimate (Lemma 2.1) to see that v0 is +bounded in C2,α on B0.95 by a dimensional constant C. Then one can differentiate the +det(v0)i¯j = 1 and use classical elliptic estimates to conclude that |v0|C4,B0.9 ≤ C′. +□ +As a consequence, we get that v0 is actually C3 close to |z|2 − 1, hence convex if γ is +small enough. More precisely: +Corollary 4.4. Let v0 be as in Lemma 4.3. Then for any 0 < γ ≤ γn with γn small +enough, we have: +|Dm(v0 − (|z|2 − 1))|B0.9 ≤ Cnγ1− m +4 , m = 1, 2, 3. +Proof. This follows from Lemma 4.3 and the interpolation estimates. +□ +In order to define sections for u, we need to show that u and v0 are sufficiently close. +This is guaranteed by the following lemma: +Lemma 4.5. Assume that det ui¯j = f in Ω and u|∂Ω = 0. Let v0 be the solution to the +Dirichlet problem (4.1). Assume that 1− ε ≤ f ≤ 1+ ε. Then we have (1+ ε) +1 +n v0 ≤ u ≤ +(1 − ε) +1 +n v0. In particular +|v0 − u| ≤ 4ε in Ω. +Proof. Since det +� +(1 + ε) +1 +n (v0)i¯j +� += 1 + ε ≥ f = det ui¯j ≥ 1 − ε = det +� +(1 − ε) +1 +n (v0)i¯j +� +and those three functions all have the same boundary value, we can use the maximum +principle to conclude that +(1 + ε) +1 +n v0 ≤ u ≤ (1 − ε) +1 +n v0. +So that +u − v0 ≤ ((1 − ε) +1 +n − 1)v0 ≤ 2 +nε|v0| ≤ 4 +nε ≤ 4ε. +In the above, we used Lemma 4.3 that |v0| ≤ 2 (if γ is small enough). The lower estimate +for u − v0 is completely similar. +□ +To define sections for the first step, we need an analogue of Lemma 3.1: +Lemma 4.6. Let v ∈ C3(B1) and |vi¯j − δij| ≤ c, |D3v| ≤ c on B0.9 where 0 < c < 1. +Let δ ≥ 0 and u0 is a function on B1 with |u0 − v| ≤ δ on B0.95. Then for small enough +c > 0 (depending only on n) and for any x0 ∈ B0.8, there is a degree 2 pluriharmonic +polynomail hv,x0(z), such that for all µ with 4δ ≤ µ ≤ 0.9(0.9 − |x0|)2 we have: +(1 − γ)Eµ(x0) ⊂ {z ∈ B1 : (u0 − hv,x0)(z) ≤ u0(x0) + µ} ⊂ (1 + γ)Eµ(x0), +and Eµ(x0) ⊂ B1. +Here γ = +2δ +µ + (3c) +3 +2 µ +1 +2 , and Eµ(x0) = {z : � +i,j vi¯j(x0)(z − +x0)i(z − x0)j ≤ µ}. + +INTERIOR W 2,p ESTIMATE FOR SMALL PERTURBATIONS TO THE COMPLEX MONGE-AMPERE EQUATION +13 +Proof. The proof is very similar to Lemma 3.1. First we can write down the Taylor +expansion of v at x0: +v(z) = v0(x0) + Re +� � +i +l0,x0,i(z − x0)i +� ++ +� +i,j +vi¯j(x0)(z − x0)i(z − x0)j ++ Re +� � +i,j +b0,x0,ij(z − x0)i(z − x0)j +� ++ O(|z − x0|3). +Define hv,x0(z) = Re +� � +i l0,x0,i(z − x0)i + � +i,j b0,x0,ij(z − x0)i(z − x0)j +� +, and use the +bound for D3v, u0 − v, we get +−2δ−c|z−x0|3 ≤ (u0 −hv,x0)(z)−u0(x0)− +� +i,j +vi¯j(x0)(z−x0)i(z − x0)j ≤ c|z−x0|3 +2δ. +Let z ∈ B1 with (u0 − hv,x0)(z) ≤ u0(x0) + µ, we get +� +i,j +vi¯j(x0)(z − x0)i(z − x0)j ≤ µ + 2δ + c|z − x0|3. +By choosing c small, we may assume that vi¯j(x0) ≥ 1 +2I, so that +1 +2|z − x0|2 ≤ µ + 2δ + 2c|z − x0|2. +Hence if 4c < 1 +2, we get +|z − x0|2 ≤ 2(µ + 2δ) ≤ 3µ. +Hence +� +i,j +vi¯j(x0)(z − x0)i(z − x0)j ≤ µ(1 + 2δ +µ + 3 +3 +2 cµ +1 +2 ). +This proves the inclusion +{z ∈ B1 : (u0 − hv,x0)(z) ≤ u0(x0) + µ} ⊂ (1 + C1γ)Eµ(x0). +Next we show that Eµ(x0) ⊂ B1. From |vi¯j −δij| ≤ c, we see that Eµ(x0) ⊂ B( +µ +1−c ) +1 +2 (x0). +Hence we just need to make sure ( µ +1−c) +1 +2 ≤ 0.9 − |x0|. If c is chosen small enough, this is +indeed true. +Now we prove the inclusion that +(1 − γ)Eµ(x0) ⊂ {z ∈ B1 : (u0 − hv,x0)(z) ≤ u0(x0) + µ}. +Assume that z ∈ (1 − γ)Eµ(x0), so that +� +i,j +vi¯j(x0)(z − x0)i(z − x0)j ≤ (1 − γ)2µ. +Hence |z − x0| ≤ √2µ, using vi¯j(x0) ≥ 1 +2I. Therefore +(u0 − hv,x0)(z) − u0(x0) ≤ +� +i,j +vi¯j(x0)(z − x0)i(z − x0)j + c|z − x0|3 + 2δ +≤ (1 − γ)µ + (2µ) +3 +2 + 2δ ≤ µ. +□ + +14 +JINGRUI CHENG, YULUN XU +Now we choose µ0 > 0 so that: +(4.2) +3 +3 +2µ +1 +2 +0 = min( 1 +20σ, 1 +20γn), µ0 < 0.9 · 0.12. +where γn is the constant given by Lemma 4.3. There is no loss of generality to assume +that σ ≤ γn and we will assume this throughout this section. +We can apply Lemma 4.6 to v0 and u and construct sections Sµ(x0) for all µ2 +0 < µ ≤ µ0. +Corollary 4.7. Let u and Ω be as stated in Theorem 1.3. Assume that γn is small +enough, µ0 is chosen according to (4.2), and 16ε ≤ µ2 +0, 8ε +µ0 ≤ +1 +20σ. For µ2 +0 < µ ≤ µ0 and +x0 ∈ B0.8 we define: +Sµ(x0) = {z ∈ B1 : (u − hv0,x0)(z) ≤ u0(x0) + µ}. +Then we have +(1 − 0.1σ)Eµ(x0) ⊂ Sµ(x0) ⊂ (1 + 0.1σ)Eµ(x0), +for any µ2 +0 < µ ≤ µ0. +Proof. This follows directly from Lemma 4.6, by choosing u0 to be u, v to be v0 in that +lemma. Choose δ = 4ε, and assume that γn stated in Theorem 1.3 is small enough so +as to make |(v0)i¯j − δij| and |D3v0| small enough on B0.9. After these choice, we may +use Lemma 4.6 to conclude that for and x0 ∈ B0.8 and 4 · 4ε ≤ µ ≤ 0.9 · 0.12, we can +conclude: +(1 − γ)Eµ(x0) ⊂ Sµ(x0) ⊂ (1 + γ)Eµ(x0). +Note that the range 16ε ≤ µ ≤ 0.9 · 0.12 contains the range 1 +2µ0 ≤ µ ≤ µ0. Moreover, +from the assumptions on the parameters, we get +γ ≤ 2 · 4ε +µ0 ++ 3 +3 +2µ +1 +2 +0 ≤ σ +20 + σ +20 ≤ σ +10. +□ +Now we need to define Sµ(x0) for µ ≤ µ2 +0. +Let ˜T1,x0 be a C-affine map such that ˜T1,x0(B√µ0(0)) = Eµ0(x0). We hope to estimate +how far ˜T1,x0 is away from identity map, in terms of how the ellipsoid Eµ0(x0) is close +to a ball. For that we need the following lemma: +Lemma 4.8. Let E ⊂ Cn be an ellipsoid, given by: +E = {z : +n +� +i,j=1 +ai¯j(z − x0)i(z − x0)j ≤ r2}, +with ai¯j being positive Hermitian matrix, det ai¯j = 1. Then there is a C-affine transform +T such that T(Br(0)) = E, det T = 1, and ||T − I|| ≤ max1≤i≤n |λ +− 1 +2 +i +− 1|, ||T −1 − 1|| ≤ +max1≤i≤n |λ +1 +2 +i − 1|, where λi are eigenvalues of ai¯j. +Proof. First we consider when ai¯j is diagnal, so that E = {z : �n +i=1 λi|zi|2 ≤ r2}, with +Πiλi = 1. Then we define T(w) = x0 + ( w1 +√λ1 , · · · , wn +√λn ). In the general case, we can take +a unitary transformation U, so that E becomes diagnal under the new coordinate, then +the desired C-affine map is given by T = U −1T ′U, where T ′ is the dilation map along +coordinate axis. Hence the result would follow from the diagnal case. +□ +As a consequence, we see that: + +INTERIOR W 2,p ESTIMATE FOR SMALL PERTURBATIONS TO THE COMPLEX MONGE-AMPERE EQUATION +15 +Lemma 4.9. Let u and Ω be as stated in Theorem 1.3. Assume that γ is small enough +depending on n. Let ˜T1,x0 be the C-affine map given by Lemma 4.8, applied to Eµ0(x0). +Then for some C′ +n > 0, we have +|| ˜T1,x0 − I|| ≤ C′ +nγ +1 +2, || ˜T −1 +1,x0 − I|| ≤ C′ +nγ +1 +2 . +Proof. Note that Eµ(x0) = {z : � +i,j(v0)i¯j(x0)(z − x0)i(z − x0)j ≤ µ}. It follows from +Corollary 4.4 that |(v0)i¯j −δij| ≤ Cnγ +1 +2 , for all γ small enough. So that 1−Cnγ +1 +2 ≤ λi ≤ +1 + Cnγ +1 +2. Therefore +|λ +1 +2 +i − 1| ≤ |(1 + Cnγ +1 +2) +1 +2 − 1| ≤ C′ +nγ +1 +2 , +which implies that || ˜T −1 +1,x0 − I|| ≤ C′ +nγ +1 +2. The estimate for || ˜T1,x0 − I|| is similar. +□ +To define Sµ(x0) for µ < µ2 +0, we need to rescale the ellipsoid Eµ0(x0) to be a unit ball. +Define the change of coordinate: +z(1) = +1 +√µ0 +˜T −1 +1,x0(z). +Then +1 +√µ0 ˜T −1 +1,x0(Eµ0(x0)) = B1(0). +Define Ωx0,1 = +1 +√µ0 ˜T −1 +1,x0(Sµ0(x0)). +According to +Corollary 4.7, we have that: +B1−0.1σ(0) ⊂ Ωx0,1 ⊂ B1+0.1σ(0), Ωx0,1 is pseudoconvex. +Define vx0,1 be the solution to the following Dirichlet problem: +det(vx0,1)i¯j = 1, +in Ωx0,1, +vx0,1 = 0 +on ∂Ωx0,1. +(4.3) +We can normalize u on Ωx0,1 to be: +(4.4) +ux0,1 = 1 +µ0 +� +u − u(x0) − hv0,x0 − µ0 +� +( ˜T1,x0(√µ0z(1))), z(1) ∈ Ωx0,1. +Then +det(ux0,1)i¯j = fx0,1 in Ωx0,1, fx0,1(z(1)) = f( ˜T1,x0(√µ0z(1))), +ux0,1 = 0 on ∂Ωx0,1. +We observe that the following holds for ux0,1 and vx0,1: +Lemma 4.10. Let vx0,1 and ux0,1 be defined by (4.3) and (4.4) respectively. Assume that +σ ≤ γn, where γn is given by Lemma 4.3. Let µ0 be defined by 4.2. Then the following +hold: +(1) vx0,1 ∈ C4(B0.95), and |Dm(vx0,1 − (|z(1)|2 − 1))|B0.9 ≤ Cnσ1− m +4 , m = 1, 2, 3, +where Cn is the same Cn in Corollary 4.4. +(2) |vx0,1 − ux0,1| ≤ 4ε in Ωx0,1. +(3) Define +˜hx0,1(z(1)) = Re +� � +i +(vx0,1)i(0)z(1) +i ++ +� +i,j +(vx0,1)ij(0)z(1) +i +z(1) +j +� +, + +16 +JINGRUI CHENG, YULUN XU +then for µ2 +0 ≤ µ ≤ µ0, we have: +(1 − 0.1σ) ˜Eµ(0) ⊂ {z(1) ∈ B1 : (ux0,1 − ˜hx0,1 − ux0,1(0))(z(1)) ≤ µ} +⊂ (1 + 0.1σ) ˜Eµ(0), +where ˜Eµ(0) = {z(1) : � +i,j(vx0,1)i¯j(0)z(1) +i +z(1) +¯j +≤ µ}. +(4) Let ˜T2,x0 be the C-affine transform given by Lemma 4.8 normalizing ˜Eµ0(x0), +then one has: +|| ˜T2,x0 − I|| ≤ C′ +nσ +1 +2 , +|| ˜T −1 +2,x0 − I|| ≤ C′ +nσ +1 +2, +where C′ +n is the constant given by Lemma 4.9. +Proof. To prove (1), we just note that since vx0,1 solves (4.3), with B1−0.1σ ⊂ Ωx0,1 ⊂ +B1+0.1σ, and σ ≤ γn, then Lemma 4.3 and Corollary 4.4 can be applied to show that +(1) holds, with γ there replaced by σ. Item (2) above follows from Lemma 4.5, since +ux0,1 solves det(ux0,1)i¯j = fx0,1 with |fx0,1 − 1| ≤ ε (since f satisfies the same). Item +(3) essentially follows from Lemma 4.6, applied to v = vx0,1, u = ux0,1, c = 1, δ = 4ε, +x0 = 0. Because of our choice of µ0 and ε, we would have γ ≤ 0.1σ. Also the range for µ +in Lemma 4.6 is 4 · 4ε ≤ µ ≤ 0.93 (with x0 = 0), which contains the range µ2 +0 ≤ µ ≤ µ0. +The proof of item (4) follows from Lemma 4.9, because of item (3). +□ +We define +˜S1,µ(0) = {z(1) ∈ B0.9 ⊂ Ωx0,1 : (ux0,1 − ˜hx0,1 − ux0,1(0))(z(1)) ≤ µ}, µ2 +0 < µ ≤ µ0. +We can now define Sµ(x0) for µ3 +0 < µ ≤ µ2 +0 by transforming back to z variable. In +other words, for µ3 +0 < µ ≤ µ2 +0, we define +Sµ(x0) = ˜T1,x0 +�√µ0 ˜S1,µ−1 +0 µ(0) +� += {z ∈ 0.9Eµ0(x0) ⊂ Sµ0(x0) : u(z) − hv0,x0(z) − µ0˜hx0,1( 1 +√µ0 +˜T −1 +1,x0(z)) ≤ u(x0) + µ}. +Next we use an induction process to define Sµ(x0), for all 0 < µ ≤ µ0. Assume that +for some k0 ≥ 2, we have defined Sµ(x0) for all µk0 +0 < µ ≤ µ0, we wish to define Sµ(x0) +for µk0+1 +0 +< µ ≤ µk0 +0 . +We make the following induction hypothesis, stated with k0: +(1) Sµ(x0) = {z ∈ Sµk−2 +0 +(x0) : u(z) − hx0,k−2(z) ≤ u(x0) + µ}, for all µk +0 < µ ≤ µk−1 +0 +, +2 ≤ k ≤ k0. Here hx0,k−2 is a pluriharmonic polynomial of degree 2. +(2) There exists a family of ellipsoids Eµ(x0), centered at x0, such that (1−0.1σ)Eµ(x0) ⊂ +Sµ(x0) ⊂ (1 + 0.1σ)Eµ(x0). Moreover, +Eµ(x0) = {z : +� +i,j +ai¯j,k−2(z − x0)i(z − x0)j ≤ µ}, +for all µk +0 < µ ≤ µk−1 +0 +, 2 ≤ k ≤ k0, and det ai¯j,k−2 = 1. +(3) There exists a sequence of C-affine coordinate change: z(k) = +1 +√µ0 ˜T −1 +k,x0(z(k−1)) for +k ≥ 2, z(1) = +1 +√µ0 ˜T −1 +x0,1(z − x0) with det ˜Tk,x0 = 1. ˜Tk,x0 maps B√µ0(0) to be the +image of Eµk +0(x0) under z(k−1). Moreover || ˜Tk,x0−I|| ≤ C′ +nσ +1 +2 , || ˜T −1 +k,x0−I|| ≤ C′ +nσ +1 +2 +for all 2 ≤ k ≤ k0 − 1. + +INTERIOR W 2,p ESTIMATE FOR SMALL PERTURBATIONS TO THE COMPLEX MONGE-AMPERE EQUATION +17 +(4) Denote Ωx0,k to be the image of Sµk +0(x0) under the coordinate z(k), then we have +B1−0.1σ ⊂ Ωx0,k ⊂ B1+0.1σ, for all 1 ≤ k ≤ k0 − 1. +(5) The image of Eµ(x0) under coordinate z(k−1) is B� +µ1−k +0 +µ(0), for µk +0 < µ ≤ µk−1 +0 +, +2 ≤ k ≤ k0. +First we observe that the above inductive hypothesis indeed hold for k0 = 2. Indeed, the +items (1) and (3) follow from Corollary 4.7. Item (2) follows from item (3) of Lemma +4.10. +Now we will construct Sµ(x0) for µk+1 +0 +≤ µ < µk0 +0 , and verify that the above inductive +hypothesis continues to hold with k0 replaced by k0 + 1. That is, we prove: +Lemma 4.11. Assume that above induction hypothesis holds with some k0 ≥ 2, then +we can construct Sµ(x0) for µk0+1 +0 +< µ ≤ µk0 +0 which satisfies the induction hypothesis +with k0 replaced by k0+1. In particular, the induction hypothesis holds for all k0 ≥ 2. +Proof. We solve the following Dirichlet problem on Ωx0,k0−1: +(4.5) +det(vx0,k0−1)i¯j = 1, +in Ωx0,k0−1, +vx0,k0−1 = 0 on ∂Ωx0,k0−1. +Define Tk,x0 = ˜T1,x0 ◦ ˜T2,x0 · · · ˜Tk,x0 so that the change of coordinate between z(k) and z +is given by z = x0 + Tk,x0(µ +k +2 +0 z(k)). +(4.6) +ux0,k0−1(z(k0−1)) = +1 +µk0−1 +0 +(u − hx0,k0−2 − u(x0) − µk0−1 +0 +)(x0 + Tk0−1,x0(µ +k0−1 +2 +0 +z(k0−1))). +Then ux0,k0−1 solves: +det(ux0,k0−1)i¯j = fx0,k0−1, +in Ωx0,k0−1, fx0,k0−1 = f(x0 + Tk0−1,x0(µ +k0−1 +2 +0 +z(k0−1))), +ux0,k0−1 = 0, +on ∂Ωx0,k0−1. +Using the same argument as in Lemma 4.10, (1), we have that vx0,k0−1 ∈ C4(B0.95), +and |Dm(vx0,k0−1 − |z(k0−1)|2 − 1)|B0.9 ≤ Cnσ1− m +4 for m = 1, 2, 3. This follows from our +inductive hypothesis that B1−0.1σ ⊂ Ωx0,k0−1 ⊂ B1+0.1σ, and an application of Lemma +4.3 and Corollary 4.4. Also we would have +|vx0,k0−1 − ux0,k0−1| ≤ 4ε on Ωx0,k0−1, +following the same argument as Lemma 4.10. +Then we may consider the Taylor expansion of vx0,k0−1 at z(k0−1) = 0: +vx0,k0−1(z(k0−1)) = vx0,k0−1(0) + Re +� � +i +liz(k0−1) +i +� ++ +� +i,j +ai¯jz(k0−1) +i +¯z(k0−1) +j ++ Re +� � +i,j +bijz(k0−1) +i +z(k0−1) +j +� ++ O(|z(k0−1)|3). +Define +(4.7) +˜hx0,k0−1(z(k0−1)) = Re( +� +i +liz(k0−1) +i +) + Re +� � +i,j +bijz(k0−1) +i +z(k0−1) +j +� +. + +18 +JINGRUI CHENG, YULUN XU +Then the argument for part (3) of Lemma 4.10 shows that: +(1 − 0.1σ) ˜Eµ(0) +⊂ ˜Sk0−1,µ := {z(k0−1) ∈ Ωx0,k0−1 : (ux0,k0−1 − ˜hx0,k0−1 − ux0,k0−1(0))(z(k0−1)) ≤ µ} +⊂ (1 + 0.1σ) ˜Eµ(0), +(4.8) +for any µ2 +0 < µ ≤ µ0, where ˜Eµ(0) = {z(k0−1) : � +i,j(vx0,k0−1)i¯j(0)z(k0−1) +i +¯z(k0−1) +j +≤ µ}. +Now we define ˜Tk0,x0 be the C-linear transforma given by Lemma 4.8 normalizing ˜Eµ0(0) +above. Then for µk0+1 +0 +< µ ≤ µk0 +0 , we define +Sµ(x0) = x0 + ˜Tk0,x0(µ +k0−1 +2 +0 +˜Sk0−1,µ−(k0−1) +0 +µ). +Similarly, we define +Eµ(x0) = x0 + ˜Tk0,x0(µ +k0−1 +2 +0 +˜Eµ−(k0−1) +0 +µ). +Using (4.6), one find that, for µk0+1 +0 +≤ µ < µk0 +0 : +Sµ(x0) = {z ∈ 0.9Eµk0−1 +0 +(x0) : u(z)−hx0,k0−2(z)−µk0−1 +0 +˜hx0,k0−1(µ +− k0−1 +2 +0 +T −1 +k0−1,x0(z−x0)) ≤ µ}. +Define +hx0,k0−1(z) = hx0,k0−2(z) + µk0−1 +0 +˜hx0,k0−1(µ +− k0−1 +2 +0 +T −1 +k0−1,x0(z − x0)). +In view of (4.7), as well as the induction hypothesis for hx0,k0−2(z), we see that hx0,k0−1(z) +is a pluriharmonic polynomial of degree 2, and hx0,k0−1(x0) = 0. This proves the induc- +tion hypothesis, part (1), for k = k0 + 1. +Part (2) simply follows from (4.8) and the fact that det(vx0,k0−1)i¯j(0) = 1, as well as +det Tk,x0 = 1. +Since |(vx0,k0−1)i¯j(0)−δij| ≤ Cnσ +1 +2 , the argument in Lemma 4.10, part (4) shows that +|| ˜Tk0,x0 − I|| ≤ C′ +nσ +1 +2 , +|| ˜T −1 +k0,x0 − I|| ≤ C′ +nσ +1 +2 . +Using ˜Tk0,x0, we can define a change of coordinates z(k0) = +1 +√µ0 ˜T −1 +k0,x0(z(k0−1)). +This +proves part (2) of the Inductive hypothesis with k = k0. Part (3) follows from (4.8) and +our choice of ˜Tk0,x0 above. +□ +Now we constructed Sµ(x0) for 0 < µ ≤ µ0, let us verify Proposition 4.1. +Part (1) has already been proved by the above argument. Indeed, we define hµ,x0(z) = +hx0,k−1(z) for µk +0 < µ ≤ µk−1 +0 +. Indeed, part (1) of Proposition 4.1 follows from part (1) +and (2) of induction hypothesis. Now we verify part (2) of Proposition 4.1. First, we +make the following observation out of the above inductive process. +Lemma 4.12. Let 0 < µ1 < µ2, then the following hold: +(1) If there is some k ≥ 2 such that µk +0 < µ1 < µ2 ≤ µk−1 +0 +, then Sµ1(x0) ⊂ Sµ2(x0), +Eµ1(x0) ⊂ Eµ2(x0), +(2) If there is some k ≥ 1 such that +1 +2µk +0 < µ2 and µ1 ≤ µk+1 +0 +, then Sµ1(x0) ⊂ +Sµ2(x0), Eµ1(x0) ⊂ Eµ2(x0). + +INTERIOR W 2,p ESTIMATE FOR SMALL PERTURBATIONS TO THE COMPLEX MONGE-AMPERE EQUATION +19 +Proof. Part (1) above is obvious, due to part (1) and (2) of induction hypothesis. +Part (2) requires more work, due to that Sµ(x0) and Eµ(x0) are discontinuous in µ +for µ → µk +0+ and µ → µk +0−. +First, without loss of generality we may assume that +µk+2 +0 +< µ1 ≤ µk+1 +0 +. +If µk +0 < µ2 ≤ µk−1 +0 +(with k ≥ 2 in this case), we know from part (5) of induction hypothesis +that under z(k−1), Eµ2(x0) is given by B� +µ1−k +0 +µ2(0). On the other hand, the image of +Eµ1(x0) under z(k+1) is given by B� +µ−1−k +0 +µ1(0). Hence if we recall the transition formula +given by part (3) of induction hypothesis, we see that: +Eµ1(x0) under z(k−1) = µ0 ˜Tk+1,x0 ◦ ˜Tk,x0(B� +µ−1−k +0 +µ1(0)) +⊂ (1 + C′ +nσ +1 +2 )2B� +µ1−k +0 +µ1(0). +(4.9) +In the first inclusion above, we used that || ˜Ti,x0|| ≤ 1 + C′ +nσ +1 +2 . Therefore +image of Sµ1(x1) under z(k−1) ⊂ B +(1+0.1σ)(1+C′nσ +1 +2 )2 +� +µ1−k +0 +µ1(0) +On the other hand, +image of Sµ2(x2) under z(k−1) ⊃ (1 − 0.1σ)B� +µ2µ1−k +0 +(0) +We will be able to show Sµ1(x0) ⊂ Sµ2(x0) if we can ensure: +(1 + C′ +nσ +1 +2 )4(1 + 0.1σ)2µ1 ≤ (1 − 0.1σ)2µ2. +This can be guaranteed if we take µ0 small enough so that (1 + C′ +nσ +1 +2)4 (1+0.1σ)2 +(1−0.1σ)2 ≤ µ−1 +0 , +since µ2 ≥ µ0µ1. +The other case is when 1 +2µk +0 < µ2 ≤ µk +0. The calculation in this case is similar to the +case when µk +0 < µ2 ≤ µk−1 +0 +, except that we need to use the coordinate z(k), and we may +conclude: +image of Sµ2(x0) under z(k) ⊃ (1 − 0.1σ)B� +µ2µ−k +0 +(0), +On the other hand, +image of Sµ1(x0) under z(k+1) ⊂ (1 + 0.1σ)B� +µ1µ−1−k +0 +(0). +Hence, using the transition between z(k) and z(k+1): +image of Sµ1(x0) under z(k) ⊂ √µ0(1 + C′ +nσ +1 +2)(1 + 0.1σ)B� +µ1µ−1−k +0 +(0). +We will have the inclusion as long as we can make sure: +(1 − 0.1σ)2µ2 ≥ (1 + C′ +nσ +1 +2)2(1 + 0.1σ)2µ1. +We will still have this since µ2 ≥ 1 +2µ1µ0 and we can take µ0 small enough. +□ +With the help of the previous lemma, we are ready to prove the almost monotonicity +of sections claimed in part (2): + +20 +JINGRUI CHENG, YULUN XU +Corollary 4.13. Let c(σ) = (1+0.1σ)2 +(1−0.1σ)2 (1 + C′ +nσ +1 +2)2 − 1 with C′ +n given by Lemma 4.10. +Then for all 0 < µ1 ≤ µ2 ≤ +µ0 +1+c(σ) and any x0 ∈ B0.8, +Sµ1(x0) ⊂ S(1+c(σ))µ2(x0). +Proof. Denote c(σ) = (1+0.1σ)2 +(1−0.1σ)2 (1 + C′ +nσ +1 +2)2 − 1. Let k ≥ 1 be such that µk+1 +0 +< µ1 ≤ µk +0 +and µ2 > µ1. There are several cases to consider: +Case 1: µk+1 +0 +< µ1 < µ2 ≤ +µk +0 +1+c(σ). Then from Lemma 4.12, we know that Sµ1(x0) ⊂ +S(1+c(σ))µ2(x0). +Case 2: +µk +0 +1+c(σ) < µ2 ≤ +µk−1 +0 +1+c(σ) (k ≥ 2 for this case). +First, under the coordinate z(k−1), we have the following inclusions: +image of S(1+c(σ))µ2(x0) under z(k−1) ⊃ B +(1−0.1σ) +� +(1+c(σ))µ2µ1−k +0 +(0) += B +(1+0.1σ)(1+C′nσ +1 +2 ) +� +µ2µ1−k +0 +(0). +(4.10) +On the other hand, if we consider the image of Sµ1(x0) under z(k), we have +image of Sµ1(x0) under z(k) ⊂ B +(1+0.1σ) +� +µ1µ−k +0 +(0) +Then we use the transition between z(k) and z(k−1): z(k) = +1 +√µ0 ˜Tk,x0(z(k−1)) to get: +(4.11) +Image of Sµ1(x0) under z(k−1) ⊂ √µ0 ˜T −1 +k,x0(B +(1+0.1σ) +� +µ1µ−k +0 +(0)) ⊂ B +(1+0.1σ)(1+C′nσ +1 +2 ) +� +µ1µ1−k +0 +(0). +Combining (4.10) and (4.11), we get that: +Sµ1(x0) ⊂ S(1+c(σ))µ2(x0). +Case 3: µ2 ≥ +µk−1 +0 +1+c(σ) (k ≥ 2 for this case). +Without loss of generality, we may assume σ small enough so that 1 + c(σ) < 2, then +the conclusion would follow from Lemma 4.12. Since µ1 ≤ µk +0 but µ2 ≥ 1 +2µk−1 +0 +. +□ +Now we verify part (3) of Proposition 4.1. +Lemma 4.14. Define Tµ,x0 = Tk,x0 := ˜T1,x0 ◦ ˜T2,x0 · · · ˜Tk,x0 for µk+1 +0 +< µ ≤ µk +0, k ≥ 1. +Then for 0 < µ1 < µ2 ≤ µ0, we have: +||T −1 +µ1,x0 ◦ Tµ2,x0||, ||T −1 +µ2,x0 ◦ Tµ1,x0|| ≤ C3,n +�µ2 +µ1 +� C3,nσ +1 +2 +log(0.1σ) . +Here C3,n is some dimensional constant. +Proof. First we find k1 ≥ k2 such that µk1+1 +0 +< µ1 ≤ µk1 +0 , µk2+1 +0 +< µ2 ≤ µk2 +0 . Then we +have: +T −1 +µ1,x0 ◦Tµ2,x0 = ˜T −1 +k1,x0 ◦ ˜T −1 +k1−1,x0 · · · ˜T −1 +k2+1,x0, T −1 +µ2,x0 ◦Tµ1,x0 = ˜Tk2+1,x0 ◦ ˜Tk2+1,x0 · · · ˜Tk1,x0. +Then we may use part (3) of induction hypothesis that: +||T −1 +µ1,x0 ◦ Tµ2,x0|| ≤ Πk1−1 +k=k2||T −1 +k,x0|| ≤ (1 + C′ +nσ +1 +2)k1−k2. + +INTERIOR W 2,p ESTIMATE FOR SMALL PERTURBATIONS TO THE COMPLEX MONGE-AMPERE EQUATION +21 +On the other hand, we easily have the bound µk2−k1+1 +0 +≤ µ2 +µ1 , hence +||T −1 +µ1,x0 ◦ Tµ2,x0|| ≤ (1 + C′ +nσ +1 +2 )− +log(µ2µ−1 +1 +) +− log(µ0) +1. +Recall our choice of µ0 made in (4.2): 3 +3 +2 µ +1 +2 +0 = +σ +20. Then the claimed estimate follows +easily. The same computation works also for T −1 +µ2,x0 ◦ Tµ1,x0. +□ +As a direct consequence, the diameter of Sµ(x0) should go to zero as µ → 0. Namely: +Corollary 4.15. Assume that σ is small enough (depending on n). Then the diameter +of Sµ(x0) goes to zero as µ → 0. This convergence is uniform for x0 ∈ B0.8. +Proof. Since Tµ,x0 = id for µ2 +0 < µ ≤ µ0, we see that, for any µ ≤ µ2 +0, +||T −1 +µ,x0||, ||Tµ,x0|| ≤ C3,n( µ +µ0 +) +C3,nσ +1 +2 +log(0.1σ) . +On the other hand Tµ,x0(B√µ(0)) = Eµ(x0) − x0, we see that +diam Eµ(x0) ≤ 2µ +1 +2 ||Tµ,x0|| ≤ 2C3,nµ +1 +2 · ( µ +µ0 +) +C3,nσ +1 +2 +log(0.1σ) . +If σ is small enough so that 1 +2 + C3,nσ +1 +2 +log(0.1σ) > 0, then the right hand side will go to zero as +µ → 0. The result follows from that Sµ(x0) ⊂ (1 + 0.1σ)Eµ(x0). +□ +4.2. Further properties of sections. Our intention will be to use the sections given +by Proposition 4.1 to replace the role of balls in the uniformly elliptic case. The most +crucial property that we need is the following “engulfing property” of sections, formulated +below: +Proposition 4.16. Assume that x1, x2 ∈ B0.8, 0 < µ1, µ2 ≤ µ0 and µ1 ≤ 4µ2. Let +σ > 0 be small enough (depending only on dimension). +Assume also that Sµ1(x1) ∩ +Sµ2(x2) ̸= ∅, then Sµ1(x1) ⊂ 10Sµ2(x2). +In Caffarelli’s proof for W 2,p estimate in the real case, we also need this “engulfing +property”, but this property is not a problem in the real case. The essential point is the +“invariance of sections under linear transformations”. To be more clear, in the real case, +the sections are simply defined as: (with u being strictly convex function) +Sµ(x0) = {x : u(x) ≤ u(x0) + ∇u(x0) · (x − x0) + µ}, +µ > 0. +Now we define v(y) = +1 +r2 u(x′ +0 + rTy) where T is linear with det T = 1. +Such a +transformation would preserve the Monge-Ampere equation. Under the change of co- +ordinates x = x′ +0 + rTy, Sµ(x0) will be transformed to a section of v centered at +y0 := T −1(1 +r(x0 − x′ +0)), with height +µ +r2. +We no longer have this property in the complex case. Indeed, if you do a similar change +of coordinates (now with T being C-linear), and you do the same construction described +in Proposition 4.1 for the function v in the variable y, then transform back to x, you will +get different sections than the direct construction in the original x coordinates. The two +definitions will differ by an addition of a pluriharmonic function. +What saves us is the following “uniqueness” property, which shows that an addition +of a pluriharmonic function will not affect the sections we get, as long as they are close +to ellipsoids. + +22 +JINGRUI CHENG, YULUN XU +Lemma 4.17. Let u be a function defined on an open set U ⊂ Cn and let h(z) be +pluriharmonic function on U such that h(0) = 0. Let 0 < γ < 1 and µ > 0 be such that: +B(1−γ)√µ(0) ⊂ {u ≤ u(0) + µ} ⊂ B(1+γ)√µ(0) ⊂ U, +(1 − γ)Eµ(0) ⊂ {u ≤ h + u(0) + µ} ⊂ (1 + γ)Eµ(0) ⊂ U. +(4.12) +In the above, Eµ(0) = {z ∈ Cn : �n +i,j=1 ai¯jzi¯zj ≤ µ} with ai¯j positive Hermitian. Then +we have: +(4.13) +B(1−γ)√µ(0) ⊂ (1 + γ)Eµ(0), +(1 − γ)Eµ(0) ⊂ B(1+γ)√µ(0). +Proof. First, by considering a unitary transformation if necessary, we may assume that +Eµ(0) = {z ∈ Cn : �n +i=1 λi|zi|2 ≤ µ} with 0 < λ1 ≤ λ2 · · · ≤ λn. Also it will suffice to +prove one of the two inclusions in (4.13), say B(1−γ)√µ(0) ⊂ (1 + γ)Eµ(0). To prove the +other inclusion, we may consider a change of coordinates: wi = √λizi, so that Eµ(0) +becomes B√µ(0), and we use v(z) := u(z)−h(z) to replace u. Then the second inclusion +would follow from the first. +We wish to argue by contradiction and assume that B(1−γ)√µ(0) is not contained in +(1 + γ)Eµ(0), then we must have: +B(1−γ)√µ(0) ∩ ∂ +� +(1 + γ)Eµ(0) +� +̸= ∅. +Note that u ≥ h(z)+u(0)+µ on (1+γ)∂Eµ(0) and u ≤ u(0)+µ in B(1−γ)√µ. Therefore, +h(z) ≤ 0 on B(1−γ)√µ ∩ (1 + γ)∂Eµ. We will show that this hypersurface, if nonempty, +actually bounds a nontrivial region, so that we get h ≤ 0 in a neighborhood of 0. Since +h is a pluriharmonic function and h(0) = 0, we can use the strong maximum principle +to get h ≡ 0. Hence (4.12) would give us B(1−γ)√µ(0) ⊂ {u ≤ h + µ} ⊂ (1 + γ)Eµ(0), +contrary to what we assume above. +First we present the argument when n = 2. We want to show that if λ2 > (1+γ)2 +(1−γ)2 , then +h ≡ 0, which would contradict (4.12). On the other hand, if λ2 ≤ (1+γ)2 +(1−γ)2 , then we would +have B(1−γ)√µ ⊂ (1 + γ)Eµ, which is another contradiction. +To see that λ2 > (1+γ)2 +(1−γ)2 implies h ≡ 0, we fix some z1,∗ and consider the cross section +between (z1,∗, z2) and B(1−γ)√µ ∩ (1 + γ)Eµ (viewed as a subset in C for z2). They are +given by: +|z1,∗|2 + |z2|2 ≤ (1 − γ)2µ, +λ1|z1,∗|2 + λ2|z2|2 ≤ (1 + γ)2µ. +(4.14) +We want to argue that, if λ2 > (1+γ)2 +(1−γ)2 , then the boundary of the cross section will be on +(1 + γ)∂Eµ ∩ B(1−γ)√µ, for all z1,∗ close to zero. Since h ≤ 0 on the boundary of cross +section, we would have h(z1,∗, z2) ≤ 0 in the interior of the section (z1,∗, z2) since h is +pluriharmonic. This is true for all z1,∗ close to 0. Hence h ≥ 0 in a neighborhood of 0 +and we can conclude from strong maximum principle that h ≡ 0 since h(0) = 0. +The boundary of the cross section is on (1+γ)∂Eµ ∩B(1−γ)√µ if and only if the second +inequality in (4.14) implies the first (with z1,∗ fixed). The first inequality is equivalent +to: +|z2|2 ≤ (1 − γ)2µ − |z1,∗|2, +whereas the second inequality is equivalent to: + +INTERIOR W 2,p ESTIMATE FOR SMALL PERTURBATIONS TO THE COMPLEX MONGE-AMPERE EQUATION +23 +|z2|2 ≤ (1 + γ)2 +λ2 +µ − λ1 +λ2 +|z1,∗|2. +In order for the second inequality to be stronger than the first, we need that: +(4.15) +(1 + γ)2 +λ2 +µ − λ1 +λ2 +|z1,∗|2 ≤ (1 − γ)2µ − |z1,∗|2. +If λ2 > (1+γ)2 +(1−γ)2 , the inequality in (4.15) is strict with z1,∗ = 0. Hence (4.15) will hold for +z1,∗ close enough to zero. This proves our earlier claim about the boundary of the cross +section and finishes the argument that λ2 > (1+γ)2 +(1−γ)2 implies h ≡ 0. If λ1 ≤ λ2 ≤ (1+γ)2 +(1−γ)2 , +then B(1−γ)√µ ⊂ (1 + γ)Eµ. Indeed, B(1−γ)√µ is given by: +|z1|2 + |z2|2 ≤ (1 − γ)2µ, +but then λ1|z1|2 + λ2|z2|2 ≤ (1+γ)2 +(1−γ)2 · (1 − γ)2µ = (1 + γ)2µ, so that (z1, z2) ∈ (1 + γ)Eµ. +Now we look at general n. As in n = 2, λn ≤ (1+γ)2 +(1−γ)2 will immediately imply that +B(1−γ)√µ ⊂ (1 + γ)Eµ (since λn is the largest eigenvalue), immediately giving what we +want to prove. We just need to show that λn > (1+γ)2 +(1−γ)2 implies h ≡ 0, and then (4.12) +will give the result. +For this we consider the cross section between (z1,∗, · · · , zn−1,∗, zn) and B(1−γ)√µ ∩ +(1 + γ)Eµ, and it is given by: +n−1 +� +i=1 +|zi,∗|2 + |zn|2 ≤ (1 − γ)2µ, +n−1 +� +i=1 +λi|zi,∗|2 + λn|zn|2 ≤ (1 + γ)2µ. +(4.16) +The boundary of the cross section is on (1 + γ)∂Eµ ∩ B(1−γ)√µ iff the second inequality +in (4.16) is stronger than the first. This would mean: +(4.17) +1 +λn +� +(1 + γ)2µ − +n−1 +� +i=1 +λi|zi,∗|2� +≤ (1 − γ)2µ − +n−1 +� +i=1 +|zi,∗|2. +Note that if λn > (1+γ)2 +(1−γ)2 , the inequality in (4.17) is strict with zi,∗ = 0, 1 ≤ i ≤ n − 1. +Hence (4.17) will continue to hold for (z1,∗, · · · , zn−1,∗) close to 0. Hence we would have +h ≤ 0 in a neighborhood of 0, and strong maximum principle would give h ≡ 0. +□ +As a consequence, we deduce that: +Corollary 4.18. Let u be a function defined on an open set U ⊂ Cn with 0 ∈ U. Let +h1(z), h2(z) be pluriharmonic functions on U such that h1(0) = h2(0) = 0. Let 0 < γ < 1 +and µ > 0 be such that: +(1 − γ)Ep,µ(0) ⊂ {u ≤ hp + u(0) + µ} ⊂ (1 + γ)Ep,µ(0) ⊂ U, +p = 1, 2. + +24 +JINGRUI CHENG, YULUN XU +In the above, Ep,µ(0) = {z ∈ Cn : � +i,j ap,i¯jzi¯zj ≤ µ} with ap,i¯j being positive Hermitian +and det ap,i¯j = 1, p = 1, 2. Let Tp be a C-linear transformation mapping B√µ to Ep,µ, +then we have: +||T −1 +1 +◦ T2|| ≤ (1 + γ)2 +(1 − γ)2 , +||T −1 +2 +◦ T1|| ≤ (1 + γ)2 +(1 − γ)2 . +Proof. We can apply a map T −1 +1 +to the above picture and reduce E1,µ to be B√µ(0). +Denote ˜E2,µ = T −1 +1 (E2,µ) = T −1 +1 +◦ T2(B√µ). Then we know that the eigenvalues of ˜E2,µ +is between (1+γ)2 +(1−γ)2 and (1−γ)2 +(1+γ)2 . This implies the result. +□ +Before we move further, we first want to explain the idea why Corollary 4.18 help +with proving engulfing property. +Suppose, say, we have Sµ(x1) ∩ Sµ(x2) ̸= ∅. +Take +x∗ ∈ Sµ(x1) ∩ Sµ(x2). We can normalize S100µ(x1) to be close to a unit ball and Sµ(x1) +will then be close to a ball with radius 0.1. Now we can define a section ˜S∗ with height +1 +100 centered at x′ +∗(image of x∗ under the new coordinate) using the new coordinate, so +that its shape is comparable to a ball. +Then we go back to the original coordinate, we get a section S∗ centered at x∗ with +height µ whose shape is comparable with Sµ(x1). Because of Corollary 4.18, S∗ and +Sµ(x∗) will also have similar shapes. Therefore we see that the shapes of Sµ(x∗) and +Sµ(x1) are comparable (Lemma 4.19). Similarly, Sµ(x∗) and Sµ(x2) are also comparable. +Hence, if we normalize Sµ(x1) to be close to a unit ball, the other section Sµ(x2) will +be close to an ellipsoid whose shape is not too eccentric. Then the engulfing property +would follow from the standard engulfing property for balls. +Lemma 4.19. Let x∗ ∈ Sµ(x0) ∩ B0.8 for some x0 ∈ B0.8 and 0 < µ ≤ µ0. Let Tµ,x∗ +and Tµ,x0 be the C-linear transfomation given by Proposition 4.1, part (3). Then for σ +chosen small enough depending only on n, +||T −1 +µ,x∗ ◦ Tµ,x0|| ≤ 1.13, +||T −1 +µ,x0 ◦ Tµ,x∗|| ≤ 1.13. +Proof. First we can find k ≥ 1 such that µk+1 +0 +< µ ≤ µk +0. +We will work under the +coordinate z(k−1), defined as z = x0 + µ +k−1 +2 +0 +Tk−1,x0(z(k−1)), where Tk−1,x0 is the C-linear +transformation mapping a ball to Eµk−1 +0 +(0). We have that +Sµ(x0) under z(k−1) = {z(k−1) : ux0,k−1(z(k−1)) ≤ ux0,k−1(0) + ˜hx0,k−1(z(k−1)) + +µ +µk−1 +0 +}, +where ux0,k−1 is the normalized u on Ωx0,k−1 (Sµk−1 +0 +(x0) under z(k−1) which is close to a +ball.) Also we have +Eµ(x0) under z(k−1) = {z(k−1) : +� +i,j +(vx0,k−1)i¯j(0)z(k−1) +i +¯z(k−1) +j +≤ +µ +µk−1 +0 +}. +Here vx0,k−1 solves the Dirichlet problem (4.5). On the other hand, denote z∗ to be the +image of x∗ under z(k−1), then we know that z∗ ∈ the image of Sµ(x0) under z(k−1). +We can find a section centered at z∗ under z(k−1) using Lemma 4.6, applied to vx0,k−1 +and ux0,k−1, δ = 4ε, x0 = z∗, µ replaced by +µ +µk−1 +0 +≤ µ0, which is between µ0 and µ2 +0, we + +INTERIOR W 2,p ESTIMATE FOR SMALL PERTURBATIONS TO THE COMPLEX MONGE-AMPERE EQUATION +25 +have that: +(1 − γ) ˜E +µ +µk−1 +0 +(z∗) ⊂ {z(k−1) ∈ B1 : (ux0,k−1 − h∗)(z(k−1)) ≤ ux0,k−1(z∗) + +µ +µk−1 +0 +} +⊂ (1 + γ) ˜E +µ +µk−1 +0 +(z∗). +(4.18) +In the above, γ = 8εµk−1 +0 +µ ++ 3µ +1 +2µ +− k−1 +2 +0 +. It is less that 0.1σ, because of our choice of µ0 in +(4.2) and our assumption that ε is small (we made a precise choice of ε in Corollary 4.7.) +Also the coefficients of ˜E +µ +µk−1 +0 +(z∗) is defined by (vx0,k−1)i¯j(z∗). Let T∗ be the C-linear +transform given by Lemma 4.8 such that T∗(B� +µµ1−k +0 +(0)) = ˜Eµµ1−k +0 +(0). Then similar to +Lemma 4.10, part (4), we would have that (using (vx0,k−1)i¯j is close to identity by σ +1 +2 ): +|| ˜T∗ − I|| ≤ C′ +nσ +1 +2 , || ˜T −1 +∗ +− I|| ≤ C′ +nσ +1 +2 . +We can tranform the picture (4.18) back to the z variable, and obtain that: +• There is an ellipsoid E∗ +µ centered at x∗, such that: +(1 − 0.1σ)E∗ +µ(x∗) ⊂ S∗ +µ(x∗) ⊂ (1 + 0.1σ)E∗ +µ(x∗) ⊂ Sµk−1 +0 +(x0), +where S∗ +µ(x∗) = {z ∈ Sµk−1 +0 +(x0) : (u − h∗)(z) ≤ u(x∗) + µ}, with h∗ being a +quadratic pluriharmonic polynomial. +• With T∗ = Tk−1,x0 ◦ ˜T∗, then we have T∗(B√µ(0)) = E∗ +µ(z∗), and ||T −1 +k−1,x0 ◦ T∗ − +I|| ≤ 0.1, ||T −1 +∗ +◦ Tk−1,x0 − I|| ≤ 0.1, if σ is small enough depending only on n. +On the other hand, with Sµ(x∗), Eµ(x∗) being given by Proposition 4.1, we also have +(1 − 0.1σ)Eµ(x∗) ⊂ Sµ(x∗) ⊂ (1 + 0.1σ)Eµ(x∗). Also we have a C-linear transform Tµ,x∗ +such that Tµ,x∗(B√µ(0)) = Eµ(x∗). Using Corollary 4.18, we have +||T −1 +∗ +◦ Tµ,x∗|| ≤ (1 + 0.1σ)2 +(1 − 0.1σ)2 , ||T −1 +µ,x∗ ◦ T∗|| ≤ (1 + 0.1σ)2 +(1 − 0.1σ)2 +Hence we get that +||T −1 +k−1,x0 ◦ Tµ,x∗||, ||T −1 +µ,x∗ ◦ Tk−1,x0|| ≤ 1.12, +if σ is small enough. +Finally, we note that Tµ,x0 = Tk−1,x0 ◦ ˜Tk,x0, and || ˜Tk,x0||, || ˜T −1 +k,x0|| ≤ 1 + C′ +nσ +1 +2 , so the +result would follow if σ is small enough depending only on n. +□ +Then the engulfing property would follow from Lemma 4.19. +Proof. (of Proposition 4.16) +First we want to reduce to when µ1 and µ2 are comparable. +Indeed, if µ1 < µ2, +we know from Proposition 4.1 that Sµ1(x1) ⊂ S(1+c(σ))µ2(x1) and c(σ) → 0 as σ → 0. +Hence we may assume that σ is small enough so that Sµ1(x1) ⊂ Sµ′ +1(x1), for some +µ2 < µ′ +1 ≤ 4µ2. Hence it will suffice to prove the following statement: +For any 0 < µ2 ≤ µ1 ≤ 4µ2, if Sµ1(x1) ∩ Sµ2(x2) ̸= ∅, we have Sµ1(x1) ⊂ 10Sµ2(x2). +First, we choose x∗ ∈ Sµ1(x1) ∩ Sµ2(x2). By choosing σ small enough, we know from +Lemma 4.19 that +||T −1 +µp,x∗ ◦ Tµp,xp|| ≤ 1.13, ||T −1 +µp,xp ◦ Tµp,x∗|| ≤ 1.13, p = 1, 2. + +26 +JINGRUI CHENG, YULUN XU +On the other hand, we know from Lemma 4.9 that, with σ small enough, we have +||T −1 +µ1,x∗ ◦ Tµ2,x∗|| ≤ 1.1, ||T −1 +µ2,x∗ ◦ Tµ1,x∗|| ≤ 1.1. +This follows from the observation that µ1 and µ2 must belong to the same level or +adjacent levels (either µk+1 +0 +< µ2 ≤ µ1 ≤ µk +0, or µk+1 +0 +< µ2 ≤ µk +0 < µ1 ≤ µk−1 +0 +). Hence +we get: +(4.19) +||T −1 +µ1,x1 ◦ Tµ2,x2||, ||T −1 +µ2,x2 ◦ Tµ1,x1|| ≤ 1.16 · 1.1 ≤ 2. +Now we show the containment of sections: +Sµ1(x1) ⊂ (1 + 0.1σ)Eµ1(x1) = Tµ1,x1(B(1+0.1σ)√µ1(x1,∗)) += Tµ2,x2 ◦ T −1 +µ2,x2 ◦ Tµ1,x1(B(1+0.1σ)√µ1(x1,∗)) ⊂ Tµ2,x2(B2(1+0.1σ)√µ1(x′ +1,∗)). +(4.20) +In the above, we denote x1,∗ and x′ +1,∗ so that Tµ1,x1(x1,∗) = x1, T −1 +µ2,x2◦Tµ1,x1(x1,∗) = x′ +1,∗. +We also used the bound (4.19). On the other hand, +Sµ2(x2) ⊂ Tµ2,x2(B(1+0.1σ)√µ2(x∗ +2)), +where x2,∗ = T −1 +µ2,x2(x2). It follows that B(1+0.1σ)√µ2(x∗ +2) ∩ B2(1+0.1σ)√µ1(x′ +1,∗) ̸= ∅. Since +µ1 ≤ 4µ2, it follows that +(4.21) +B2(1+0.1σ)√µ1(x′ +1,∗) ⊂ B(1+0.1σ)9√µ2(x∗ +2). +Hence it follows from (4.20) that: +Sµ1(x1) ⊂ Tµ2,x2(B(1+0.1σ)9√µ2(x∗ +2)) ⊂ Tµ2,x2(B9.1√µ2(x∗ +2)) += 9.1Eµ2(x2) ⊂ 10Sµ2(x2). +□ +Finally let us include the following inclusion result for future reference. +Lemma 4.20. Let σ be small enough depending on n. Then for any 0 < µ ≤ +µ0 +121, we +have +10Sµ(x0) ⊂ S121µ(x0) ⊂ 12Sµ(x0). +Proof. Let k ≥ 1 be such that µk+1 +0 +< µ ≤ µk +0. There are two cases to consider: +If 121µ ≤ µk +0, then E121µ(x0) and Eµ(x0) are defined by the same coefficients, hence +10Sµ(x0) ⊂ 10(1 + 0.1σ)Eµ(x0). +On the other hand, +(1 − 0.1σ)E121µ(x0) = (1 − 0.1σ)11Eµ(x0) ⊂ S121µ(x0). +Hence 10Sµ(x0) ⊂ S121µ(x0), as long as 10(1+ 0.1σ) ≤ 11(1− 0.1σ). On the other hand, +S121µ(x0) ⊂ (1+0.1σ)E121µ(x0) = 11(1+0.1σ)Eµ(x0) ⊂ 12(1−0.1σ)Eµ(x0) ⊂ 12Sµ(x0). +If µk +0 < 121µ ≤ µk−1 +0 +, then under z(k−1), E121µ(x0) becomes a ball with radius +11 +� +µµ1−k +0 +, hence S121µ(x0) contains the ball B +11(1−0.1σ) +� +µµ1−k +0 +(0) and is contained in +B +11(1+0.1σ) +� +µµ1−k +0 +under z(k−1). On the other hand, z(k) and z(k−1) differs by a coordi- +nate change ˜Tk,x0, for which we have the bound || ˜Tk,x0|| ≤ 1+C′ +nσ +1 +2 , || ˜T −1 +k,x0|| ≤ 1+C′ +nσ +1 +2 . + +INTERIOR W 2,p ESTIMATE FOR SMALL PERTURBATIONS TO THE COMPLEX MONGE-AMPERE EQUATION +27 +Therefore, Eµ(x0) is contained in B +(1+C′nσ +1 +2 ) +� +µµ1−k +0 +(0) , and contains B√ +µµ1−k +0 +1+C′nσ +1 +2 +(0) under +z(k−1). Therefore, working under z(k−1), one has +image of 10Sµ(x0) in z(k−1) ⊂ image of 10(1 + 0.1σ)Eµ(x0) under z(k−1) +⊂ B +10(1+0.1σ)(1+C′nσ +1 +2 ) +� +µµ1−k +0 +(0) ⊂ (1 − 0.1σ)B +11 +� +µµ1−k +0 +(0) ⊂ image of S121µ(x0) under z(k−1). +The above inequality holds as long as 10(1 + 0.1σ)(1 + C′ +nσ +1 +2 ) ≤ 11(1 − 0.1σ). On the +other hand, still working under z(k−1): +Image of S121µ(x0) in z(k−1) ⊂ 11(1 + 0.1σ)B� +µµ1−k +0 +(0) +⊂ 11(1 + 0.1σ)(1 + C′ +nσ +1 +2)×Eµ(x0) in z(k−1) ⊂ 12(1 − 0.1σ)×Eµ(x0) in z(k−1) +⊂ 12×Sµ(x0) in z(k−1). +In the above, we need to require that 11(1 + 0.1σ)(1 + C′ +nσ +1 +2) ≤ 12(1 − 0.1σ). +□ +As a consequence, we also have the following version of engulfing property: +Corollary 4.21. Under the assumptions of Proposition 4.16, we have: +Sµ1(x0) ⊂ S121µ2(x0). +5. Some measure theoretic lemmas +In this section we will prove a covering lemma which will be used in the W 2,p estimate. +In the following, m always denotes the standard Lebesgue measure. +Lemma 5.1. Let Sµα(xα) ⊂ Rd be a family of sets. Assume that 0 < µα ≤ µ0 for all α +and ∪αSµα(xα) is bounded. Assume that the volume of Sµα(xα) is comparable to that of +a standard ball with radius √µα. I.e. there exists a uniform constant C such that +1 +C m(B√µα(0)) ≤ m(Sµα(xα)) ≤ Cm(B√µα(0)) +Assume that Sµα(xα) satisfies the following engulfing property: +For any Sµα1(xα1) and Sµα2(xα2) with Sµα1(xα1) ∩ Sµα2(xα2) ̸= ∅, if √µα1 ≤ 2√µα2, +then Sµα1(xα1) ⊂ 10Sµα2(xα2). +Let X be a measurable set with X ⊂ ∪αSµα(xα), then one can choose a sequence (finite +or infinite) Sµi(xi), such that: +(1) Sµi(xi) are all disjoint. +(2)X ⊂ ∪i10Sµi(xi). +Proof. The proof for this lemma is very similar to the standard Vitali’s covering lemma +in measure theory. +First, we choose a set Sµ1(x1) with √µ1 > 1 +2 supα +√µα. Then we consider all Sµα(xα) +which does not intersect Sµ1(x1), and you choose Sµ2(x2) so that √µ2 > 1 +2 sup{√µα : +Sµα(xα) ∩ Sµ1(x1) = ∅}. Then you consider all Sµα(xα) which does not intersect with +Sµ1(x1) or Sµ2(x2) and you choose Sµ3(x3) among those so that √µ3 > 1 +2sup of √µα +among all Sµα(xα) which don’t intersect with Sµ1(x1) or Sµ2(x2). +We continue this +process. + +28 +JINGRUI CHENG, YULUN XU +This process may stop in finite steps. If this happens, then we get a finite sequence +Sµ1(x1), Sµ2(x2), · · · , SµN (xN) such that they are mutually disjoint, and all Sµα(xα) +must intersect with one of them. Let i0 be the first index such that Sµi(xi)∩Sµα(xα) ̸= ∅, +then Sµi0−1(xi0−1) ∩ Sµα(xα) = ∅, and due to our inductive choice, √µα ≤ 2√µi0. Using +the engulfing property, we get: +Sµα(xα) ⊂ 10Sµi0(xi0). +Therefore, X ⊂ ∪αSµα(xα) ⊂ ∪N +i=110Sµi(xi). +The other possibility is that we find an infinite sequence of {Sµi(xi)}∞ +i=1. They are +mutually disjoint because of our construction. Then we must have that µi → 0, since +∪αSµα(xα) is bounded and the volume of Sµα(xα) is comparable to that of a standard +ball with radius √µα. In particular, if you define di = sup{µα : Sµα(xα) ∩ Sµj(xj) = +∅, 1 ≤ j ≤ i}, then we have di → 0. It follows that any Sµα(xα), there exists Sµi0(xi0) +such that √µα ≤ 2√µi0 such that Sµα(xα)∩Sµi0(xi0) ̸= ∅. Hence if you use the engulfing +property, you see that Sµα(xα) ⊂ 10Sµi0(xi0). +□ +Using this, we can follow the usual proof of Lebesgue differentiation theorem to con- +clude that: +Lemma 5.2. Let {Sµ(x)}0<µ≤µ0, x∈B0.8 be a family of sets such that: +(1) Sµ(x) ⊂ B1, for all 0 < µ ≤ µ0, and x ∈ B0.8, +(2) There is C > 0, such that for all 0 < µ ≤ µ0 and all x ∈ B0.8, +1 +C m(B√µ(0)) ≤ +m(Sµ(x)) ≤ Cm(B√µ(0)), +(3) For any Sµ1(x1) and Sµ2(x2), if √µ1 ≤ 2√µ2 and Sµ1(x1) ∩ Sµ2(x2) ̸= ∅, then +Sµ1(x1) ⊂ 10Sµ2(x2), +(4) diam Sµ(x) tends to 0 as µ → 0, uniformly for x ∈ B0.8. +Let f : B0.8 → R be an L1 function. Then for m-a.e. x ∈ B0.8, we have: +lim +sup +x∈Sµα(xα), µα→0 +1 +m(Sµα(xα)) +� +Sµα(xα) +|f(y) − f(x)|dm(y) = 0. +In particular, for all measurable set A, we have: +lim +inf +x∈Sµα(xα), µα→0 +m(Sµα(xα) ∩ A) +m(Sµα(xα)) += 1, a.e. x ∈ A. +Proof. The proof of this lemma follows the proof of the standard Lebesgue differentiation +theorem. +First, we define the maximal function: given f ∈ L1(B0.8), +M(f)(x) = +sup +x∈Sµ(x′), 0<µ≤µ0, x′∈B0.8 +1 +m(Sµ(x′)) +� +Sµ(x′) +f(y)dm(y). +As in the proof of Lebesgue differentiation theorem, the result would follow from the +following estimate: +(5.1) +m{x ∈ B0.8 : M(|f|)(x) > t} ≤ Cn +||f||L1 +t +, ∀t > 0. +Indeed, for t > 0, we can define: +Ωt := {x ∈ B0.8 : lim +sup +x∈Sµα(xα), µα→0 +1 +m(Sµα(xα)) +� +Sµα(xα) +|f(y) − f(x)|dm(y) > t}. + +INTERIOR W 2,p ESTIMATE FOR SMALL PERTURBATIONS TO THE COMPLEX MONGE-AMPERE EQUATION +29 +We only need to show that m(Ωt) = 0 for any t > 0. +On the other hand, for any +g ∈ C( ¯B0.8), we have +1 +m(Sµα(xα)) +� +Sµα(xα) +|f(y) − f(x)|dm(y) ≤ M(|f − g|)(x) ++ +1 +m(Sµα(xα)) +� +Sµα(xα) +|g(y) − g(x)|dm(y) + |f − g|(x). +(5.2) +Therefore, +Ωt ⊂ {x : M(|f − g|) > t +3} ∪ {x : |f − g|(x) > t +3}. +Here we implicitly used item (4) of the assumption, so that the middle term in (5.2) +tends to zero as µα → 0. Hence +m(Ωt) ≤ m{x : M(|f − g|) > t +3}+ m{x : |f − g|(x) > t +3} ≤ Cn +3||f − g||L1 +t ++ ||f − g||L1 +t +. +We can then choose a sequence gj ∈ C( ¯B0.8) such that gj → f in L1, so that we may +conclude that m(Ωt) = 0. +Now it only remains to show (5.1). Denote the set on the left hand side to be E, then +for any x ∈ E, we can find 0 < µx ≤ µ0 and yx ∈ B0.8 with x ∈ Sµx(yx), such that: +tm(Sµx(yx)) ≤ +� +Sµx(yx) +|f(y)|dm(y). +Hence we get a covering of E: E ⊂ ∪x∈ESµx(yx). Now we are in a position to apply +Lemma 5.1 to choose a countable sequence Sµi(xi), which is mutually disjoint, and +E ⊂ ∪i10Sµi(xi). Hence +m(E) ≤ m(∪i10Sµi(xi)) ≤ +� +i +m(10Sµi(xi)) = 102n � +i +m(Sµi(xi)) +≤ 102n � +i +1 +t +� +Sµi(xi) +|f(y)|dm(y) ≤ 102n ||f||L1 +t +. +This proves (5.1). +□ +Another lemma we will need is: +Lemma 5.3. Let {Sµ(x)}0<µ≤µ0, x∈B0.8 satisfy the assumptions of Lemma 5.2, and we +assume additionally: +(5.3) +10Sµ(x) ⊂ S121µ(x) ⊂ 12Sµ(x), for any x ∈ B0.8, 0 < µ ≤ µ0 +121. +Let X, Y ⊂ B0.8 be two measurable sets. Let 0 < ¯ε < 1. Assume that: +(1) For any x0 ∈ B0.8, m(Sµ(x0) ∩ X) < ¯εm(Sµ(x0)) for any µ0 +4 ≥ µ ≥ µ0 +484. +(2) For any Sµ(x) with m(Sµ(x)∩X) ≥ ¯εm(Sµ(x)) and µ ≤ µ0 +2 , one has Sµ(x) ⊂ Y . +Then +m(X) ≤ 122n¯εm(Y ). +Proof. For a.e. any x0 ∈ X, we know from the previous lemma that: +(5.4) +lim +µ→0 +m(X ∩ Sµ(x)) +m(Sµ(x)) += 1. + +30 +JINGRUI CHENG, YULUN XU +Hence, if we define µ′ +x = sup{0 < µ ≤ 1 +4µ0 : m(X ∩Sµ(x)) ≥ ¯εSµ(x)}, then 0 < µ′ +x ≤ µ0 +484 +for x ∈ X satisfying (5.4). For such x, we may choose µx, such that 5 +6µ′ +x ≤ µx ≤ µ′ +x, and +that m(X ∩ Sµx(x)) ≥ ¯εSµx(x). We may assume without loss of generality that (5.4) +holds for all x ∈ X, then we get a covering of X: X ⊂ ∪x∈XSµx(x) (otherwise we get a +covering of X modulo a measure zero set). +Then we may use Lemma 5.1 to obtain a countable sequence Sµi(xi), such that Sµi(xi) +are mutually disjoint, with X ⊂ ∪i10Sµi(xi). Hence +m(X) ≤ +� +i +m(X ∩ 10Sµi(xi)) ≤ +� +i +m(X ∩ S121µi(xi)) ≤ +� +i +¯εm(S121µi(xi)) +≤ ¯ε +� +i +m(12Sµi(xi)) ≤ ¯ε122n � +i +m(Sµi(xi)) ≤ 122n¯εm(Y ). +In the second inequality, we used (5.3). +In the third inequality, we used that µ0 +4 ≥ 121µi > µ′ +xi, since our choice of µx guaran- +tees µi ≥ 5 +6µ′ +xi. Therefore m(X ∩ S121µi(xi)) < ¯εm(S121µi(xi)). +In the forth inequality, we used (5.3) again. +In the last inequality, we used that Sµi(xi) are disjoint, and contained in Y , due to +assumption (2) of this lemma. +□ +6. The W 2,p estimate +Definition 6.1. We define Dk to be the set of z0 ∈ ¯B0.8 such that for any 0 < µ ≤ µ0, +Sµ(z0) ⊂ B(z0, +� +10kµ). Define Ak = ¯B0.8 − Dk. +In the above, one should think of Dk to be the “good set” and Ak the “bad set”. +Roughly speaking, Dk is the set on which λi(z0) ≥ 10−k. So that heuristically, we +can conclude that λi(z0) ≤ 10k(n−1) since the equation is Πiλi = f. To see this picture, +we can pretend that Sµ(z0) ≈ {z : � +i,j ui¯j(z0)(z − z0)i(z − z0)j ≤ µ}. The requirement +that this ellipsoid is contained in B√ +10kµ(z0) implies that +1 +√λi ≤ 10 +k +2 . This of course +needs to be made rigorous since our solution u is merely a viscosity solution (hence only +continuous.) +In order to show the W 2,p estimates, there are roughly two steps. +(1) For every k ≥ 1, m(Ak ∩ Brk(0)) ≤ (122n¯ε)k−1m(B0.7), for k ≥ 1, by choosing σ +and ε chosen sufficiently small and rk = rk−1 − 1 +102−k, r0 = 0.7. +(2) Show that for a.e. x ∈ Dk, there is a paraboloid with opening Mk(n−1) +0 +touching +u from above at x, and a paraboloid with opening M−k +0 +touching u from below at +x. This is the viscosity interpretation of D2u(x) ≤ M(k−1)n +0 +and ui¯j(x) ≥ M−k +0 . +We will carry out steps (1) and (2) in the following two subsections. For the convenience +of argument, we will assume that u ∈ C2(Ω) ∩ C(¯Ω), and obtain a quantitative W 2,p +bound on B 1 +2 . Then the general case would follow from an approximation argument. +6.1. Power decay of the measure of bad set. The plan is to use Lemma 5.3 with +X = Ak+1 ∩ Brk+1(0), Y = Ak ∩ Brk(0). Note that the sections Sµ(x) satisfy all the +assumptions of that lemma. Therefore, we just have to show the following two things: +(1) Choosing M0 large enough depending on ¯ε and n, we have m(Sµ(x0) ∩ A1 ∩ +Br1(x0)) < ¯εm(Sµ(x0)) for any µ0 +2 ≥ µ ≥ µ0 +242. + +INTERIOR W 2,p ESTIMATE FOR SMALL PERTURBATIONS TO THE COMPLEX MONGE-AMPERE EQUATION +31 +(2) For all Sµ(x0) with m(Sµ(x0) ∩ Ak+1 ∩ Brk+1(0)) ≥ ¯εm(Sµ(x0)) and µ ≤ µ0 +2 , one +has Sµ(x0) ⊂ Ak ∩ Brk(0), by choosing σ and ε small enough. +We will start with the following lemma, which is the analogue of Lemma 6 in [3]. +Lemma 6.2. Let u0 be a C2 solution to det(u0)i¯j = f0 in Ω with B1−γ(0) ⊂ Ω ⊂ B1+γ. +We also assume that |f0 − 1| < ε on Ω. Let v0 be the solution of det(v0)i¯j = 1 on Ω and +v0|∂Ω = 0. Then there exists a dimensional constant C6,n such that for all γ and ε small +enough (depending only on dimension), +m({x ∈ B 1 +2 : Γ(u0 − 1 +2v0) = u0 − 1 +2v0}) +m(B 1 +2 ) +≥ 1 − C6,nε +1 +2 − C6,nγ +1 +2 . +In the above, Γ(u0 − 1 +2v0) is the convex envelope of u0 − 1 +2v0 in B0.9, defined as: +Γ(u0 − 1 +2v0)(z) = sup{l(z) : l(z) is affine and l ≤ u0 − 1 +2v0 on B0.9}. +Proof. The proof follows similar lines as Lemma 6 of [3]. First, we may use Lemma 4.3 +and Corollary 4.4 to conclude that, with γ chosen small enough: +(6.1) +|Dm(v0 − (|z|2 − 1))|B0.9 ≤ Cnγ1− m +4 , m = 0, 1, 2, 3. +In particular, we know that v0 is strictly convex from (6.1), after choosing γ small enough. +Also by maximum principle, +(1 + 3ε)v0 ≤ (1 + ε) +1 +n v0 ≤ u0 ≤ (1 − ε) +1 +n v0 ≤ (1 − 3ε)v0. +So we get: +(1 +2 + 3ε)v0 ≤ Γ(u0 − 1 +2v0) ≤ (1 +2 − 3ε)v0. +In the following, we will simply denote Γ(u0 − 1 +2v0) by Γ. We make the following claim: +Claim 6.3. +(6.2) +∇ +� +(1 +2 − 3ε)v0 +� +(B 1 +2− +√ +96ε) ⊂ ∇Γ(B 1 +2). +First we use Claim 6.3 to finish the proof and then prove the Claim 6.3 itself. +Indeed, we have: +m(∇Γ(B 1 +2 )) ≥ m +� +∇(1 +2 − 3ε)v0 +� +(B 1 +2− +√ +96ε) = +� +B 1 +2 − +√ +96ε +det +� +(1 +2 − 3ε)D2v0 +� +dm(x) +≥ (1 +2 − 3ε)2n(2 − Cnγ +1 +2)2nm(B 1 +2 − +√ +96ε) ≥ m(B 1 +2 )(1 − C5,nε +1 +2 − C5,nγ +1 +2). +(6.3) +In the equality of the first line above, we used that v0 is strictly convex on B0.9, hence +∇v0 is injective. +In the first inequality of the second line, we used (6.1) with m = 2 to get that +|D2(v0 − |z|2)| ≤ C′ +nγ +1 +2. + +32 +JINGRUI CHENG, YULUN XU +On the other hand, we have, using the Lemma 6.4 below: +m(∇Γ(B 1 +2)) ≤ +� +B 1 +2 ∩{Γ=u0− 1 +2v0} +� +2(1 + ε) +1 +n − det(D2(1 +2v0)) +1 +2n � +dm(x) +≤ +� +B 1 +2 ∩{Γ=u0− 1 +2v0} +� +2(1 + ε) +1 +n − (1 − C′ +nγ +1 +2) +�2ndm(x) +≤ (1 + C5,nγ +1 +2 + C5,nε)2nm(B 1 +2 ∩ {Γ = u0 − 1 +2v0}). +(6.4) +The result follows from combing (6.3) and (6.4). +Now it only remains to prove Claim 6.3. +Indeed, let p = ∇(1 +2 − 3ε)v0(x0), with +x0 ∈ B 1 +2 − +√ +96ε. Denote lx0 = (1 +2 − 3ε)v0(x0) + p · (x − x0). We just need to show that: +{z ∈ B0.9 : Γ(z) < lx0(z)} ⊂ {(z ∈ B0.9 : (1 +2 + 3ε)v0 ≤ lx0} ⊂ B√ +96ε(x0). +That is, the minimum of Γ − lx0 is achieved in the interior of B√ +96ε(x0) ⊂ B 1 +2 , giving +p ∈ ∇Γ(B 1 +2). The first inclusion above is obvious since Γ ≥ (1 +2 +3ε)v0. To see the second +inclusion, we need the following calculation: +First, we note that, for z ∈ B0.9, we have: +v0(z) ≥ v0(x0) + ∇v0(x0) · (z − x0) + (2 − Cnγ +1 +2 )|z − x0|2. +In the above, we used (6.1) again. Therefore, +{z ∈ B0.9 : (1 +2 + 3ε)v0 ≤ lx0} ⊂ {z ∈ B0.9 : (1 +2 + 3ε)(v0(x0) + ∇v0(x0) · (z − x0) ++ (2 − Cnγ +1 +2 )|z − x0|2} ≤ (1 +2 − 3ε)(v0(x0) + ∇v0(x0) · (z − x0))} +So that the above implies +(6.5) +(1 +2 + 3ε)(2 − Cnγ +1 +2)|z − x0|2 ≤ −6ε(v0(x0) + ∇v0(x0) · (z − x0)). +In the above, we may assume |v0(x0)| ≤ 1.1 and |∇v0(x0)| ≤ 2, and we get |z − x0|2 ≤ +6ε(2+2·2) +( 1 +2 +3ε)(2−Cnγ +1 +2 ) ≤ 96ε, and Claim 6.3 is proved. +□ +In the above proof, we used the following lemma, which is the analogue of Lemma 5 +in [3]. +Lemma 6.4. Let u0 and v0 be as given by Lemma 6.2, with γ small enough so that (6.1) +holds. Denote Γ = Γ(u0 − 1 +2v0) as defined by Lemma 6.2. Then we have: +M(Γ) ≤ +� +2(1 + ε) +1 +n − det(D2(1 +2v0)) +1 +2n �2nχΓ=u0− 1 +2 v0 on B0.8. +Here M(Γ) is the (real) Monge-Ampere measure of a convex function, defined as +M(Γ)(E) = m(∂Γ(E)). +Proof. Since u0 is C2, then we know that Γ is C1,1 on B0.8. Hence for any Borel set +E ⊂ B0.8, one would have: M(Γ)(E) = +� +E det(D2Γ)(x)dm(x). Moreover, it is a general + +INTERIOR W 2,p ESTIMATE FOR SMALL PERTURBATIONS TO THE COMPLEX MONGE-AMPERE EQUATION +33 +fact that M(Γ) is concentrated on the contact set {Γ = u0 − 1 +2v0}. On such a set, we +would have: +D2u0 ≥ D2Γ + D2(1 +2v0), a.e. +Therefore +det(D2Γ) +1 +2n + det(D2(1 +2v0)) +1 +2n ≤ det(D2u0) +1 +2n ≤ +� +22n det(u0)2 +i¯j +� 1 +2n ≤ 2(1 + ε) +1 +n . +In the above, we used the concavity of the function A �→ det +1 +d (A), restricted to positive +definite d × d symmetric matrices. +□ +As a corollary to Lemma 6.2, we get: +Corollary 6.5. Let u0 be as Lemma 6.2 and Cn be from (6.1). Define E to be the subset +in B0.8 such that x0 ∈ E if and only if there is a paraboloid with opening 1 +2(1 − Cnγ +1 +2 ) +touching u0 from below at x0 in B0.9. Then we have: +m(E ∩ B 1 +2 ) +m(B 1 +2 ) +≥ 1 − C6,nε +1 +2 − C6,nγ +1 +2, +where C6,n is from Lemma 6.2. +Proof. We just need to show that Γ(u0 − 1 +2v0) = u0 − 1 +2v0 has the property described in +this lemma. Let x0 ∈ B 1 +2 with Γ(u0 − 1 +2v0)(x0) = (u0 − 1 +2v0)(x0). Since Γ is a convex +function on B0.9, we may find p ∈ Cn which defines a supporting plane for Γ, then we +have: +(u0 − 1 +2v0)(z) ≥ Γ(u0 − 1 +2v0)(z) ≥ (u0 − 1 +2v0)(x0) + p · (z − x0), z ∈ B0.9. +On the other hand, from (6.1), we see that: +v0(z) ≥ v0(x0) + ∇v0(x0) · (z − x0) + (1 − Cnγ +1 +2 )|z − x0|2, z ∈ B0.9. +Hence on B0.9 : +u0(z) ≥ u0(x0) + (p + 1 +2∇v0(x0)) · (z − x0) + 1 +2(1 − Cnγ +1 +2 )|z − x0|2. +The above has equality at z = x0 and the right hand side defines a paraboloid with +opening 1 +2(1 − Cnγ +1 +2 ). +□ +Having a paraboloid touching below at a point x0 is a very strong condition, and it +will imply the control of the shape of Sµ(x0) on all scales of µ, together with a control +on the associated pluriharmonic function hµ,x0. More precisely +Lemma 6.6. Let u be a function on B0.9. Let x0 ∈ B0.8 be such that there is a paraboloid +with opening κ > 0, touching u from below at x0 in B0.9. +Let 0 < ˜γ < 1, µ > 0, +and A = ai¯j be positive Hermitian with det A = 1. Define E(x0) = {z : � +i,j ai¯j(z − +x0)i(z − x0)j ≤ µ}. Let h(z) be a degree 2 pluriharmonic polynomial with h(x0) = 0. +(1 − 1 +2˜γ)E(x0) ⊂ {z ∈ B0.9 : (u − h)(z) ≤ u(x0) + µ} ⊂⊂ B0.9. + +34 +JINGRUI CHENG, YULUN XU +Then we have the following estimates for A and h(z): +κ(1 − ˜γ)2 ≤ λi(A) ≤ κ1−n(1 − ˜γ)2(1−n), 1 ≤ i ≤ n. +− +1 +(1 − ˜γ)2n κ1−nI2n ≤ D2h ≤ +n − 1 +(1 − ˜γ)2n κ1−nI2n. +Proof. Denote S(x0) = {z ∈ B0.9 : (u − h)(z) ≤ u(x0) + µ}. First we see that +� +i,j +ai¯j(z − x0)i(z − x0)j > (1 − ˜γ)2µ on (1 − 1 +2˜γ)∂E(x0). +Hence +(6.6) +1 +(1 − ˜γ)2 +� +i,j +ai¯j(z − x0)i(z − x0)j + u(x0) > (u − h)(z), z ∈ (1 − 1 +2˜γ)∂E(x0). +On the other hand, we use that u is being touched below by a paraboloid, we get +u(z) ≥ κ|z − x0|2 + lx0(z), z ∈ B0.9. +Here lx0(z) is an affine function with lx0(x0) = u(x0). Hence +(6.7) +1 +(1 − ˜γ)2 +� +i,j +ai¯j(z−x0)i(z − x0)j+u(x0) > κ|z−x0|2+lx0(z)−h(z), z ∈ (1− 1 +2˜γ)∂E(x0). +However, the above inequality has equality with z = x0. Hence LHS of (6.7)−RHS of +(6.7) has a minimum in the interior of S(x0). Denote this point to be x′ +0. Taking the +complex Hessian at x′ +0, we see that, +1 +(1 − ˜γ)2 ai¯j ≥ κI. +That is λi(A) ≥ κ(1 − ˜γ)2. On the other hand, since Πiλi(A) = 1, we see that λi(A) ≤ +κ1−n(1 − ˜γ)2(1−n). +Next we take the full Hessian of (6.7) at x′ +0, we get: +1 +(1 − ˜γ)2 D2� � +i,j +ai¯j(z − x0)i(z − x0)j +� +≥ 2κI − D2h. +So that +D2h ≥ − +1 +(1 − ˜γ)2n κ1−nI. +On the other hand, since h is harmonic, we get that: +D2h ≤ +n − 1 +(1 − ˜γ)2n κ1−n. +□ +As a consequence, we get that: +Corollary 6.7. Let u0 be as stated in Lemma 6.2, with γ and ε small enough as required +by that lemma. Define D to be the subset of B0.8 such that x0 ∈ D if and only Sµ(x0) ⊂ +B√M1µ(x0) for any 0 < µ ≤ µ0, where M1 = 2(1 + 0.1σ)2(1 − Cnγ +1 +2)−1(1 − 0.2σ)−2, + +INTERIOR W 2,p ESTIMATE FOR SMALL PERTURBATIONS TO THE COMPLEX MONGE-AMPERE EQUATION +35 +where Sµ(x0) is the family of sections constructed in Section 2 (applied to u0). Then we +have: +m(B 1 +2 ∩ D) +m(B 1 +2 ) +≥ 1 − C6,nγ +1 +2 − C6,nε +1 +2 . +Proof. First, from Corollary 6.5, we just need to show that E ∩ B 1 +2 ⊂ D ∩ B 1 +2 . Indeed, +let x0 ∈ E such that there is a paraboloid with opening 1 +2(1 − Cnγ +1 +2), with Cn coming +from (6.1). On the other hand, our construction in Section 2 gives Sµ(x0), 0 < µ ≤ µ0 +with +(1 − 0.1σ)Eµ(x0) ⊂ Sµ(x0) ⊂ (1 + 0.1σ)Eµ(x0), +where Sµ(x0) = {z : u(z) ≤ hµ,x0(z) + u(x0) + µ} and Eµ(x0) = {� +i,j aµ,x0,i¯j(z − +x0)i(z − x0)j ≤ µ}.. +Now we are in a position to use Lemma 6.6 (we could assume +µ0 small enough earlier so that Sµ(x0) ⊂ B0.9 with 0 < µ ≤ µ0, x0 ∈ B0.8), with +κ = 1 +2(1 − Cnγ +1 +2), ˜γ = 0.2σ, A = aµ,x0,i¯j, to get: +1 +2(1 − Cnγ +1 +2)(1 − 0.2σ)2In ≤ aµ,x0,i¯j ≤ 2n−1(1 − Cnγ +1 +2 )1−n(1 − 0.2σ)2(1−n)In. +This would imply that Eµ(x0) ⊂ B(1−Cnγ +1 +2 )− 1 +2 (1−0.2σ)−1√2µ(x0). +So that Sµ(x0) ⊂ +B√M1µ(x0) for all 0 < µ ≤ µ0. +□ +Now we are ready to prove the first statement made in the beginning of this subsection. +More precisely, +Proposition 6.8. Let 0 < ¯ε < 1. If σ and ε are small enough depending only on n and +¯ε, we have: +m(Sµ(x0) ∩ A1 ∩ Br1(x0)) < ¯εm(Sµ(x0)), for any µ0 +484 ≤ µ ≤ µ0 +4 . +Proof. We fix some µ between +µ0 +484 and +µ0 +4 . +We may also assume µ0 ≤ +1 +484 so that +µ2 +0 < µ ≤ µ0, so that E4µ(x0), Eµ(x0), and Eµ0(x0) have the same coefficients. +Recall from Section 2 that ˜T1,x0 is the coordinate change such that z = x0 + +√4µ ˜T1,x0(w) will make E4µ(x0) becomes B1 under w. +Denote ˜Ω to be the image of +S4µ(x0) under w. Then from (1 − 0.1σ)E4µ(x0) ⊂ S4µ(x0) ⊂ (1 + 0.1σ)E4µ(x0), we see +that: +(6.8) +B1−0.1σ ⊂ ˜Ω ⊂ B1+0.1σ. +Also we define ˜u = +1 +4µ(u−h4µ,x0)(x0+√4µ ˜T1,x0(w)), then ˜u, ˜Ω will fullfil the assumptions +we made in Lemma 6.2. Now we are in a position to apply Corollary 6.7 to conclude +that: +(6.9) +m(B 1 +2 ∩ ˜D) +m(B 1 +2) +≥ 1 − C6.nσ +1 +2 − C6,nε +1 +2. +Here ˜D is the subset of B0.8 (under the w variable) such that w0 ∈ ˜D if and only if +˜S˜µ(w0) ⊂ B√M1˜µ(w0) for any 0 < ˜µ ≤ µ0, where M1 = +2(1+0.1σ)2 +(1−0.1Cnσ +1 +2 )(1−0.2σ)2 is given by +Corollary 6.7, but with γ replaced by 0.1σ because of (6.8). Here ˜S˜µ(w0) is the section +given by the construction of Section 2, but carried out for ˜u. First, since Eµ(x0) and + +36 +JINGRUI CHENG, YULUN XU +E4µ(x0) have the same coefficients, Eµ(x0) is now B 1 +2 under w. Therefore, if we define +˜Ω1 to be the image of Sµ(x0) under w, we would get that: +(6.10) +m(˜Ω1 ∩ ˜D) +m(˜Ω1) +≥ 1 − C7,nσ +1 +2 − C7,nε +1 +2. +Next we would like to translate the set ˜D back to z variable, and show that: +(6.11) +the image of ˜D under z variable ⊂ D1, +where D1 is defined in Definition 6.1. This would finish the proof. +In order to show (6.11), we take w0 ∈ ˜D, then we have ˜S˜µ(w0) ⊂ B√M1˜µ(w0) for +0 < ˜µ ≤ µ0, and (1 − 0.1σ) ˜E˜µ(w0) ⊂ ˜S˜µ(w0) ⊂ (1 + 0.1σ) ˜E˜µ(w0). Now we switch back +to z coordinates, then the above inclusions become: +S′ +4µ˜µ(z0) ⊂ x0 + +� +4µ ˜T1,x0(B√M1˜µ(w0)) ⊂ B(1+Cnσ +1 +2 )√4M1µ˜µ(z0), +(1 − 0.1σ)E′ +4µ˜µ(z0) ⊂ S′ +4µ˜µ(z0) ⊂ (1 + 0.1σ)E′ +4µ˜µ(z0). +In the above, z0 is the image of w0 under z and S′ +4µ˜µ is of the form {u−h ≤ u(z0)+4µ˜µ}, +and E′ +4µ˜µ is an ellipsoid having the same volume as B√4µ˜µ. Now we may use Corollary +4.18 to conclude that (1−0.1σ)E4µ˜µ(z0) ⊂ (1+0.1σ)E′ +4µ˜µ(z0), where E4µ˜µ is the ellipsoid +given by Section 2, but constructed directly with z0 (under z coordinate). From this we +see: +S4µ˜µ(z0) ⊂ (1 + 0.1σ)E4µ˜µ(z0) ⊂ (1 + 0.1σ)2 +1 − 0.1σ E′ +4µ˜µ(z0) ⊂ B(1+Cnσ +1 +2 ) (1+0.1σ)2 +(1−0.1σ)2 +√4M1µ˜µ. +This would give the control of Sµ′(z0) for µ′ ≤ 4µµ0, and 4µµ0 ≥ µ2 +0 +121. +For µ′ ≥ µ2 +0 +121, we may assume without loss of generality that µ3 +0 < µ′ ≤ µ2 +0, then +Sµ′(z0) ⊂ (1 + 0.1σ)Eµ′(z0) = z0 + (1 + 0.1σ)T2,z0(B√µ′(0)) ⊂ B(1+0.1σ)(1+Cnσ +1 +2 )2√µ′(z0). +Hence we see that if we have: +(6.12) +10 ≥ max +� +(1 + Cnσ +1 +2) +� +M1, (1 + 0.1σ)(1 + Cnσ +1 +2 )2� +, +we can conclude that z0 ∈ D1, thereby finishing the proof. This is indeed true if we +choose σ small enough depending only on n. Moreover, we need to take σ and ε so that +in (6.10), we have C7,nσ +1 +2 + C7,nε +1 +2 < ¯ε. +□ +Now we move on to prove Statement (2) made in the beginning of Subsection 6.1. +First, we need to show that, if m(Sµ(x0) ∩ Ak+1 ∩ Brk+1(0)) ≥ ¯εm(Sµ(x0)), then +Sµ(x0) ⊂ Brk(0), with σ and ε small enough. +For this we observe that: +Lemma 6.9. Let 0 < µ ≤ µ0 +4 and x0 ∈ B0.8 such that S4µ(x0) ⊂ B0.8. Assume that +there exists Λ > 0 such that ||Tµ1,x0|| ≤ Λ for all 4µ ≤ µ1 ≤ µ0. Then for σ > 0, ε > 0 +chosen small enough depending only on n and ¯ε, we have +m(Sµ(x0) ∩ D2Λ) +m(Sµ(x0)) +≥ 1 − ¯ε +2. +Here D2Λ is the set of z0 such that Sµ′(z0) ⊂ B2Λ√µ′(z0) for all 0 < µ′ ≤ µ0. + +INTERIOR W 2,p ESTIMATE FOR SMALL PERTURBATIONS TO THE COMPLEX MONGE-AMPERE EQUATION +37 +Proof. We consider the section S4µ(x0) with µk+1 +0 +< 4µ ≤ µk +0. From the assumption we +see that ||T4µ,x0|| ≤ Λ. +Similar to the proof of Proposition 6.8, we consider the change of coordinates z = +x0 + √4µT4µ,x0(w), so that E4µ(x0) gets transformed to be a unit ball. We denote ˜Ω to +be the image of S4µ(x0) under w and ˜Ω1 to be the image of Sµ(x0). Similar to the proof +of Proposition 6.8. +m(˜Ω1 ∩ ˜D) +m(˜Ω1) +≥ 1 − C7,nσ +1 +2 − C7,nε +1 +2. +Here ˜D is the subset of B0.8 under the w variable such that w0 ∈ ˜D if and only if +˜S˜µ(w0) ⊂ B√M1˜µ(w0) for 0 < ˜µ ≤ µ0, where M1 = +2(1+0.1σ)2 +(1−0.1Cnσ +1 +2 )(1−0.2σ)2 . We repeat the +argument in the proof of Proposition 6.8 below (6.11), transform this containment back +to z variable and conclude that +(6.13) +Sµ′(z0) ⊂ B(1+Cnσ +1 +2 ) (1+0.1σ)2 +(1−0.1σ)2 Λ√M1µ′, 0 < µ′ ≤ 4µµ0. +Now it only remains to control Sµ′(z0) for µ′ > 4µµ0. +First, if µ′ ≥ 4µ, then we have z0 ∈ Sµ′(x0). Then from Lemma 4.19, we know that: +||T −1 +µ′,x0 ◦ Tµ′,z0|| ≤ 1.13, +||T −1 +µ′,z0 ◦ Tµ′,x0|| ≤ 1.13. +Hence +||Tµ′,z0|| ≤ 1.13Λ, µ′ ≥ 2µ. +Therefore, we have: +(6.14) +Sµ′(z0) ⊂ (1 + 0.1σ)Eµ′(z0) ⊂ B1.13(1+0.1σ)Λ√µ′(z0), µ′ ≥ 2µ. +Finally, for 2µµ0 < µ′ ≤ 2µ, we note that T2µ,x0 and Tµ′,x0 differ at most by ˜Tk+1,x0, +whose norm is bounde by 1 + Cnσ +1 +2, hence we have: +(6.15) +Sµ′(z0) ⊂ B1.13(1+0.1σ)(1+Cnσ +1 +2 )Λ√µ′(z0), 2µµ0 < µ′ < 2µ. +Hence if one combines (6.13)-(6.15) and define +(6.16) +M2 = max +� +(1 + Cnσ +1 +2 )2 (1 + 0.1σ)4 +(1 − 0.1σ)4 M1, 1.16(1 + 0.1σ)2(1 + Cnσ +1 +2)2� +, +then for any z0 in the image of ˜D in the z coordinate, Sµ′(z0) ⊂ B√ +M2Λ2µ′(z0), 0 < µ′ ≤ +µ0. Also we note that the right hand side of (6.16) is less than 4, if σ is small enough +depending only on n. +□ +As a consequence, we get that +Corollary 6.10. Let 0 < µ ≤ µ0 +4 and x0 ∈ B0.8 such that S4µ(x0) ⊂ B0.8. Assume that +for some k0 ≥ 1, m(Sµ(x0) ∩ Ak0+1 ∩ Brk0+1(0)) ≥ ¯εm(Sµ(x0)). Assume also that σ and +ε are chosen small enough depending on ¯ε and n. Then we have: +µ ≤ 10 +− +| log(µ0)| +2 log(1+Cnσ +1 +2 ) +k0 +. + +38 +JINGRUI CHENG, YULUN XU +Proof. Let k1 ≥ 1 be such that µk1+1 +0 +< 4µ ≤ µk1 +0 . Then T4µ,x0 = Tk1,x0 and we have the +estimate: +||Tk,x0||, ||T −1 +k,x0|| ≤ (1 + Cnσ +1 +2 )k1, 1 ≤ k ≤ k1. +Therefore, we may use Lemma 6.9, with Λ = (1 + Cnσ +1 +2)k1, to conclude that: +m(Sµ(x0) ∩ D2Λ) +m(Sµ(x0)) +≥ 1 − ¯ε +2. +From our assumption that m(Sµ(x0) ∩ Ak0+1 ∩ Brk0+1(0)) ≥ ¯εm(Sµ(x0)), we must have +Dk0+1 ⊂ D2Λ (note that we either have Dk0+1 ⊂ D2Λ or D2Λ ⊂ Dk0+1 by definition), in +other words, +(6.17) +10k0+1 ≤ 4(1 + Cnσ +1 +2 )2k1. +So that +µ ≤ 1 +4µk1 +0 ≤ 10 +− +| log(µ0)| +2 log(1+Cnσ +1 +2 ) +k0 +. +□ +As a further corollary, we see that +Corollary 6.11. Assume that for some k0 ≥ 1, m(Sµ(x0) ∩ Ak0+1 ∩ Brk0+1(0)) ≥ +¯εm(Sµ(x0)) for some x0 ∈ B0.8 and µ ≤ µ0 +4 , then Sµ(x0) ⊂ Brk0(0), if σ and ε are +chosen small enough depending on n and ¯ε. +Proof. By our assumption, Sµ(x0) ∩ B0.7 ̸= ∅, hence we know that S4µ(x0) ⊂ B0.8 (by +choosing σ, hence µ0 small enough.) Also we denote k1 so that µk1+1 +0 +< 4µ ≤ µk1 +0 . Then +we may use Corollary 6.10 to obtain that: +diam Sµ(x0) ≤ diam E4µ(x0) ≤ (4µ) +1 +2 ||T4µ,x0|| ≤ µk1 +0 (1 + Cnσ +1 +2 )k1 +≤ +� +µ0(1 + Cnσ +1 +2 ) +� +log 10 +2 log(1+Cnσ +1 +2 ) +k0 +≤ 1 +40 · 2−k0. +(6.18) +The above is true if we choose σ small enough depending on n (according to our choice +of µ0 made in (4.2), µ0 ≤ σ2.) +On the other hand, rk0 − rk0+1 = +1 +10 · 2−k0−1. +Since Sµ(x0) ∩ Brk0+1(0) ̸= ∅, the +conclusion follows from (6.18). +□ +The remaining part of Statement (2) is to show that Sµ(x0) ⊂ Ak. This would directly +follow from the following observation: +Lemma 6.12. Let 0 < µ ≤ µ0 +4 and x0 ∈ B0.8 be such that S4µ(x0) ⊂ B0.8 and Sµ(x0) ∩ +Dk0 ̸= ∅, for some k0 ≥ 1. Then we have: +m(Sµ(x0) ∩ Dk0+1) ≥ (1 − ¯ε +2)m(Sµ(x0)), +provided that σ is small enough depending only on ¯ε and n. +Proof. From the assumption, we can find x∗ ∈ Sµ(x0) ∩ Dk0, which means that +Sµ′(x∗) ⊂ B√ +10k0µ′(x∗) for any 0 < µ′ ≤ µ0. Since (1 − 0.1σ)Eµ′(x∗) ⊂ Sµ′(x∗) and +Tµ′,x∗(B√µ′(0)) = Eµ′(0), we see that +||Tµ′,x∗|| ≤ +√ +10k0 +1 − 0.1σ , +0 < µ′ ≤ µ0. + +INTERIOR W 2,p ESTIMATE FOR SMALL PERTURBATIONS TO THE COMPLEX MONGE-AMPERE EQUATION +39 +Since x∗ ∈ Sµ(x0), we know that x∗ ∈ Sµ′(x0) for all µ′ ≥ 4µ. Hence we may use Lemma +4.19 to conclude that: +||T −1 +µ′,x∗ ◦ Tµ′,x0|| ≤ 1.13, ||T −1 +µ′,x0 ◦ Tµ′,x∗|| ≤ 1.13. +Hence we obtain that +||Tµ′,x0|| ≤ 1.13√ +10k0 +1 − 0.1σ , 4µ ≤ µ′ ≤ µ0. +Now we are in a position to use Lemma 6.9 with Λ = 1.13√ +10k0 +1−0.1σ . Then we conclude that: +m(Sµ(x0) ∩ D2Λ) +m(Sµ(x0)) +≥ 1 − ¯ε +2. +Here D2Λ is the set of z0 such that Sµ′(z0) ⊂ B2Λ√µ′(z0) for all 0 < µ′ ≤ µ0. We will be +done if we can ensure that D2Λ ⊂ Dk0+1, and we just need: +4Λ2 = 4 · 1.16 · 10k0 +(1 − 0.1σ)2 ≤ 10k0+1. +This is clear if σ is small enough. +□ +Now we are ready to show the Statement (2) in the beginning of Subsection 6.1. +Proposition 6.13. Let 0 < µ ≤ µ0 +4 , x0 ∈ B0.8 be such that m(Sµ(x0)∩Ak0+1∩Brk0+1) ≥ +¯εm(Sµ(x0)) for some k0 ≥ 1, then Sµ(x0) ⊂ Ak0 ∩ Brk0(0), if σ and ε are small enough +depending on ¯ε and n. +Proof. Corollary 6.11 already implies Sµ(x0) ⊂ Brk(0), and we just have to show +Sµ(x0) ⊂ Ak0. +If not, namely Sµ(x0) ∩ Dk0 ̸= ∅, then Lemma 6.12 would give us: +m(Sµ(x0) ∩ Dk0+1) ≥ (1 − ¯ε +2)m(Sµ(x0)). +This is in contradiction with our assumption that: +m(Sµ(x0) ∩ Ak0+1 ∩ Brk0+1) ≥ +¯εm(Sµ(x0)). +□ +Now we are ready to show that: +Theorem 6.1. Let Ak be define by Definition 6.1. Let 0 < ¯ε < 1 be given. Let σ > 0 +and ε > 0 be small enough depending on n and ε, we have: +m(Ak ∩ Brk(0)) ≤ m(B0.7)(122n¯ε)k−1, k ≥ 1. +Here rk = rk−1 − 1 +10 · 2−k, r0 = 0.7. In particular, m(Ak ∩ B0.6) ≤ m(B0.7)(122n¯ε)k−1. +Proof. For k = 1, the above estimate is trivial. +For k ≥ 1, we have the following estimate holds: +m(Ak+1 ∩ Brk+1(0)) ≤ 122n¯εm(Ak ∩ Brk(0)). +This follows from taking X = Ak+1∩Brk+1(0), Y = Brk(0)∩Brk(0) in Lemma 5.3, where +the two assumptions of that lemma indeed hold, due to Proposition 6.8 and 6.13. +□ + +40 +JINGRUI CHENG, YULUN XU +6.2. Control of the second derivatives on the good set and completion of proof. +In this section, we show that the derivatives are controlled on the good sets Dk. We +wish to emphasize that we are assuming u ∈ C2(Ω), only for the sake of convenience of +argument, but the regularity of u does not go into the quantitative estimates. +We start with the following lemma: +Lemma 6.14. Let u ∈ C2(B1) solving det ui¯j = f with |f − 1| < ε. Let x0 ∈ Dk for +some k ≥ 1. Then +1 +10k In ≤ ui¯j(x0) ≤ 2 · 10(n−1)kIn. +Proof. From Definition 6.1, we know that Sµ(x0) ⊂ B√ +10kµ(x0). Let 0 < c < 1, then we +have, on ∂Sµ(x0), +(6.19) +u − hµ,x0 = u(x0) + µ ≥ u(x0) + |x − x0|2 +10k +> u(x0) + c|x − x0|2 +10k +. +In the above, we noted that |x − x0|2 ≤ 10kµ on ∂Sµ(x0). +On the other hand, (6.19) achieves equality when x = x0. Hence the function x �→ +(u − hµ,x0)(x) − c|x−x0|2 +10k +achieves minimum in the interior of Sµ(x0), say xµ. Then we +have: +(6.20) +ui¯j(xµ) ≥ +c +10k In. +Since det ui¯j ≤ 1 + ε, we see that +ui¯j(xµ) ≤ (1 + ε)c1−n10k(n−1)In. +Since diam Sµ(x0) → 0 and xµ ∈ Sµ(x0), we see that xµ → x0 as µ → 0. Hence we +conclude that: +c +10k In ≤ ui¯j(x0) ≤ 2 · c1−n10k(n−1)In. +Since 0 < c < 1 is arbitrary, we can make c → 1 to conclude the result. +□ +As a consequence, we get that: +Proposition 6.15. Let p > 1 be given. Let u be given by Theorem 1.3 and is C2(B1). +Let γ be small enough depending only on n, and ε small enough depending on p and n, +then we have: +� +B0.6 +� +(∆u)p + (truωE)p� +≤ Cp,n. +Here ∆u = � +i ui¯i, truωE = � +i +1 +ui¯i . +Proof. Note that on Dk, we have: +∆u ≤ 2n · 10(n−1)k, truωE ≤ n · 10k. + +INTERIOR W 2,p ESTIMATE FOR SMALL PERTURBATIONS TO THE COMPLEX MONGE-AMPERE EQUATION +41 +Hence {∆u > 2n · 10(n−1)k} and {truωE > n · 10k} are contained in Ak for k ≥ 1. +� +B0.6∩{∆u>2n·10n−1} +(∆u)p ≤ +∞ +� +k=1 +� +B0.6∩{2n·10(n−1)k<∆u≤2n·10(n−1)(k+1)} +(∆u)p +≤ +∞ +� +k=1 +� +2n · 10(n−1)(k+1)�pm(Ak ∩ B0.6) ≤ +∞ +� +k=1 +� +2n · 10(n−1)(k+1)�p · m(B0.7)(122n¯ε)k−1 += (2n · 102(n−1))pm(B0.7) +∞ +� +k=1 +(10(n−1)p122n¯ε)k−1. +In order for the above sum to be finite, we can choose ¯ε > 0 so that 10(n−1)p122n¯ε = 1 +2 so +that the above integral ≤ 2(2n · 102(n−1))pm(B0.7). In the above, we used Theorem 6.1. +In order for the theorem to apply, we need to choose ε and σ small enough depending +only on ¯ε and n, so the choice of ε and σ eventually depend on p and n. +The estimation for truωE is completely similar. +□ +From Proposition 6.15, we can get full second order estimate by applying the Lp- +estimate for the Laplacian. That is, we get: +Corollary 6.16. Under the assumption of Proposition 6.15, we have: +||u||W 2,p(B 1 +2 ) ≤ Cp,n, +if γ is small enough depending only on n, and ε small enough depending only on p and +n. +Proof. The result would follow from Proposition 6.15 and the classical W 2,p estimates +(Gilbarg-Trudinger [8], Chapter 9): for any u ∈ C2(B0.6) and 1 < p < ∞, +||u||W 2,p(B 1 +2 ) ≤ C(||u||Lp(B0.6) + ||∆u||Lp(B0.6)). +□ +7. Some corollaries of the main theorem +First we prove Corollary 1.1. +Proof. (of Corollary 1.1) First, we wish to use the following Lemma 7.1 to conclude that +ϕ is close to zero. Indeed, by taking g = 1, ψ = 0, p = 2 in the following, we find that: +||ϕ||L∞ ≤ cε +1 +2(n+4) . +Here c depends only on the background metric. +Now we take some point p0 ∈ M and take normal coordinates (z1, · · · , zn) at p0 +so that gi¯j(p0) = δij and ∇g(p0) = 0. We can choose local potential ρ(z), such that +ω0 = √−1∂ ¯∂ρ near p0, say on B1(p0) (under local coordinates z). +So that on this +neighborhood, the equation can be written as: +(7.1) +det((ρ + ϕ)i¯j) = f det(gi¯j), +in B1. +In order to use Theorem 1.4, we need to zoom in (7.1) at p0 at a suitable scale so that +the right hand side is close to a contant. + +42 +JINGRUI CHENG, YULUN XU +Denote u = ρ + ϕ. Let 0 < r0 < 1, we perform a change of variable z = r0w. Next we +define +˜ur0(w) = 1 +r2 +0 +u(r0w), ˜ρr0 = 1 +r2 +0 +ρ(r0w), ˜ϕr0(w) = 1 +r2 +0 +ϕ(r0w). +Assume that for some C0 > 0, we have +1 +C0 I ≤ ρi¯j ≤ C0I, |D3ρ| ≤ C0 on B1, then we see +that with the same C0 > 1: +1 +C0 +I ≤ (˜ρr0)wi ¯wj ≤ C0I, |D3 +w ˜ρr0| ≤ C0, +for |w| < 1. +Also +|˜ur0 − ˜ρr0| ≤ 1 +r2 +0 +||ϕ||L∞ ≤ c 1 +r2 +0 +ε +1 +2(n+4) . +Also on {|w| < 1}, we have +det(˜ur0)i¯j = f det(gi¯j)(r0w), +and we can estimate how close the right hand side is from 1: +|f det(gi¯j) − 1| ≤ |f − 1| det gi¯j(r0w) + f| det(gi¯j)(r0w) − 1| ≤ C1ε + C1r0. +Here C1 depends only on the background metric and n. +Hence in order to apply Theorem 1.4, we need to make sure that: +C1ε + C1r0 ≤ εp,n, +c 1 +r2 +0 +ε +1 +2(n+4) ≤ δ0. +Here εp,n and δ0 are determined by Theorem 1.4. Hence we need to choose r0 first so +that C1r0 = 1 +2εp,n. Then we fix this choice, and choose ε small enough so as to make +sure C1ε ≤ 1 +2εp,n and c 1 +r2 +0 ε +1 +2(n+4) ≤ δ0. Then Theorem 1.4 gives W 2,p estimate for ˜ur0 in +B 1 +2. Scaling back to u, we get W 2,p estimate for u in B 1 +2 r0(p0). +□ +In the above proof, we used the following stability estimate due to Kolodziej [11] +Lemma 7.1. ([11]) Let (M, ω0) be a compact K¨ahler manifold. Let ϕ ∈ PSH(M, ω0) +and ψ ∈ PSH(M, ω0) be the solution to the complex Monge-Ampere equations: +(ω0 + +√ +−1∂ ¯∂ϕ)n = fωn +0 , sup +M +ϕ = 0, +(ω0 + +√ +−1∂ ¯∂ψ)n = gωn +0 , sup +M +ψ = 0. +Assume that there is some c0 > 0, p > 1 such that +||f||Lp ≤ c0, ||g||Lp ≤ c0. +Then we have +sup +M +|ϕ − ψ| ≤ c(c0, p)||f − g|| +1 +n+4 +Lp . +Now we prove Corollary 1.2. The argument is similar to Corollary 1.2. + +INTERIOR W 2,p ESTIMATE FOR SMALL PERTURBATIONS TO THE COMPLEX MONGE-AMPERE EQUATION +43 +Proof. (of Corollary 1.2) We first use Lemma 3.4 with q = 2 to estimate the difference +between ϕ and ϕ0. We have that +sup +Ω +|u − u0| ≤ ε + cε +1 +2n ≤ c′ε +1 +2n . +Here c depends only on n and diam Ω. The rest of the argument is similar to Corollary +1.1. In other words, we take any point z0 ∈ Ω′ and consider rescaling ˜ur0(w) = 1 +r2 +0 u(z0 + +r0w), and we similarly consider ˜u0,r0 = 1 +r2 +0 u0(z0 + r0w). Then for suitable chosen r0, we +will have that ˜ur0 satisfy the hypothesis of Theorem 1.4. +□ +Next, we prove the Liouville theorem Corollary 1.3. +Proof. (of Corollary 1.3) +Let r > 1, we consider: +ur(w) = 1 +r2 u(rw). +Then we know that for r large enough, one has +|ur(w) − |w|2| ≤ 2ε, w ∈ B1. +On the other hand, we have 1 − ε ≤ det ua¯b ≤ 1 + ε, hence from Theorem 1.4 we obtain: +||ur||W 2,p(B 1 +2 ) ≤ Cp,n, +as long as we choose ε small enough depending on n and p. On the other hand, ur will +also satisfy the scalar flat equation, hence we may use the following Lemma 7.2, and +choose p = pn, then we obtain that: +||ur||C2,α(B 1 +4 ) ≤ C. +Here C is a uniform constant independent of r. Rescaling back to u, it gives: +|D2u(x) − D2u(y)|rα ≤ C|x − y|α, ∀x, y ∈ B r +2 . +One can fix x, y and let r → ∞ and conclude that D2u is a constant, hence u is a +quadratic polynomial. +□ +In the following, we used the following estimate of the scalar flat equation, which +originates from Chen-Cheng [6], Corollary 6.2. Note that this is from the preprint version +on arxiv, which was deleted in the published version. +Lemma 7.2. ([6]) Let u ∈ C4(B1) ∩ PSH(B1) be a bounded solution to the scalar flat +equation: +n +� +i,j=1 +ui¯j∂i¯j +� +log det ua¯b +� += 0. +Assume that for some p > 3n(n − 1), we have ∆u ∈ Lp(B1), � +i +1 +ui¯i ∈ Lp(B1). Then for +any 0 < α < 1, +||u||C2,α(B 1 +2 ) ≤ C. +Here C depends on n, p, ||u||L∞(B1), ||∆u||Lp(B1), || � +i +1 +ui¯i ||Lp(B1). +Now let us prove the Schauder type estimate, Corollary 1.4. + +44 +JINGRUI CHENG, YULUN XU +Proof. (of Corollary 1.4) First, we just need to prove that u ∈ W 2,p for p large enough, +depending only on n and α. Indeed, W 2,p embeds into C1,1− 2n +p for p > 2n (keep in mind +that the real dimension is 2n.) In order to apply Lemma 7.3, we just need to choose p +large, so that +1 − 2n +p > 1 − +α +n(2 + α) − 1. +Now we fix this p and it only remains to show u ∈ W 2,p. The argument is similar +to Corollary 1.1 and 1.2. +We need to do rescaling, so that we are in the situation +of Theorem 1.4. +In other words, we take z0 ∈ B 1 +2 , and consider rescaling ˜ur0(w) = +1 +r2 +0(f(z0)) +1 +n u(z0 + r0w). Similarly, we define ˜wr0(w) = +1 +r2 +0(f(z0)) +1 +n w0(z0 + r0w). Then for +|w| < 1, the closeness between ˜ur0 and ˜wr0 becomes: +|˜ur0 − ˜wr0| ≤ +δ0 +r2 +0(f(z0)) +1 +n +≤ δ0 +C +1 +n +1 +r2 +0 +. +On the other hand, +det(˜ur0)i¯j = f(z0 + r0w) +f(z0) +, +and we can estimate: +|f(z0 + r0w) +f(z0) +− 1| ≤ +1 +f(z0)rα +0 K ≤ C +1 +n +1 Krα +0 . +Hence, in order for Theorem 1.4 to apply, we just need to guarantee: +δ0 +C +1 +n +1 +r2 +0 +≤ δ′ +0, +C +1 +n +1 Krα +0 ≤ εp,n. +Here δ′ +0 is the required closeness from the solution to the smooth background, given by +Theorem 1.4, and εp,n is the required small ε under the above choice of p. +Hence we should first choose r0 so that C +1 +n +1 Krα +0 = 1 +2εp,n. Then with r0 fixed, we +choose δ0 small enough so as to make sure δ0 +C +1 +n +1 +r2 +0 ≤ δ′ +0. Then we may use Theorem 1.4 +to conclude that u is in W 2,p on B 1 +2 r0(z0). +□ +Lemma 7.3. ([12]) Let Ω be a domain in Cn and u ∈ PSH(Ω)∩C(Ω) be a weak solution +of the complex Monge-Ampere equation in Ω with 0 < λ ≤ f ∈ Cα(Ω) for some constant +λ and some α ∈ (0, 1). If u ∈ C1,β(Ω) with β ∈ (β0, 1), where +β0 = β0(n, α) = 1 − +α +n(2 + α) − 1. +Then u ∈ C2,α(Ω). Furthermore the C2,α norm of u in any relatively compact subset is +estimable in terms of n, α, β, λ, ||u||C1,β (Ω), ||f||Cα(Ω) and the distance of the set to ∂Ω. + +INTERIOR W 2,p ESTIMATE FOR SMALL PERTURBATIONS TO THE COMPLEX MONGE-AMPERE EQUATION +45 +References +[1] E. Bedford and B. A. Taylor: The Dirichlet problem for a complex Monge-Ampere equation. Invent. +Math. vol 37 (1976), 1-44. +[2] Z. Blocki: Interior regularity of the degenerate Monge-Ampere equation. Bull. of the Aust. Math. +Soc. vol 68(2003), 81-92. +[3] L. A. Caffarelli: Interior W 2,p estimates for solutions of the Monge-Amp`ere equation. Ann. of Math. +vol 131, issue 1(1990), 135-150. +[4] L. A. Caffarelli, L. Nirenberg and J. Spruck: The Dirichlet problem for nonlinear second order elliptic +equations. I. Monge-Ampere equations. Comm. Pure. Appl. Math. vol 37(1984), 369-402. +[5] L. A. Caffarelli, J. J. Kohn, L. Nirenbger, and J. Spruck: The Dirichlet problem for nonlinear second +order elliptic equations. II. Complex Monge-Ampere and uniformly elliptic equations. Comm. Pure. +Appl. Math. vol 38(1985), 209-252. +[6] X.-X. Chen and J. Cheng: On the existence of constant scalar curvature K¨ahler metrics (I): Apriori +estimates. J. Amer. Math. Soc, vol 34, no. 4(2021), 909-936. (arXiv: 1712.06679) +[7] Dinew S and Ko�lodziej S. A priori estimates for complex Hessian equations. Analysis & PDE, (2014), +7(1), 227-244. +[8] D. Gilbarg and N. S. Trudinger: Elliptic partial differential equations of second order. Vol. 224. No. +2. Berlin: springer, (1977). +[9] W. Y. He: On the regularity of complex Monge-Ampere equation. Proc. Amer. Math. Soc. vol 140, +no. 5(2012), 1719-1727. +[10] N. M. Ivochikina: Construction of aprioru bounds for convex solutions of the Monge-Ampere equa- +tion by integral methods. Ukrian. Math. J. vol 30(1978), 32-38. +[11] S. Kolodziej: The Monge-Ampere equation on compact K¨ahler manifolds. Indiana. Univ. Math. J. +vol 52, no. 3(2003), 667-686. +[12] Chao, Li, Jiayu Li, and Xi Zhang. A C2,α estimate of the complex Monge–Amp`ere equation. Journal +of Functional Analysis , vol 275, no. 1 (2018), 149-169. +[13] Pogorelov. A. V: The regularity of the generalized solutions of the equation det(∂2u/∂xi∂xj) = +ϕ(x1, x2, ..., xn) > 0. Dokl. Akad. Nauk SSSR. Vol. 200. (1971), 534-537. +[14] Pogorelov. A. V: The Minkowski multidimensional problem. J. Wiley, New York, 1978. +[15] O. Savin: Small Perturbation Solutions for Elliptic Equations. Comm. in PDE, vol 73, issue 4(2007), +557-578. +[16] F. Schulz: A C2 estimate for solutions of complex Monge-Ampere equations. J. Reine. Angew. Math. +vol 348(1984), 88-93. +[17] Yu +Wang: +A +Liouville +Theorem +for +the +Complex +Monge-Ampere +Equation. +Preprint, +arXiv:1303.2403 (2013). +[18] S. -T. Yau: On the Ricci curvature if a compact K¨ahler manifold and the complex Monge-Amp`ere +equation, I. Comm. Pure. Appl. Math. vol 31(1978), 339-441. + diff --git a/sdAzT4oBgHgl3EQfBPql/content/tmp_files/load_file.txt b/sdAzT4oBgHgl3EQfBPql/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f4068ef00a2eb561d711fa8828e1ed9d96ed49b1 --- /dev/null +++ b/sdAzT4oBgHgl3EQfBPql/content/tmp_files/load_file.txt @@ -0,0 +1,1828 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf,len=1827 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='00940v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='AP] 3 Jan 2023 INTERIOR W 2,p ESTIMATE FOR SMALL PERTURBATIONS TO THE COMPLEX MONGE-AMPERE EQUATION JINGRUI CHENG, YULUN XU Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let w0 be a bounded, C3, strictly plurisubharmonic function defined on B1 ⊂ Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then w0 has a neighborhood in L∞(B1) with the following property: for any continuous, plurisubharmonic function u in this neighborhood solving 1 − ε ≤ MA(u) ≤ 1 + ε, one has u ∈ W 2,p(B 1 2 ), as long as ε > 0 is small enough depending only on n and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' This partially generalizes Caffarelli’s interior W 2,p estimates for real Monge-Ampere to the complex version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Introduction Monge-Ampere equations are second-order partial differential equations whose lead- ing term is the determinant of the Hessian of a real unknown function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' The Hessian is required to be positive or at least nonnegative, so that the equations are elliptic or degenerate elliptic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Monge-Ampere equations can be divided into real or complex, de- pending on whether one is considering real Hessian or complex Hessian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' In the real case, the Hessian is uij, so that the positivity of the Hessian is a convexity condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' In the complex case, the Hessian is ui¯j, and its positivity is a plurisubharmonicity condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' For both real and complex Monge-Ampere, the existence and regularity theory with smooth data has been well established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' In the real case, it is proved by Caffarelli- Nirenberg-Spruck [4] on smooth strictly convex domains on Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' In the complex case, the foundations of an existence and regularity theory were laid out by Yau [18] in the setting of a compact K¨ahler manifold, and by Caffarelli-Kohn-Nirenberg-Spruck [5], in the setting of a smooth pseudo-convex domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Another important aspect about the Monge-Ampere equations is their apriori esti- mates, starting with interior ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' In general, the results known for the real case is much stronger than the complex case, due to the fact that the solution being convex gives much more stringent constraint than being plurisubharmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' For example, the interior gradient estimate for real Monge-Ampere equation is more or less a trivial matter (if we know the solution is bounded), since the underlying solution considered is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' This is not the case for complex Monge-Ampere, and the boundedness of plurisubharmonic function only gives the gradient being in L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Arguably the most important estimate of Monge-Ampere is to get second derivative estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' If we get such estimates in L∞, then the equation becomes unformly elliptic and the standard theory can apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' On the other hand, the bad news is that for both real and complex Monge-Ampere equations, there are no purely interior C2 estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Indeed, having a convex solution to det uij = 1 in a domain doesn’t imply u ∈ C2 in the interior, due to a counterexample by Pogorelov [14] (a counterexample for the complex version is given by He in [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' In general, one needs to impose some boundary conditions Date: Nov 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' 1 2 JINGRUI CHENG, YULUN XU (say, u = 0 on the boundary), in order to conclude that D2u is bounded in the interior (Pogorelov’s estimate [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Based on that, Caffarelli’s proved the following interior W 2,p estimate when the right hand side is a small perturbation of a constant: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let Ω ⊂ Rn be a convex domain such that B1 ⊂ Ω ⊂ Bn and u is a weak solution to det uij = f with |f − 1| < ε and u = 0 on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then for any 1 < p < ∞, if ε > 0 is small enough depending only on p and n, then ||u||W 2,p(B 1 2 ) can be bounded by a constant depending only on p and n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' In the above theorem, the weak solution is defined using the measure of the im- age of the gradient mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' One could also replace the boundary condition by a strict convexity assumption, meaning that the supporting plane of a convex function touches the function only at one point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' That is, we have: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let Ω ⊂ Rn be a bounded convex domain, and u is a weak solution to det uij = f with |f − 1| < ε which is strictly convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then for any compact subdomain Ω′, one has ||u||W 2,p(Ω′) ≤ C, where C depends on p, n, dist(Ω′, ∂Ω), the modulus of strict convexity of u, as long as ε is small enough depending only on p and n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='2 actually follows from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Indeed, for any x0 ∈ Ω′, we can take lx0 to be a linear function touching u from below, then Sc := {u(x) ≤ lx0(x) + c} will be contained in Ω for c small enough, due to the strict convexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then one can normalize Sc to be in the situation of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' The goal of this paper is to generalize (partially) Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='2 to the complex Monge-Ampere equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' More precisely, we show that Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let Ω ⊂ Cn be a bounded domain with B1−γ0 ⊂ Ω ⊂ B1+γ0 for some γ0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let u ∈ C2(Ω) ∩ PSH(Ω) ∩ C(¯Ω) be such that 1 − ε ≤ det ui¯j ≤ 1 + ε in Ω and u = 0 on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Given 1 < p < ∞, if γ0 is small enough depending only on n, and ε small enough depending only on n and p, then ||u||W 2,p(B 1 2 ) ≤ C, || � i 1 ui¯i ||Lp(B 1 2 ) ≤ C, where the constant C depends only on n and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' This theorem should be understood as the analogue of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' More generally, we have: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let w0 be a smooth function in the unit ball such that for some C0 > 1: 1 C0 I ≤ (w0)zi¯zj ≤ C0I, |D3w0| ≤ C0 in B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then there exists δ0 > 0 small enough, depending only on C0 and n, such that for all u ∈ PSH(B1) ∩ C(B1) with |u − w0| ≤ δ0 on B1, solving 1 − ε ≤ MA(u) ≤ 1 + ε, we have u ∈ W 2,p(B 1 2 ) and � i 1 ui¯i ∈ Lp(B 1 2 ), as long as ε is small enough depending only on n and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' In the above, MA(u) is the complex Monge-Ampere operator defined for continuous plurisubharmonic functions, in the Bedford-Taylor sense (see [1]), so that MA(u) = det ui¯j when u ∈ C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' INTERIOR W 2,p ESTIMATE FOR SMALL PERTURBATIONS TO THE COMPLEX MONGE-AMPERE EQUATION3 If one compares Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='3 to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1, or Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='4 to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='2, the biggest difference is that we have to assume our solution is close to a smooth plurisub- harmonic function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' The reason we have to make this assumption is related to whether one has Pogorelov type estimates for complex Monge-Ampere equations, which has been open until now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Note that the interior C2 estimates for the complex Monge-Ampere equation with zero boundary values were studied by F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Schulz in [16], using the integral approach of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Ivochikina [10] for real Monge-Ampere equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' However, the proof in [16] is not complete, which was first pointed out by Blocki [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' We will comment more on the technical aspect later, but for now, let us present some direct consequences of our main theorem, which seems new and interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' First we observe that Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='4 would give us the following result in the manifold setting: Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let (M, ω0) be a compact K¨ahler manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let ϕ ∈ PSH(M, ω0)∩C(M) be the solution to: (ω0 + √ −1∂ ¯∂ϕ)n = fωn 0 , ω0 + √ −1∂ ¯∂ϕ > 0, where |f −1| < ε and � M fωn 0 = � M ωn 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let 1 < p < ∞, then we have that ϕ ∈ W 2,p(M) as long as ε is small enough depending only on p, n and the background metric ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Similar results would also hold for the setting of bounded domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' In other words, we have: Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let Ω ⊂ Cn be a bounded domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let u0 ∈ C3(Ω) ∩ C(¯Ω) ∩ PSH(Ω) be the solution to det(u0)i¯j = f0 > 0 in Ω and u0|∂Ω = ϕ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let u ∈ C(¯Ω) ∩ PSH(Ω) be the solution to MA(u) = f and u|∂Ω = ϕ such that |f − f0| < ε, |ϕ − ϕ0| < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let Ω′ be a compact subdomain of Ω, then we have u ∈ W 2,p(Ω′), as long as ε is small enough depending only on Ω′, Ω, the C3 bound and complex Hessian lower bound of u0 in a neighborhood of Ω′, p and n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' One more application of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='4 is the following Liouville theorem for entire scalar flat metric on Cn, which is a generalization of a result by Yu Wang [17]: Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let u be a C2 plurisubharmonic function on Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Denote ωu = √−1∂ ¯∂u and assume that ωu is scalar flat, namely n � i,j=1 ui¯j∂i¯j � log det ua¯b � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then there exists εn > 0 small enough depending only on n, such that we can deduce u is quadratic, provided that: (1) limr→∞ supBr |u(z)−|z|2| r2 ≤ εn, (2) 1 − εn ≤ det ua¯b ≤ 1 + εn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' The result by Yu Wang [17] is a special case of the above Corollary with det ua¯b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Finally we observe the following C2,α estimate for complex Monge-Ampere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' For K > 0, 0 < α < 1 and C1 > 1, we define the following class of functions: F(K, α, C1) = {f is defined on B1 : ||f||α,B1 ≤ K, 1 C1 ≤ f ≤ C1 on B1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' 4 JINGRUI CHENG, YULUN XU Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let w0 be as in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then there exists δ0 > 0, depending only on C0, n, K, α, C1, such that for any u ∈ PSH(B1) ∩ C(B1) with |u − w0| ≤ δ0 on B1 and solving MA(u) = f for some f ∈ F(K, α, C1), we have u ∈ C2,α(B 1 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Next we would like to explain the ideas of proof for Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' The heart of the idea is from Caffarelli’s paper [3] which we explain first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Since the solution u is strictly convex, we may consider sections of u of the form Sc(x0) := {(u − l)(x) ≤ u(x0) + c} which is strictly contained in Ω, where l(x) is the supporting linear function of u at x0 and c > 0 is called the “height” of the section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Now we solve det wij = 1 on this open set, equaling u on the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' From Pogorelov estimate, we know that w is smooth in the interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' By doing Taylor expansion for w, we find that the sections of w will be close to ellipsoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' On the other hand, since f is close to 1, we also have u is very close to w by maximum principle, hence the sections of u are close to ellipsoids as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' If the shape of the ellipsoids are comparable to a ball for heights going to 0, then the second derivatives are under control at that point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' The whole point of W 2,p estimate is then to estimate the measure of the set where the shape of such ellipsoids loses control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' (which is reflected by the opening of the paraboloid touching u from below) For this purpose, we will need a version of Vitali’s covering lemma, but adapted to sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' To establish the covering lemma for sections, a crucial property we need is the following engulfing property: If Sµ1(x1) ∩ Sµ2(x2) ̸= ∅, with µ1 ≤ µ2, then Sµ1(x1) ⊂ S10µ2(x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' This property would be a result of compactness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Indeed, if u were the standard solution 1 2|x|2, we would have Sµ1(x1) = B√µ1(x1), Sµ2(x2) = B√µ2(x2) and the engulfing property indeed holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' We can still expect this property if u is close to a quadratic polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' We follow similar lines of argument in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' The first hurdle we face is to take sections with u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Unlike the convex function, given x0 ∈ Ω, it is not clear whether one can find a pluriharmonic function h, for which {u − h < u(x0) + c} is compactly contained in Ω for c > 0 small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Even though this is not clear in general, we show that, however, it is indeed possible if u is close to a smooth plurisubharmonic function whose complex Hessian has a lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Next we need to solve the Dirichlet problem det wi¯j = 1, w = u on the boundary of a small section of u, similar to what we did in the real case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' The problem we are facing now, is that we do not know if w is smooth, since Pogorelov’s estimate is not known for the complex Monge-Ampere equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' However, if u is close to a smooth, strictly plurisubharmonic function, then the section defined by u will be close to an ellipsoid from the very beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' This will allow us to use Savin’s perturbation result to conclude that w is indeed smooth in the interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' In this paper, we use an induction process to construct sections Sµ(x0) for µ > 0 and small, which takes the form {(u − hµ,x0)(z) ≤ u(x0) + µ}, where hµ,x0(z) is pluriharmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Moreover, we also show that Sµ(x0) remains close to an ellipsoid in the induction process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' A drawback with our construction is that it is highly non-canonical, since it relies on solving Dirichlet problem on a sequence of smaller and smaller sections “centered at” x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' This construction of Sµ(x0) does not commute with linear transformations we use to normalize the ellipsoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' We explain this matter in greater detail in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='2, under Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' The fact that Sµ(x0) is non-canonical makes it apparently very hard to relate Sµ(x0) to Sµ(x1), even if x0 and x1 are very close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' This would make it seeming impossible to INTERIOR W 2,p ESTIMATE FOR SMALL PERTURBATIONS TO THE COMPLEX MONGE-AMPERE EQUATION5 prove the engulfing property for the Sµ(x0) we constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' (let us also comment that the hµ,x0(z) above is unbounded in general as µ → 0, which is totally different from the real case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=') However, one surprising thing we observed is that, if a section in the form {u − h ≤ u(x0) + µ} happens to be close to an ellipsoid, then the shape of this ellipsoid is “unique” in a quatitative sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' This key observation allows us to show the engulfing property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' The above discussion shows the importance to understand whether we have Pogorelov estimates for complex Monge-Ampere equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' In particular, we are motivated to make the following definition: Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let Ω ⊂ Cn be a bounded domain and Ω′ ⊂ Ω be compactly contained in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' We say that (Ω, Ω′) has the Pogorelov property if: (1) There exists u ∈ PSH(Ω) ∩ C(¯Ω), solving MA(u) = 1 in Ω in the sense of Bedford-Taylor, and u = 0 on ∂Ω, (2) The solution u is C2 (hence C∞) on Ω′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' If one carefully checks the argument of the present paper, what we really proved is the following result: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let u ∈ PSH(B1) ∩ C(B1) solve 1 − ε ≤ MA(u) ≤ 1 + ε in B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Assume that there exists finitely many Pogorelov pairs (Ωi, Ω′ i), 1 ≤ i ≤ N, such that (1) Each Ωi is of the form {u − hi < ci} for some pluriharmonic function hi and ci ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' (2) ¯B 1 2 ⊂ ∪N i=1Ω′ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then we have u ∈ W 2,p(B 1 2), as long as ε is small enough, depending only on n, p, the lower and upper Hessian bound and C3 bound for ui on Ω′ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Here ui is the solution to MA(ui) = 1 on Ωi and ui = 0 on ∂Ωi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' For now, it seems mysterious to characterize when the assumptions (1) and (2) above hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' We are only able to verify such assumptions when u is close to a smooth function for the moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' For example, in the setting of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='3, Ω = {u < 0}, and (Ω, B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='6) has the Pogorelov property, as long as Ω is close to a unit ball, thanks to Savin’s C2,α estimates for small perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' To conclude the Introduction, we will explain the organization of the rest of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' In Section 2, we include some definitions, notations and some preliminary results we will use again and again in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' In Section 3, we show how to reduce Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='4 to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' The later is a special case of the former by taking w0 = |z|2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Section 4-6 below are devoted to the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' In Section 4, we construct sections Sµ(x0) for u which are of the form {(u−hµ,x0)(z) ≤ u(x0) + µ}, where hµ,x0 is pluriharmonic and Sµ(x0) is close to an ellipsoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' The second half of this section focuses on the engulfing property of sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' In Section 5, we prove some measure-theoretic lemmas which will be needed to estimate the “bad” set where the second derivative loses control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' These lemmas are all standard results for balls, but we have to adapt them to Sµ(x0) we constructed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' The engulfing property of Sµ(x0) is crucially used in establishing these lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' 6 JINGRUI CHENG, YULUN XU In Section 6, we verify that the “bad sets” fits in the assumptions of the measure theoretic lemmas in Section 5, and obtain the power decay of the measure of the bad set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Contrary to the real case, we first obtain control for the mixed Hessian ui¯j, then the full W 2,p estimate follows from the classical Lp estimate for Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' In Section 7, we discuss some implications of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' In particular, we give detailed proofs for Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='2 ,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='3 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' preliminaries The key result we will need again and again is the following lemma: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let u be a viscosity solution to det(ui¯j) = 1 in B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Suppose that ||u − w||L∞ ≤ δ, where w is a smooth solution to det(wi¯j) = 1 in B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' If δ is small enough depending only on the smoothness of w and n, we have ||u||C4(B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='99) ≤ C, where C has the same dependence as δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' The small perturbation theorem of Savin is for more general fully nonlinear elliptic equations, and applies to equations of the form F(D2u, x) = 0, where F(r, x) : S ×B1 → R is C2 in the x variable, elliptic in the r variable, and uniformly elliptic only for r in a neighborhood of 0, with F(0, x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then for any solution to F(D2u, x) = 0 with ||u||L∞ small enough, we would get C2,α estimate in the interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1 follows from the general perturbation theorem of Savin in [15] by writing the complex Monge-Ampere operator in the real form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' The details can be found in Yu Wang [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Also once we get C2,α estimate the above equation, it is straightforward to apply standard elliptic estimates to improve the estimate to C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' In this paper, we will mostly apply Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1 with w = |z|2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Another thing we will need is the following interpolation lemma: We need the following interpolation estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let u be a C4 function in Br0(0) with |D4u|L∞(Br0) ≤ C, and |u|L∞(Br0) ≤ µ, then for any 0 < λ < r0, one has |u(0)| ≤ µ, |Du(0)| ≤ C(λ3 + µ λ), |D2u(0)| ≤ C(λ2 + µ λ2 ), |D3u(0)| ≤ C(λ + µ λ3 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' From the Taylor expansion, we find that, for x ∈ Br0, the following estimate holds: |u(x) − � |α|≤3 Dαu(0) α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' xα| ≤ CnC|x|4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' If we restrict to |x| ≤ λ, we find that | � |α|≤3 Dαu(0) α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' xα| ≤ CnCλ4 + µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' This is equivalent to: sup |y|≤1 | � |α|≤3 Dαu(0) α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' λ|α|yα| ≤ CnCλ4 + µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' INTERIOR W 2,p ESTIMATE FOR SMALL PERTURBATIONS TO THE COMPLEX MONGE-AMPERE EQUATION7 Hence we would have: |Dmu(0)|λm ≤ C′ nCλ4 + µ, m = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' □ In the present paper, we will frequently use differentiation with respect to complex variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Following the usual conventions, we define: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1) ∂zi = 1 2(∂xi − √ −1∂yi), ∂¯zi = 1 2(∂xi + √ −1∂yi), zi = xi + √ −1yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' So that we find the Laplacian operator can be written as: ∆ = 4 n � i=1 ∂zi¯zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' A notion we will encounter again and again is pluriharmonic function, which we explain below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let Ω ⊂ Cn be a domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let h ∈ C2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' We say that h is plurihar- monic if hzi¯zj(z) = 0 for all 1 ≤ i, j ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Note that h being pluriharmonic will imply h being harmonic, but not the other way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' One can also see that if h is the real part of a holomorphic function, then h is pluriharmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Another definition we need is: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let T : Cn → Cn be a C-linear transformation, we define ||T|| to be the operator norm of T, namely: ||T|| = sup |z|≤1 |T(z)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' In the proof of engulfing property of sections, we will need to frequently consider dilation maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Hence we introduce the following definition to make the notations simpler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let E ⊂ Cn be a set and x0 ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' We will sometimes denote E to be E(x0) to indicate it is a “pointed set”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let c > 0, we define: cE(x0) = {x0 + c(y − x0) : y ∈ E(x0)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Namely cE(x0) is the image of the dilation map centered at x0 by a factor c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Reduction of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='4 to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='3 In this section, we will show how to use Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='3 to deduce Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' To see the implication in an intuitive way, we can take any point x0 ∈ B 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' After subtracting a pluriharmonic function hx0, we will see that {w − hx0(z) ≤ w0(x0) + µ} will be close to an ellipsoid (centered at x0) when µ is small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' The same would be true for {u − hx0(z) ≤ µ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' After normalization the ellipsoid to a unit ball, we are in the situation of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='3 and we get u ∈ W 2,p in a neighborhood of x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' We will make this idea precise in the rest of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' 8 JINGRUI CHENG, YULUN XU Let w0 be as in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Take any point x0 ∈ B 1 2, we can write down the Taylor expansion of w0 at x0: w0 = w0(x0) + Re( � i lx0,i(z − x0)i) + � i,j ax0,i¯j(z − x0)i(z − x0)j + Re( � i,j bx0,ij(z − x0)i(z − x0)j) + O(|z − x0|3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1) Define hx0(z) = Re(� i lx0,i(z − x0)i) + Re(� i,j bx0,ij(z − x0)i(z − x0)j), First we want to show that if we have another function u0, such that |u0 − w0| ≤ δ, then the section {z : u0 − hx0(z) ≤ u0(x0) + µ} will be close to an ellipsoid if µ is small, but much larger than δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' More precisely Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let w0 be as stated in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Namely we assume that w0 ∈ C3(B1), and 1 C0I ≤ (w0)i¯j ≤ C0I, |D3w0| ≤ C0 on B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let δ ≥ 0 and u0 is a function on B1 with |u0 − w0| ≤ δ on B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then there exists C1 > 0 large enough and µ0 > 0 small enough depending only on C0, such that for all µ with 4C1δ ≤ µ ≤ µ0, we have: (1 − C1γ)Eµ(x0) ⊂ {z ∈ B 1 2C2 0 (x0) : (u0 − hx0)(z) ≤ u0(x0) + µ} ⊂ (1 + C1γ)Eµ(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Moreover, (u0 − hx0)(z) = u0(x0) + µ on ∂{z ∈ B 1 2C2 0 (x0) : (u0 − hx0)(z) ≤ u0(x0) + µ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Here γ = δ µ + µ 1 2 and Eµ(x0) = {z ∈ Cn : �n i,j=1 ax0,ij(z − x0)i(z − x0)j ≤ µ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1), we see that on B1: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='2) − C0|z − x0|3 ≤ w0 − w0(x0) − hx0(z) − � i,j ax0,ij(z − x0)i(z − x0)j ≤ C0|z − x0|3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Since |u0 − w0| ≤ 2δ0 on B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='95, we see that for any x0 ∈ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='95: −2δ − C0|z − x0|3 ≤ u0 − u0(x0) − hx0(z) − � i,j ax0,ij(z − x0)i(z − x0)j ≤ C0|z − x0|3 + 2δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let z ∈ B 1 2C2 0 (x0) and (u0 − hx0)(z) ≤ u0(x0) + µ, we get −2δ − C0|z − x0|3 ≤ µ − � i,j ax0,ij(z − x0)i(z − x0)j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Since ax0,ij ≥ 1 C0 I, we get 1 C0 |z − x0|2 ≤ � i,j ax0,ij(z − x0)i(z − x0)j ≤ µ + 2δ + C0 · 1 2C2 0 |z − x0|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' So that |z − x0| ≤ � 2C0(µ + 2δ) ≤ � 3C0µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' So that � i,j ax0,ij(z − x0)i(z − x0)j ≤ u0 − u0(x0) − hx0(z) + 2δ + C0|z − x0|3 ≤ µ + 2δ + C0(3C0µ) 3 2 ≤ (1 + C1γ)µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' INTERIOR W 2,p ESTIMATE FOR SMALL PERTURBATIONS TO THE COMPLEX MONGE-AMPERE EQUATION9 This proves the inclusion {z ∈ B 1 2C2 0 (x0) : (u0 − hx0)(z) ≤ u0(x0) + µ} ⊂ (1 + C1γ)Eµ(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Now we prove the other inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let z ∈ (1 − C1γ)Eµ(x0), which implies � i,j ax0,ij(z − x0)i(z − x0)j ≤ (1 − C1γ)2µ ≤ (1 − C1γ)µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' So that |z − x0|2 ≤ C0 � i,j ax0,ij(z − x0)i(z − x0)j ≤ C0(1 − C1γ)µ < 1 2C2 0 , as long as µ ≤ µ0 with µ0 small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Moreover u0 − u0(x0) − hx0(z) ≤ � i,j ax0,ij(z − x0)i(z − x0)j + C0|z − x0|3 + 2δ ≤ (1 − C1γ)µ + C0(3C0µ) 3 2 + 2δ ≤ µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' The last inequality would hold if we take C1 to be large enough depending on C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' This proves the inclusion: (1 − C1γ)Eµ(x0) ⊂ {z ∈ B 1 2C2 0 (x0) : (u0 − hx0)(z) ≤ u0(x0) + µ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' □ Now we are ready to verify the implication from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='3 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let u and w0 be as stated in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let µ > 0 and x0 ∈ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let Tµ,x0 be a C-affine transformation such that Tµ,x0(B√µ(0)) = Eµ(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Define (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='3) uµ,x0(ζ) = 1 µ| det Tµ,x0| 2 n (u − hx0 − µ)(Tµ,x0(√µζ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Since Eµ(x0) is defined in terms of ax0,ij, with 1 C0 ≤ ax0,ij ≤ C0I, it is easy to see that: ||Tµ,x0|| ≤ C2, ||T −1 µ,x0|| ≤ C2, 1 C2 ≤ | det Tµ,x0|2 ≤ C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Here C2 is a large enough constant depending only on C0 and n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Define Ωµ = T −1 µ,x0({z ∈ B 1 2C2 0 : (u − hx0)(z) ≤ u(x0) + µ}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then by straightforward calculation and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1, we can see the following: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' There is µ0 > 0 small enough depending only on C0 such that for all 4C1δ0 ≤ µ ≤ µ0 (with C1 > 0 being the constant given by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1), we have (1) B1−C1γ ⊂ Ωµ ⊂ B1+C1γ, with γ = δ0 µ + µ 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' (2) det(uµ,x0)ζi ¯ζj = f(Tµ,x0(√µζ)) in Ωµ, uµ,x0 = 0 on ∂Ωµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' The renormalized function uµ,x0 fits in the assumptions for Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='3 after suitably choosing the parameters, and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='4 follows as a direct consequence: Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='4 holds, if we assume Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='3 and u ∈ C2(B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' 10 JINGRUI CHENG, YULUN XU Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' We wish to apply Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='3 to each uµ,x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' In order to do so, we just need: C1γ = C1(δ0 µ + µ 1 2 ) ≤ γ0(n), |f(Tµ,x0(√µζ)) − 1| ≤ ε(n, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Here γ0(n) and ε(n, p) are the constants given by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' So we could just take µ so that 2C1µ 1 2 ≤ 1 2γ0(n) and also µ ≤ µ0 (given by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' With this µ, we can take δ0 so that C1 δ0 µ ≤ 1 2γ0(n) and also that 4C1δ0 ≤ µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' We fix this choice from now on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Since we assumed that Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='3 holds, we conclude that: ||uµ,x0||W 2,p(B 1 2) ≤ C, where C is a constant depending only on n and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='3) we may go back to u and obtain that ||u||W 2,p(E 1 2 µ(x0)) ≤ C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Here C′ depends on C0, p and n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Note that µ is already chosen which depends only on C0 and n, and hx0 is defined using w0, which can be bounded in terms of C0 and n as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Note that E 1 2µ(x0) contains Br0(x0) for some r0 > 0 small enough (depending only on C0) for any x0 ∈ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' The result of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='4 would follow right away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' □ Next we can use an approximation argument to remove the assumption that u ∈ C2(B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' (of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='4, without assuming u ∈ C2(B1)) First, we can find fk ∈ C∞(B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='9) such that fk → f in L2(B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='9) and |fk − 1| ≤ ε (since |f − 1| ≤ ε, one can see that the standard smoothing by convolution will preserve this property).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' We can also find a sequence of gk ∈ C∞(∂B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='9), such that gk → u uniformly on ∂B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='9 (since u is assumed to be continuous).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let vk be the solution to the Dirichlet problem: det(vk)i¯j = fk in B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='9, vk = gk on ∂B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' From Caffarelli-Kohn-Nirenberg-Spruck [5], we know that vk ∈ C∞( ¯B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Also from the following Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='4, we know that vk → u uniformly on ¯B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Hence for large enough k, vk will fullfil the assumption of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='4, and each vk is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Hence we may use Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='3 to conclude that ||vk||W 2,p(B 1 2 ) ≤ C, where C depends only on C0, n and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' In particular, C is uniform in k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Passing to the limit, we see that u ∈ W 2,p(B 1 2), with the same bound C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' □ In the above, we used the following stability estimate from Dinew-Kolodziej [7] to deduce the uniform convergence of the approximation sequence: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' ([7]) Let ωE be the Euclidean K¨ahler form on Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let q > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Consider u, v ∈ PSH(Ω) ∩ C(¯Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Assume that MA(u) = f, MA(v) = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' INTERIOR W 2,p ESTIMATE FOR SMALL PERTURBATIONS TO THE COMPLEX MONGE-AMPERE EQUATION 11 for some f, g ∈ Lq(Ω, dV ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then sup Ω (v − u) ≤ sup ∂Ω (v − u) + c(q, n, diam(Ω))||f − g|| 1 n Lq(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' construction of sections From now on we will focus on the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Our first step is to construct sections of u which are close to ellipsoids via an induction process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Next, we prove some fine properties of the sections which ensures that they are good differentiation basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Inductive construction of sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let us summarize our construction into the following proposition: Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let Ω and u be as stated in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='3, with γ0 small enough depending only on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let 0 < σ < 1 be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then there exists ε > 0 depending only on σ and n, such that if |f − 1| ≤ ε, the following hold: (1) There exists µ0 > 0 small enough depending only on n and σ, such that for all x0 ∈ B0,8 and all µ ≤ µ0, there exists a degree 2 pluriharmonic polynomial hµ,x0(z) with hµ,x0(x0) = 0, such that (1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)Eµ(x0) ⊂ Sµ(x0) := {z ∈ Ω : (u − hµ,x0)(z) ≤ u(x0) + µ} ⊂ (1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)Eµ(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' In the above, Eµ(x0) = {z ∈ Cn : �n i,j=1 aµ,x0,ij(z − x0)i(z − x0)j ≤ µ}, with aµ,x0,ij being positive Hermitian and det aµ,x0,ij = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' (2) There is a function c(σ) : σ ∈ (0, 1) → R>0, such that for any x0 ∈ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8 and any 0 < µ1 ≤ µ2 ≤ µ0 1+c(σ), one has Sµ1(x0) ⊂ S(1+c(σ))µ2(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Moreover, 0 < c(σ) ≤ C2,nσ 1 2 for some dimensional constant C2,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' (3) There is a dimensional constant C3,n > 0 such that for all 0 < µ ≤ µ0 and any x0 ∈ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8, there exists a C-linear transformation Tµ,x0, such that | det Tµ,x0| = 1, Tµ,x0(B√µ(0)) = Eµ(x0), Tµ0,x0 = id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Moreover, for any 0 < µ1 < µ2 ≤ µ0 and any x0 ∈ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8: ||Tµ1,x0 ◦ T −1 µ2,x0|| ≤ C3,n(µ2 µ1 ) C3,nσ 1 2 − log(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ) , ||Tµ2,x0 ◦ T −1 µ1,x0|| ≤ C3,n(µ2 µ1 ) C3,nσ 1 2 − log(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' We will make a choice of σ later on, depending on the value of p in the W 2,p estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' (The larger p is, the smaller σ needs to be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=') So that the choice of ε eventually depends only on p and n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' We fix some 0 < σ < 1, and describe the construction of Sµ(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' First we solve the following Dirichlet problem on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' det((v0)i¯j) = 1 in Ω v0 = 0 on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1) To start the process, we need that v0 is smooth in the interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' This is guaranteed by the fact that Ω is close to B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' More precisely, we have: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let Ω ⊂ Cn be a bounded domain and B1−γ(0) ⊂ Ω ⊂ B1+γ(0) for some 0 < γ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let v0 be the solution to the Dirichlet problem in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1), then |z|2 − 1 − 3γ ≤ v0 ≤ |z|2 − 1 + 3γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' 12 JINGRUI CHENG, YULUN XU Moreover, there exists γn > 0 small enough, such that if γ ≤ γn, we have v0 ∈ C4( ¯B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='9) with ||v0 − (|z|2 − 1)||C4,B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='9 ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Here C depends only on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' From the assumption, we see that 1 − γ ≤ |z| ≤ 1 + γ on ∂Ω, hence |z|2 − 1 − 3γ ≤ v0 ≤ |z|2 − 1 + 3γ, on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Note that both |z|2 − 1 − 3γ and |z|2 − 1 + 3γ satisfy det ui¯j = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Hence from maximum principle, we see that |z|2 − 1 − 3γ ≤ v0 ≤ |z|2 − 1 + 3γ in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' If γ is small enough, then we may use Savin’s estimate (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1) to see that v0 is bounded in C2,α on B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='95 by a dimensional constant C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then one can differentiate the det(v0)i¯j = 1 and use classical elliptic estimates to conclude that |v0|C4,B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='9 ≤ C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' □ As a consequence, we get that v0 is actually C3 close to |z|2 − 1, hence convex if γ is small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' More precisely: Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let v0 be as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then for any 0 < γ ≤ γn with γn small enough, we have: |Dm(v0 − (|z|2 − 1))|B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='9 ≤ Cnγ1− m 4 , m = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' This follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='3 and the interpolation estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' □ In order to define sections for u, we need to show that u and v0 are sufficiently close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' This is guaranteed by the following lemma: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Assume that det ui¯j = f in Ω and u|∂Ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let v0 be the solution to the Dirichlet problem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Assume that 1− ε ≤ f ≤ 1+ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then we have (1+ ε) 1 n v0 ≤ u ≤ (1 − ε) 1 n v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' In particular |v0 − u| ≤ 4ε in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Since det � (1 + ε) 1 n (v0)i¯j � = 1 + ε ≥ f = det ui¯j ≥ 1 − ε = det � (1 − ε) 1 n (v0)i¯j � and those three functions all have the same boundary value, we can use the maximum principle to conclude that (1 + ε) 1 n v0 ≤ u ≤ (1 − ε) 1 n v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' So that u − v0 ≤ ((1 − ε) 1 n − 1)v0 ≤ 2 nε|v0| ≤ 4 nε ≤ 4ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' In the above, we used Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='3 that |v0| ≤ 2 (if γ is small enough).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' The lower estimate for u − v0 is completely similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' □ To define sections for the first step, we need an analogue of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let v ∈ C3(B1) and |vi¯j − δij| ≤ c, |D3v| ≤ c on B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='9 where 0 < c < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let δ ≥ 0 and u0 is a function on B1 with |u0 − v| ≤ δ on B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then for small enough c > 0 (depending only on n) and for any x0 ∈ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8, there is a degree 2 pluriharmonic polynomail hv,x0(z), such that for all µ with 4δ ≤ µ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='9(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='9 − |x0|)2 we have: (1 − γ)Eµ(x0) ⊂ {z ∈ B1 : (u0 − hv,x0)(z) ≤ u0(x0) + µ} ⊂ (1 + γ)Eµ(x0), and Eµ(x0) ⊂ B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Here γ = 2δ µ + (3c) 3 2 µ 1 2 , and Eµ(x0) = {z : � i,j vi¯j(x0)(z − x0)i(z − x0)j ≤ µ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' INTERIOR W 2,p ESTIMATE FOR SMALL PERTURBATIONS TO THE COMPLEX MONGE-AMPERE EQUATION 13 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' The proof is very similar to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' First we can write down the Taylor expansion of v at x0: v(z) = v0(x0) + Re � � i l0,x0,i(z − x0)i � + � i,j vi¯j(x0)(z − x0)i(z − x0)j + Re � � i,j b0,x0,ij(z − x0)i(z − x0)j � + O(|z − x0|3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Define hv,x0(z) = Re � � i l0,x0,i(z − x0)i + � i,j b0,x0,ij(z − x0)i(z − x0)j � , and use the bound for D3v, u0 − v, we get −2δ−c|z−x0|3 ≤ (u0 −hv,x0)(z)−u0(x0)− � i,j vi¯j(x0)(z−x0)i(z − x0)j ≤ c|z−x0|3 +2δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let z ∈ B1 with (u0 − hv,x0)(z) ≤ u0(x0) + µ, we get � i,j vi¯j(x0)(z − x0)i(z − x0)j ≤ µ + 2δ + c|z − x0|3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' By choosing c small, we may assume that vi¯j(x0) ≥ 1 2I, so that 1 2|z − x0|2 ≤ µ + 2δ + 2c|z − x0|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Hence if 4c < 1 2, we get |z − x0|2 ≤ 2(µ + 2δ) ≤ 3µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Hence � i,j vi¯j(x0)(z − x0)i(z − x0)j ≤ µ(1 + 2δ µ + 3 3 2 cµ 1 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' This proves the inclusion {z ∈ B1 : (u0 − hv,x0)(z) ≤ u0(x0) + µ} ⊂ (1 + C1γ)Eµ(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Next we show that Eµ(x0) ⊂ B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' From |vi¯j −δij| ≤ c, we see that Eµ(x0) ⊂ B( µ 1−c ) 1 2 (x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Hence we just need to make sure ( µ 1−c) 1 2 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='9 − |x0|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' If c is chosen small enough, this is indeed true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Now we prove the inclusion that (1 − γ)Eµ(x0) ⊂ {z ∈ B1 : (u0 − hv,x0)(z) ≤ u0(x0) + µ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Assume that z ∈ (1 − γ)Eµ(x0), so that � i,j vi¯j(x0)(z − x0)i(z − x0)j ≤ (1 − γ)2µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Hence |z − x0| ≤ √2µ, using vi¯j(x0) ≥ 1 2I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Therefore (u0 − hv,x0)(z) − u0(x0) ≤ � i,j vi¯j(x0)(z − x0)i(z − x0)j + c|z − x0|3 + 2δ ≤ (1 − γ)µ + (2µ) 3 2 + 2δ ≤ µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' □ 14 JINGRUI CHENG, YULUN XU Now we choose µ0 > 0 so that: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='2) 3 3 2µ 1 2 0 = min( 1 20σ, 1 20γn), µ0 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='9 · 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' where γn is the constant given by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' There is no loss of generality to assume that σ ≤ γn and we will assume this throughout this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' We can apply Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='6 to v0 and u and construct sections Sµ(x0) for all µ2 0 < µ ≤ µ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let u and Ω be as stated in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Assume that γn is small enough, µ0 is chosen according to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='2), and 16ε ≤ µ2 0, 8ε µ0 ≤ 1 20σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' For µ2 0 < µ ≤ µ0 and x0 ∈ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8 we define: Sµ(x0) = {z ∈ B1 : (u − hv0,x0)(z) ≤ u0(x0) + µ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then we have (1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)Eµ(x0) ⊂ Sµ(x0) ⊂ (1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)Eµ(x0), for any µ2 0 < µ ≤ µ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' This follows directly from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='6, by choosing u0 to be u, v to be v0 in that lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Choose δ = 4ε, and assume that γn stated in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='3 is small enough so as to make |(v0)i¯j − δij| and |D3v0| small enough on B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' After these choice, we may use Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='6 to conclude that for and x0 ∈ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8 and 4 · 4ε ≤ µ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='9 · 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='12, we can conclude: (1 − γ)Eµ(x0) ⊂ Sµ(x0) ⊂ (1 + γ)Eµ(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Note that the range 16ε ≤ µ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='9 · 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='12 contains the range 1 2µ0 ≤ µ ≤ µ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Moreover, from the assumptions on the parameters, we get γ ≤ 2 · 4ε µ0 + 3 3 2µ 1 2 0 ≤ σ 20 + σ 20 ≤ σ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' □ Now we need to define Sµ(x0) for µ ≤ µ2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let ˜T1,x0 be a C-affine map such that ˜T1,x0(B√µ0(0)) = Eµ0(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' We hope to estimate how far ˜T1,x0 is away from identity map, in terms of how the ellipsoid Eµ0(x0) is close to a ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' For that we need the following lemma: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let E ⊂ Cn be an ellipsoid, given by: E = {z : n � i,j=1 ai¯j(z − x0)i(z − x0)j ≤ r2}, with ai¯j being positive Hermitian matrix, det ai¯j = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then there is a C-affine transform T such that T(Br(0)) = E, det T = 1, and ||T − I|| ≤ max1≤i≤n |λ − 1 2 i − 1|, ||T −1 − 1|| ≤ max1≤i≤n |λ 1 2 i − 1|, where λi are eigenvalues of ai¯j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' First we consider when ai¯j is diagnal, so that E = {z : �n i=1 λi|zi|2 ≤ r2}, with Πiλi = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then we define T(w) = x0 + ( w1 √λ1 , · · · , wn √λn ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' In the general case, we can take a unitary transformation U, so that E becomes diagnal under the new coordinate, then the desired C-affine map is given by T = U −1T ′U, where T ′ is the dilation map along coordinate axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Hence the result would follow from the diagnal case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' □ As a consequence, we see that: INTERIOR W 2,p ESTIMATE FOR SMALL PERTURBATIONS TO THE COMPLEX MONGE-AMPERE EQUATION 15 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let u and Ω be as stated in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Assume that γ is small enough depending on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let ˜T1,x0 be the C-affine map given by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8, applied to Eµ0(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then for some C′ n > 0, we have || ˜T1,x0 − I|| ≤ C′ nγ 1 2, || ˜T −1 1,x0 − I|| ≤ C′ nγ 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Note that Eµ(x0) = {z : � i,j(v0)i¯j(x0)(z − x0)i(z − x0)j ≤ µ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' It follows from Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='4 that |(v0)i¯j −δij| ≤ Cnγ 1 2 , for all γ small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' So that 1−Cnγ 1 2 ≤ λi ≤ 1 + Cnγ 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Therefore |λ 1 2 i − 1| ≤ |(1 + Cnγ 1 2) 1 2 − 1| ≤ C′ nγ 1 2 , which implies that || ˜T −1 1,x0 − I|| ≤ C′ nγ 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' The estimate for || ˜T1,x0 − I|| is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' □ To define Sµ(x0) for µ < µ2 0, we need to rescale the ellipsoid Eµ0(x0) to be a unit ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Define the change of coordinate: z(1) = 1 √µ0 ˜T −1 1,x0(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then 1 √µ0 ˜T −1 1,x0(Eµ0(x0)) = B1(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Define Ωx0,1 = 1 √µ0 ˜T −1 1,x0(Sµ0(x0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' According to Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='7, we have that: B1−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ(0) ⊂ Ωx0,1 ⊂ B1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ(0), Ωx0,1 is pseudoconvex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Define vx0,1 be the solution to the following Dirichlet problem: det(vx0,1)i¯j = 1, in Ωx0,1, vx0,1 = 0 on ∂Ωx0,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='3) We can normalize u on Ωx0,1 to be: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='4) ux0,1 = 1 µ0 � u − u(x0) − hv0,x0 − µ0 � ( ˜T1,x0(√µ0z(1))), z(1) ∈ Ωx0,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then det(ux0,1)i¯j = fx0,1 in Ωx0,1, fx0,1(z(1)) = f( ˜T1,x0(√µ0z(1))), ux0,1 = 0 on ∂Ωx0,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' We observe that the following holds for ux0,1 and vx0,1: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let vx0,1 and ux0,1 be defined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='3) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='4) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Assume that σ ≤ γn, where γn is given by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let µ0 be defined by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then the following hold: (1) vx0,1 ∈ C4(B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='95), and |Dm(vx0,1 − (|z(1)|2 − 1))|B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='9 ≤ Cnσ1− m 4 , m = 1, 2, 3, where Cn is the same Cn in Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' (2) |vx0,1 − ux0,1| ≤ 4ε in Ωx0,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' (3) Define ˜hx0,1(z(1)) = Re � � i (vx0,1)i(0)z(1) i + � i,j (vx0,1)ij(0)z(1) i z(1) j � , 16 JINGRUI CHENG, YULUN XU then for µ2 0 ≤ µ ≤ µ0, we have: (1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ) ˜Eµ(0) ⊂ {z(1) ∈ B1 : (ux0,1 − ˜hx0,1 − ux0,1(0))(z(1)) ≤ µ} ⊂ (1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ) ˜Eµ(0), where ˜Eµ(0) = {z(1) : � i,j(vx0,1)i¯j(0)z(1) i z(1) ¯j ≤ µ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' (4) Let ˜T2,x0 be the C-affine transform given by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8 normalizing ˜Eµ0(x0), then one has: || ˜T2,x0 − I|| ≤ C′ nσ 1 2 , || ˜T −1 2,x0 − I|| ≤ C′ nσ 1 2, where C′ n is the constant given by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' To prove (1), we just note that since vx0,1 solves (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='3), with B1−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ ⊂ Ωx0,1 ⊂ B1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ, and σ ≤ γn, then Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='3 and Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='4 can be applied to show that (1) holds, with γ there replaced by σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Item (2) above follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='5, since ux0,1 solves det(ux0,1)i¯j = fx0,1 with |fx0,1 − 1| ≤ ε (since f satisfies the same).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Item (3) essentially follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='6, applied to v = vx0,1, u = ux0,1, c = 1, δ = 4ε, x0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Because of our choice of µ0 and ε, we would have γ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Also the range for µ in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='6 is 4 · 4ε ≤ µ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='93 (with x0 = 0), which contains the range µ2 0 ≤ µ ≤ µ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' The proof of item (4) follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='9, because of item (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' □ We define ˜S1,µ(0) = {z(1) ∈ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='9 ⊂ Ωx0,1 : (ux0,1 − ˜hx0,1 − ux0,1(0))(z(1)) ≤ µ}, µ2 0 < µ ≤ µ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' We can now define Sµ(x0) for µ3 0 < µ ≤ µ2 0 by transforming back to z variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' In other words, for µ3 0 < µ ≤ µ2 0, we define Sµ(x0) = ˜T1,x0 �√µ0 ˜S1,µ−1 0 µ(0) � = {z ∈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='9Eµ0(x0) ⊂ Sµ0(x0) : u(z) − hv0,x0(z) − µ0˜hx0,1( 1 √µ0 ˜T −1 1,x0(z)) ≤ u(x0) + µ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Next we use an induction process to define Sµ(x0), for all 0 < µ ≤ µ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Assume that for some k0 ≥ 2, we have defined Sµ(x0) for all µk0 0 < µ ≤ µ0, we wish to define Sµ(x0) for µk0+1 0 < µ ≤ µk0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' We make the following induction hypothesis, stated with k0: (1) Sµ(x0) = {z ∈ Sµk−2 0 (x0) : u(z) − hx0,k−2(z) ≤ u(x0) + µ}, for all µk 0 < µ ≤ µk−1 0 , 2 ≤ k ≤ k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Here hx0,k−2 is a pluriharmonic polynomial of degree 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' (2) There exists a family of ellipsoids Eµ(x0), centered at x0, such that (1−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)Eµ(x0) ⊂ Sµ(x0) ⊂ (1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)Eµ(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Moreover, Eµ(x0) = {z : � i,j ai¯j,k−2(z − x0)i(z − x0)j ≤ µ}, for all µk 0 < µ ≤ µk−1 0 , 2 ≤ k ≤ k0, and det ai¯j,k−2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' (3) There exists a sequence of C-affine coordinate change: z(k) = 1 √µ0 ˜T −1 k,x0(z(k−1)) for k ≥ 2, z(1) = 1 √µ0 ˜T −1 x0,1(z − x0) with det ˜Tk,x0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' ˜Tk,x0 maps B√µ0(0) to be the image of Eµk 0(x0) under z(k−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Moreover || ˜Tk,x0−I|| ≤ C′ nσ 1 2 , || ˜T −1 k,x0−I|| ≤ C′ nσ 1 2 for all 2 ≤ k ≤ k0 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' INTERIOR W 2,p ESTIMATE FOR SMALL PERTURBATIONS TO THE COMPLEX MONGE-AMPERE EQUATION 17 (4) Denote Ωx0,k to be the image of Sµk 0(x0) under the coordinate z(k), then we have B1−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ ⊂ Ωx0,k ⊂ B1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ, for all 1 ≤ k ≤ k0 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' (5) The image of Eµ(x0) under coordinate z(k−1) is B� µ1−k 0 µ(0), for µk 0 < µ ≤ µk−1 0 , 2 ≤ k ≤ k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' First we observe that the above inductive hypothesis indeed hold for k0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Indeed, the items (1) and (3) follow from Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Item (2) follows from item (3) of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Now we will construct Sµ(x0) for µk+1 0 ≤ µ < µk0 0 , and verify that the above inductive hypothesis continues to hold with k0 replaced by k0 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' That is, we prove: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Assume that above induction hypothesis holds with some k0 ≥ 2, then we can construct Sµ(x0) for µk0+1 0 < µ ≤ µk0 0 which satisfies the induction hypothesis with k0 replaced by k0+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' In particular, the induction hypothesis holds for all k0 ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' We solve the following Dirichlet problem on Ωx0,k0−1: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='5) det(vx0,k0−1)i¯j = 1, in Ωx0,k0−1, vx0,k0−1 = 0 on ∂Ωx0,k0−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Define Tk,x0 = ˜T1,x0 ◦ ˜T2,x0 · · · ˜Tk,x0 so that the change of coordinate between z(k) and z is given by z = x0 + Tk,x0(µ k 2 0 z(k)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='6) ux0,k0−1(z(k0−1)) = 1 µk0−1 0 (u − hx0,k0−2 − u(x0) − µk0−1 0 )(x0 + Tk0−1,x0(µ k0−1 2 0 z(k0−1))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then ux0,k0−1 solves: det(ux0,k0−1)i¯j = fx0,k0−1, in Ωx0,k0−1, fx0,k0−1 = f(x0 + Tk0−1,x0(µ k0−1 2 0 z(k0−1))), ux0,k0−1 = 0, on ∂Ωx0,k0−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Using the same argument as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='10, (1), we have that vx0,k0−1 ∈ C4(B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='95), and |Dm(vx0,k0−1 − |z(k0−1)|2 − 1)|B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='9 ≤ Cnσ1− m 4 for m = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' This follows from our inductive hypothesis that B1−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ ⊂ Ωx0,k0−1 ⊂ B1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ, and an application of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='3 and Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Also we would have |vx0,k0−1 − ux0,k0−1| ≤ 4ε on Ωx0,k0−1, following the same argument as Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then we may consider the Taylor expansion of vx0,k0−1 at z(k0−1) = 0: vx0,k0−1(z(k0−1)) = vx0,k0−1(0) + Re � � i liz(k0−1) i � + � i,j ai¯jz(k0−1) i ¯z(k0−1) j + Re � � i,j bijz(k0−1) i z(k0−1) j � + O(|z(k0−1)|3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Define (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='7) ˜hx0,k0−1(z(k0−1)) = Re( � i liz(k0−1) i ) + Re � � i,j bijz(k0−1) i z(k0−1) j � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' 18 JINGRUI CHENG, YULUN XU Then the argument for part (3) of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='10 shows that: (1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ) ˜Eµ(0) ⊂ ˜Sk0−1,µ := {z(k0−1) ∈ Ωx0,k0−1 : (ux0,k0−1 − ˜hx0,k0−1 − ux0,k0−1(0))(z(k0−1)) ≤ µ} ⊂ (1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ) ˜Eµ(0), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8) for any µ2 0 < µ ≤ µ0, where ˜Eµ(0) = {z(k0−1) : � i,j(vx0,k0−1)i¯j(0)z(k0−1) i ¯z(k0−1) j ≤ µ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Now we define ˜Tk0,x0 be the C-linear transforma given by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8 normalizing ˜Eµ0(0) above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then for µk0+1 0 < µ ≤ µk0 0 , we define Sµ(x0) = x0 + ˜Tk0,x0(µ k0−1 2 0 ˜Sk0−1,µ−(k0−1) 0 µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Similarly, we define Eµ(x0) = x0 + ˜Tk0,x0(µ k0−1 2 0 ˜Eµ−(k0−1) 0 µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='6), one find that, for µk0+1 0 ≤ µ < µk0 0 : Sµ(x0) = {z ∈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='9Eµk0−1 0 (x0) : u(z)−hx0,k0−2(z)−µk0−1 0 ˜hx0,k0−1(µ − k0−1 2 0 T −1 k0−1,x0(z−x0)) ≤ µ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Define hx0,k0−1(z) = hx0,k0−2(z) + µk0−1 0 ˜hx0,k0−1(µ − k0−1 2 0 T −1 k0−1,x0(z − x0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' In view of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='7), as well as the induction hypothesis for hx0,k0−2(z), we see that hx0,k0−1(z) is a pluriharmonic polynomial of degree 2, and hx0,k0−1(x0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' This proves the induc- tion hypothesis, part (1), for k = k0 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Part (2) simply follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8) and the fact that det(vx0,k0−1)i¯j(0) = 1, as well as det Tk,x0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Since |(vx0,k0−1)i¯j(0)−δij| ≤ Cnσ 1 2 , the argument in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='10, part (4) shows that || ˜Tk0,x0 − I|| ≤ C′ nσ 1 2 , || ˜T −1 k0,x0 − I|| ≤ C′ nσ 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Using ˜Tk0,x0, we can define a change of coordinates z(k0) = 1 √µ0 ˜T −1 k0,x0(z(k0−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' This proves part (2) of the Inductive hypothesis with k = k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Part (3) follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8) and our choice of ˜Tk0,x0 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' □ Now we constructed Sµ(x0) for 0 < µ ≤ µ0, let us verify Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Part (1) has already been proved by the above argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Indeed, we define hµ,x0(z) = hx0,k−1(z) for µk 0 < µ ≤ µk−1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Indeed, part (1) of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1 follows from part (1) and (2) of induction hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Now we verify part (2) of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' First, we make the following observation out of the above inductive process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let 0 < µ1 < µ2, then the following hold: (1) If there is some k ≥ 2 such that µk 0 < µ1 < µ2 ≤ µk−1 0 , then Sµ1(x0) ⊂ Sµ2(x0), Eµ1(x0) ⊂ Eµ2(x0), (2) If there is some k ≥ 1 such that 1 2µk 0 < µ2 and µ1 ≤ µk+1 0 , then Sµ1(x0) ⊂ Sµ2(x0), Eµ1(x0) ⊂ Eµ2(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' INTERIOR W 2,p ESTIMATE FOR SMALL PERTURBATIONS TO THE COMPLEX MONGE-AMPERE EQUATION 19 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Part (1) above is obvious, due to part (1) and (2) of induction hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Part (2) requires more work, due to that Sµ(x0) and Eµ(x0) are discontinuous in µ for µ → µk 0+ and µ → µk 0−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' First, without loss of generality we may assume that µk+2 0 < µ1 ≤ µk+1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' If µk 0 < µ2 ≤ µk−1 0 (with k ≥ 2 in this case), we know from part (5) of induction hypothesis that under z(k−1), Eµ2(x0) is given by B� µ1−k 0 µ2(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' On the other hand, the image of Eµ1(x0) under z(k+1) is given by B� µ−1−k 0 µ1(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Hence if we recall the transition formula given by part (3) of induction hypothesis, we see that: Eµ1(x0) under z(k−1) = µ0 ˜Tk+1,x0 ◦ ˜Tk,x0(B� µ−1−k 0 µ1(0)) ⊂ (1 + C′ nσ 1 2 )2B� µ1−k 0 µ1(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='9) In the first inclusion above, we used that || ˜Ti,x0|| ≤ 1 + C′ nσ 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Therefore image of Sµ1(x1) under z(k−1) ⊂ B (1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)(1+C′nσ 1 2 )2 � µ1−k 0 µ1(0) On the other hand, image of Sµ2(x2) under z(k−1) ⊃ (1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)B� µ2µ1−k 0 (0) We will be able to show Sµ1(x0) ⊂ Sµ2(x0) if we can ensure: (1 + C′ nσ 1 2 )4(1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)2µ1 ≤ (1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)2µ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' This can be guaranteed if we take µ0 small enough so that (1 + C′ nσ 1 2)4 (1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)2 (1−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)2 ≤ µ−1 0 , since µ2 ≥ µ0µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' The other case is when 1 2µk 0 < µ2 ≤ µk 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' The calculation in this case is similar to the case when µk 0 < µ2 ≤ µk−1 0 , except that we need to use the coordinate z(k), and we may conclude: image of Sµ2(x0) under z(k) ⊃ (1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)B� µ2µ−k 0 (0), On the other hand, image of Sµ1(x0) under z(k+1) ⊂ (1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)B� µ1µ−1−k 0 (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Hence, using the transition between z(k) and z(k+1): image of Sµ1(x0) under z(k) ⊂ √µ0(1 + C′ nσ 1 2)(1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)B� µ1µ−1−k 0 (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' We will have the inclusion as long as we can make sure: (1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)2µ2 ≥ (1 + C′ nσ 1 2)2(1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)2µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' We will still have this since µ2 ≥ 1 2µ1µ0 and we can take µ0 small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' □ With the help of the previous lemma, we are ready to prove the almost monotonicity of sections claimed in part (2): 20 JINGRUI CHENG, YULUN XU Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let c(σ) = (1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)2 (1−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)2 (1 + C′ nσ 1 2)2 − 1 with C′ n given by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then for all 0 < µ1 ≤ µ2 ≤ µ0 1+c(σ) and any x0 ∈ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8, Sµ1(x0) ⊂ S(1+c(σ))µ2(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Denote c(σ) = (1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)2 (1−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)2 (1 + C′ nσ 1 2)2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let k ≥ 1 be such that µk+1 0 < µ1 ≤ µk 0 and µ2 > µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' There are several cases to consider: Case 1: µk+1 0 < µ1 < µ2 ≤ µk 0 1+c(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='12, we know that Sµ1(x0) ⊂ S(1+c(σ))µ2(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Case 2: µk 0 1+c(σ) < µ2 ≤ µk−1 0 1+c(σ) (k ≥ 2 for this case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' First, under the coordinate z(k−1), we have the following inclusions: image of S(1+c(σ))µ2(x0) under z(k−1) ⊃ B (1−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ) � (1+c(σ))µ2µ1−k 0 (0) = B (1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)(1+C′nσ 1 2 ) � µ2µ1−k 0 (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='10) On the other hand, if we consider the image of Sµ1(x0) under z(k), we have image of Sµ1(x0) under z(k) ⊂ B (1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ) � µ1µ−k 0 (0) Then we use the transition between z(k) and z(k−1): z(k) = 1 √µ0 ˜Tk,x0(z(k−1)) to get: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='11) Image of Sµ1(x0) under z(k−1) ⊂ √µ0 ˜T −1 k,x0(B (1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ) � µ1µ−k 0 (0)) ⊂ B (1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)(1+C′nσ 1 2 ) � µ1µ1−k 0 (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Combining (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='10) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='11), we get that: Sµ1(x0) ⊂ S(1+c(σ))µ2(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Case 3: µ2 ≥ µk−1 0 1+c(σ) (k ≥ 2 for this case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Without loss of generality, we may assume σ small enough so that 1 + c(σ) < 2, then the conclusion would follow from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Since µ1 ≤ µk 0 but µ2 ≥ 1 2µk−1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' □ Now we verify part (3) of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Define Tµ,x0 = Tk,x0 := ˜T1,x0 ◦ ˜T2,x0 · · · ˜Tk,x0 for µk+1 0 < µ ≤ µk 0, k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then for 0 < µ1 < µ2 ≤ µ0, we have: ||T −1 µ1,x0 ◦ Tµ2,x0||, ||T −1 µ2,x0 ◦ Tµ1,x0|| ≤ C3,n �µ2 µ1 � C3,nσ 1 2 log(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Here C3,n is some dimensional constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' First we find k1 ≥ k2 such that µk1+1 0 < µ1 ≤ µk1 0 , µk2+1 0 < µ2 ≤ µk2 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then we have: T −1 µ1,x0 ◦Tµ2,x0 = ˜T −1 k1,x0 ◦ ˜T −1 k1−1,x0 · · · ˜T −1 k2+1,x0, T −1 µ2,x0 ◦Tµ1,x0 = ˜Tk2+1,x0 ◦ ˜Tk2+1,x0 · · · ˜Tk1,x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then we may use part (3) of induction hypothesis that: ||T −1 µ1,x0 ◦ Tµ2,x0|| ≤ Πk1−1 k=k2||T −1 k,x0|| ≤ (1 + C′ nσ 1 2)k1−k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' INTERIOR W 2,p ESTIMATE FOR SMALL PERTURBATIONS TO THE COMPLEX MONGE-AMPERE EQUATION 21 On the other hand, we easily have the bound µk2−k1+1 0 ≤ µ2 µ1 , hence ||T −1 µ1,x0 ◦ Tµ2,x0|| ≤ (1 + C′ nσ 1 2 )− log(µ2µ−1 1 ) − log(µ0) +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Recall our choice of µ0 made in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='2): 3 3 2 µ 1 2 0 = σ 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then the claimed estimate follows easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' The same computation works also for T −1 µ2,x0 ◦ Tµ1,x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' □ As a direct consequence, the diameter of Sµ(x0) should go to zero as µ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Namely: Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Assume that σ is small enough (depending on n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then the diameter of Sµ(x0) goes to zero as µ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' This convergence is uniform for x0 ∈ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Since Tµ,x0 = id for µ2 0 < µ ≤ µ0, we see that, for any µ ≤ µ2 0, ||T −1 µ,x0||, ||Tµ,x0|| ≤ C3,n( µ µ0 ) C3,nσ 1 2 log(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' On the other hand Tµ,x0(B√µ(0)) = Eµ(x0) − x0, we see that diam Eµ(x0) ≤ 2µ 1 2 ||Tµ,x0|| ≤ 2C3,nµ 1 2 · ( µ µ0 ) C3,nσ 1 2 log(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' If σ is small enough so that 1 2 + C3,nσ 1 2 log(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ) > 0, then the right hand side will go to zero as µ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' The result follows from that Sµ(x0) ⊂ (1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)Eµ(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Further properties of sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Our intention will be to use the sections given by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1 to replace the role of balls in the uniformly elliptic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' The most crucial property that we need is the following “engulfing property” of sections, formulated below: Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Assume that x1, x2 ∈ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8, 0 < µ1, µ2 ≤ µ0 and µ1 ≤ 4µ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let σ > 0 be small enough (depending only on dimension).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Assume also that Sµ1(x1) ∩ Sµ2(x2) ̸= ∅, then Sµ1(x1) ⊂ 10Sµ2(x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' In Caffarelli’s proof for W 2,p estimate in the real case, we also need this “engulfing property”, but this property is not a problem in the real case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' The essential point is the “invariance of sections under linear transformations”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' To be more clear, in the real case, the sections are simply defined as: (with u being strictly convex function) Sµ(x0) = {x : u(x) ≤ u(x0) + ∇u(x0) · (x − x0) + µ}, µ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Now we define v(y) = 1 r2 u(x′ 0 + rTy) where T is linear with det T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Such a transformation would preserve the Monge-Ampere equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Under the change of co- ordinates x = x′ 0 + rTy, Sµ(x0) will be transformed to a section of v centered at y0 := T −1(1 r(x0 − x′ 0)), with height µ r2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' We no longer have this property in the complex case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Indeed, if you do a similar change of coordinates (now with T being C-linear), and you do the same construction described in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1 for the function v in the variable y, then transform back to x, you will get different sections than the direct construction in the original x coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' The two definitions will differ by an addition of a pluriharmonic function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' What saves us is the following “uniqueness” property, which shows that an addition of a pluriharmonic function will not affect the sections we get, as long as they are close to ellipsoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' 22 JINGRUI CHENG, YULUN XU Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let u be a function defined on an open set U ⊂ Cn and let h(z) be pluriharmonic function on U such that h(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let 0 < γ < 1 and µ > 0 be such that: B(1−γ)√µ(0) ⊂ {u ≤ u(0) + µ} ⊂ B(1+γ)√µ(0) ⊂ U, (1 − γ)Eµ(0) ⊂ {u ≤ h + u(0) + µ} ⊂ (1 + γ)Eµ(0) ⊂ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='12) In the above, Eµ(0) = {z ∈ Cn : �n i,j=1 ai¯jzi¯zj ≤ µ} with ai¯j positive Hermitian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then we have: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='13) B(1−γ)√µ(0) ⊂ (1 + γ)Eµ(0), (1 − γ)Eµ(0) ⊂ B(1+γ)√µ(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' First, by considering a unitary transformation if necessary, we may assume that Eµ(0) = {z ∈ Cn : �n i=1 λi|zi|2 ≤ µ} with 0 < λ1 ≤ λ2 · · · ≤ λn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Also it will suffice to prove one of the two inclusions in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='13), say B(1−γ)√µ(0) ⊂ (1 + γ)Eµ(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' To prove the other inclusion, we may consider a change of coordinates: wi = √λizi, so that Eµ(0) becomes B√µ(0), and we use v(z) := u(z)−h(z) to replace u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then the second inclusion would follow from the first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' We wish to argue by contradiction and assume that B(1−γ)√µ(0) is not contained in (1 + γ)Eµ(0), then we must have: B(1−γ)√µ(0) ∩ ∂ � (1 + γ)Eµ(0) � ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Note that u ≥ h(z)+u(0)+µ on (1+γ)∂Eµ(0) and u ≤ u(0)+µ in B(1−γ)√µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Therefore, h(z) ≤ 0 on B(1−γ)√µ ∩ (1 + γ)∂Eµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' We will show that this hypersurface, if nonempty, actually bounds a nontrivial region, so that we get h ≤ 0 in a neighborhood of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Since h is a pluriharmonic function and h(0) = 0, we can use the strong maximum principle to get h ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Hence (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='12) would give us B(1−γ)√µ(0) ⊂ {u ≤ h + µ} ⊂ (1 + γ)Eµ(0), contrary to what we assume above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' First we present the argument when n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' We want to show that if λ2 > (1+γ)2 (1−γ)2 , then h ≡ 0, which would contradict (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' On the other hand, if λ2 ≤ (1+γ)2 (1−γ)2 , then we would have B(1−γ)√µ ⊂ (1 + γ)Eµ, which is another contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' To see that λ2 > (1+γ)2 (1−γ)2 implies h ≡ 0, we fix some z1,∗ and consider the cross section between (z1,∗, z2) and B(1−γ)√µ ∩ (1 + γ)Eµ (viewed as a subset in C for z2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' They are given by: |z1,∗|2 + |z2|2 ≤ (1 − γ)2µ, λ1|z1,∗|2 + λ2|z2|2 ≤ (1 + γ)2µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='14) We want to argue that, if λ2 > (1+γ)2 (1−γ)2 , then the boundary of the cross section will be on (1 + γ)∂Eµ ∩ B(1−γ)√µ, for all z1,∗ close to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Since h ≤ 0 on the boundary of cross section, we would have h(z1,∗, z2) ≤ 0 in the interior of the section (z1,∗, z2) since h is pluriharmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' This is true for all z1,∗ close to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Hence h ≥ 0 in a neighborhood of 0 and we can conclude from strong maximum principle that h ≡ 0 since h(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' The boundary of the cross section is on (1+γ)∂Eµ ∩B(1−γ)√µ if and only if the second inequality in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='14) implies the first (with z1,∗ fixed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' The first inequality is equivalent to: |z2|2 ≤ (1 − γ)2µ − |z1,∗|2, whereas the second inequality is equivalent to: INTERIOR W 2,p ESTIMATE FOR SMALL PERTURBATIONS TO THE COMPLEX MONGE-AMPERE EQUATION 23 |z2|2 ≤ (1 + γ)2 λ2 µ − λ1 λ2 |z1,∗|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' In order for the second inequality to be stronger than the first, we need that: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='15) (1 + γ)2 λ2 µ − λ1 λ2 |z1,∗|2 ≤ (1 − γ)2µ − |z1,∗|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' If λ2 > (1+γ)2 (1−γ)2 , the inequality in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='15) is strict with z1,∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Hence (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='15) will hold for z1,∗ close enough to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' This proves our earlier claim about the boundary of the cross section and finishes the argument that λ2 > (1+γ)2 (1−γ)2 implies h ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' If λ1 ≤ λ2 ≤ (1+γ)2 (1−γ)2 , then B(1−γ)√µ ⊂ (1 + γ)Eµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Indeed, B(1−γ)√µ is given by: |z1|2 + |z2|2 ≤ (1 − γ)2µ, but then λ1|z1|2 + λ2|z2|2 ≤ (1+γ)2 (1−γ)2 · (1 − γ)2µ = (1 + γ)2µ, so that (z1, z2) ∈ (1 + γ)Eµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Now we look at general n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' As in n = 2, λn ≤ (1+γ)2 (1−γ)2 will immediately imply that B(1−γ)√µ ⊂ (1 + γ)Eµ (since λn is the largest eigenvalue), immediately giving what we want to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' We just need to show that λn > (1+γ)2 (1−γ)2 implies h ≡ 0, and then (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='12) will give the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' For this we consider the cross section between (z1,∗, · · · , zn−1,∗, zn) and B(1−γ)√µ ∩ (1 + γ)Eµ, and it is given by: n−1 � i=1 |zi,∗|2 + |zn|2 ≤ (1 − γ)2µ, n−1 � i=1 λi|zi,∗|2 + λn|zn|2 ≤ (1 + γ)2µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='16) The boundary of the cross section is on (1 + γ)∂Eµ ∩ B(1−γ)√µ iff the second inequality in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='16) is stronger than the first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' This would mean: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='17) 1 λn � (1 + γ)2µ − n−1 � i=1 λi|zi,∗|2� ≤ (1 − γ)2µ − n−1 � i=1 |zi,∗|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Note that if λn > (1+γ)2 (1−γ)2 , the inequality in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='17) is strict with zi,∗ = 0, 1 ≤ i ≤ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Hence (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='17) will continue to hold for (z1,∗, · · · , zn−1,∗) close to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Hence we would have h ≤ 0 in a neighborhood of 0, and strong maximum principle would give h ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' □ As a consequence, we deduce that: Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let u be a function defined on an open set U ⊂ Cn with 0 ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let h1(z), h2(z) be pluriharmonic functions on U such that h1(0) = h2(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let 0 < γ < 1 and µ > 0 be such that: (1 − γ)Ep,µ(0) ⊂ {u ≤ hp + u(0) + µ} ⊂ (1 + γ)Ep,µ(0) ⊂ U, p = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' 24 JINGRUI CHENG, YULUN XU In the above, Ep,µ(0) = {z ∈ Cn : � i,j ap,i¯jzi¯zj ≤ µ} with ap,i¯j being positive Hermitian and det ap,i¯j = 1, p = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let Tp be a C-linear transformation mapping B√µ to Ep,µ, then we have: ||T −1 1 T2|| ≤ (1 + γ)2 (1 − γ)2 , ||T −1 2 T1|| ≤ (1 + γ)2 (1 − γ)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' We can apply a map T −1 1 to the above picture and reduce E1,µ to be B√µ(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Denote ˜E2,µ = T −1 1 (E2,µ) = T −1 1 T2(B√µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then we know that the eigenvalues of ˜E2,µ is between (1+γ)2 (1−γ)2 and (1−γ)2 (1+γ)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' This implies the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' □ Before we move further, we first want to explain the idea why Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='18 help with proving engulfing property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Suppose, say, we have Sµ(x1) ∩ Sµ(x2) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Take x∗ ∈ Sµ(x1) ∩ Sµ(x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' We can normalize S100µ(x1) to be close to a unit ball and Sµ(x1) will then be close to a ball with radius 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Now we can define a section ˜S∗ with height 1 100 centered at x′ ∗(image of x∗ under the new coordinate) using the new coordinate, so that its shape is comparable to a ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then we go back to the original coordinate, we get a section S∗ centered at x∗ with height µ whose shape is comparable with Sµ(x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Because of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='18, S∗ and Sµ(x∗) will also have similar shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Therefore we see that the shapes of Sµ(x∗) and Sµ(x1) are comparable (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Similarly, Sµ(x∗) and Sµ(x2) are also comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Hence, if we normalize Sµ(x1) to be close to a unit ball, the other section Sµ(x2) will be close to an ellipsoid whose shape is not too eccentric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then the engulfing property would follow from the standard engulfing property for balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let x∗ ∈ Sµ(x0) ∩ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8 for some x0 ∈ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8 and 0 < µ ≤ µ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let Tµ,x∗ and Tµ,x0 be the C-linear transfomation given by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1, part (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then for σ chosen small enough depending only on n, ||T −1 µ,x∗ ◦ Tµ,x0|| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='13, ||T −1 µ,x0 ◦ Tµ,x∗|| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' First we can find k ≥ 1 such that µk+1 0 < µ ≤ µk 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' We will work under the coordinate z(k−1), defined as z = x0 + µ k−1 2 0 Tk−1,x0(z(k−1)), where Tk−1,x0 is the C-linear transformation mapping a ball to Eµk−1 0 (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' We have that Sµ(x0) under z(k−1) = {z(k−1) : ux0,k−1(z(k−1)) ≤ ux0,k−1(0) + ˜hx0,k−1(z(k−1)) + µ µk−1 0 }, where ux0,k−1 is the normalized u on Ωx0,k−1 (Sµk−1 0 (x0) under z(k−1) which is close to a ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=') Also we have Eµ(x0) under z(k−1) = {z(k−1) : � i,j (vx0,k−1)i¯j(0)z(k−1) i ¯z(k−1) j ≤ µ µk−1 0 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Here vx0,k−1 solves the Dirichlet problem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' On the other hand, denote z∗ to be the image of x∗ under z(k−1), then we know that z∗ ∈ the image of Sµ(x0) under z(k−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' We can find a section centered at z∗ under z(k−1) using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='6, applied to vx0,k−1 and ux0,k−1, δ = 4ε, x0 = z∗, µ replaced by µ µk−1 0 ≤ µ0, which is between µ0 and µ2 0, we INTERIOR W 2,p ESTIMATE FOR SMALL PERTURBATIONS TO THE COMPLEX MONGE-AMPERE EQUATION 25 have that: (1 − γ) ˜E µ µk−1 0 (z∗) ⊂ {z(k−1) ∈ B1 : (ux0,k−1 − h∗)(z(k−1)) ≤ ux0,k−1(z∗) + µ µk−1 0 } ⊂ (1 + γ) ˜E µ µk−1 0 (z∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='18) In the above, γ = 8εµk−1 0 µ + 3µ 1 2µ − k−1 2 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' It is less that 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ, because of our choice of µ0 in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='2) and our assumption that ε is small (we made a precise choice of ε in Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=') Also the coefficients of ˜E µ µk−1 0 (z∗) is defined by (vx0,k−1)i¯j(z∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let T∗ be the C-linear transform given by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8 such that T∗(B� µµ1−k 0 (0)) = ˜Eµµ1−k 0 (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then similar to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='10, part (4), we would have that (using (vx0,k−1)i¯j is close to identity by σ 1 2 ): || ˜T∗ − I|| ≤ C′ nσ 1 2 , || ˜T −1 ∗ − I|| ≤ C′ nσ 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' We can tranform the picture (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='18) back to the z variable, and obtain that: There is an ellipsoid E∗ µ centered at x∗, such that: (1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)E∗ µ(x∗) ⊂ S∗ µ(x∗) ⊂ (1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)E∗ µ(x∗) ⊂ Sµk−1 0 (x0), where S∗ µ(x∗) = {z ∈ Sµk−1 0 (x0) : (u − h∗)(z) ≤ u(x∗) + µ}, with h∗ being a quadratic pluriharmonic polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' With T∗ = Tk−1,x0 ◦ ˜T∗, then we have T∗(B√µ(0)) = E∗ µ(z∗), and ||T −1 k−1,x0 ◦ T∗ − I|| ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1, ||T −1 ∗ Tk−1,x0 − I|| ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1, if σ is small enough depending only on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' On the other hand, with Sµ(x∗), Eµ(x∗) being given by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1, we also have (1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)Eµ(x∗) ⊂ Sµ(x∗) ⊂ (1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)Eµ(x∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Also we have a C-linear transform Tµ,x∗ such that Tµ,x∗(B√µ(0)) = Eµ(x∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Using Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='18, we have ||T −1 ∗ Tµ,x∗|| ≤ (1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)2 (1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)2 , ||T −1 µ,x∗ ◦ T∗|| ≤ (1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)2 (1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)2 Hence we get that ||T −1 k−1,x0 ◦ Tµ,x∗||, ||T −1 µ,x∗ ◦ Tk−1,x0|| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='12, if σ is small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Finally, we note that Tµ,x0 = Tk−1,x0 ◦ ˜Tk,x0, and || ˜Tk,x0||, || ˜T −1 k,x0|| ≤ 1 + C′ nσ 1 2 , so the result would follow if σ is small enough depending only on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' □ Then the engulfing property would follow from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' (of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='16) First we want to reduce to when µ1 and µ2 are comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Indeed, if µ1 < µ2, we know from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1 that Sµ1(x1) ⊂ S(1+c(σ))µ2(x1) and c(σ) → 0 as σ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Hence we may assume that σ is small enough so that Sµ1(x1) ⊂ Sµ′ 1(x1), for some µ2 < µ′ 1 ≤ 4µ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Hence it will suffice to prove the following statement: For any 0 < µ2 ≤ µ1 ≤ 4µ2, if Sµ1(x1) ∩ Sµ2(x2) ̸= ∅, we have Sµ1(x1) ⊂ 10Sµ2(x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' First, we choose x∗ ∈ Sµ1(x1) ∩ Sµ2(x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' By choosing σ small enough, we know from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='19 that ||T −1 µp,x∗ ◦ Tµp,xp|| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='13, ||T −1 µp,xp ◦ Tµp,x∗|| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='13, p = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' 26 JINGRUI CHENG, YULUN XU On the other hand, we know from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='9 that, with σ small enough, we have ||T −1 µ1,x∗ ◦ Tµ2,x∗|| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1, ||T −1 µ2,x∗ ◦ Tµ1,x∗|| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' This follows from the observation that µ1 and µ2 must belong to the same level or adjacent levels (either µk+1 0 < µ2 ≤ µ1 ≤ µk 0, or µk+1 0 < µ2 ≤ µk 0 < µ1 ≤ µk−1 0 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Hence we get: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='19) ||T −1 µ1,x1 ◦ Tµ2,x2||, ||T −1 µ2,x2 ◦ Tµ1,x1|| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='16 · 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1 ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Now we show the containment of sections: Sµ1(x1) ⊂ (1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)Eµ1(x1) = Tµ1,x1(B(1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)√µ1(x1,∗)) = Tµ2,x2 ◦ T −1 µ2,x2 ◦ Tµ1,x1(B(1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)√µ1(x1,∗)) ⊂ Tµ2,x2(B2(1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)√µ1(x′ 1,∗)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='20) In the above, we denote x1,∗ and x′ 1,∗ so that Tµ1,x1(x1,∗) = x1, T −1 µ2,x2◦Tµ1,x1(x1,∗) = x′ 1,∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' We also used the bound (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' On the other hand, Sµ2(x2) ⊂ Tµ2,x2(B(1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)√µ2(x∗ 2)), where x2,∗ = T −1 µ2,x2(x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' It follows that B(1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)√µ2(x∗ 2) ∩ B2(1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)√µ1(x′ 1,∗) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Since µ1 ≤ 4µ2, it follows that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='21) B2(1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)√µ1(x′ 1,∗) ⊂ B(1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)9√µ2(x∗ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Hence it follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='20) that: Sµ1(x1) ⊂ Tµ2,x2(B(1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)9√µ2(x∗ 2)) ⊂ Tµ2,x2(B9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1√µ2(x∗ 2)) = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1Eµ2(x2) ⊂ 10Sµ2(x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' □ Finally let us include the following inclusion result for future reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let σ be small enough depending on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then for any 0 < µ ≤ µ0 121, we have 10Sµ(x0) ⊂ S121µ(x0) ⊂ 12Sµ(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let k ≥ 1 be such that µk+1 0 < µ ≤ µk 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' There are two cases to consider: If 121µ ≤ µk 0, then E121µ(x0) and Eµ(x0) are defined by the same coefficients, hence 10Sµ(x0) ⊂ 10(1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)Eµ(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' On the other hand, (1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)E121µ(x0) = (1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)11Eµ(x0) ⊂ S121µ(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Hence 10Sµ(x0) ⊂ S121µ(x0), as long as 10(1+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ) ≤ 11(1− 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' On the other hand, S121µ(x0) ⊂ (1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)E121µ(x0) = 11(1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)Eµ(x0) ⊂ 12(1−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)Eµ(x0) ⊂ 12Sµ(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' If µk 0 < 121µ ≤ µk−1 0 , then under z(k−1), E121µ(x0) becomes a ball with radius 11 � µµ1−k 0 , hence S121µ(x0) contains the ball B 11(1−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ) � µµ1−k 0 (0) and is contained in B 11(1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ) � µµ1−k 0 under z(k−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' On the other hand, z(k) and z(k−1) differs by a coordi- nate change ˜Tk,x0, for which we have the bound || ˜Tk,x0|| ≤ 1+C′ nσ 1 2 , || ˜T −1 k,x0|| ≤ 1+C′ nσ 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' INTERIOR W 2,p ESTIMATE FOR SMALL PERTURBATIONS TO THE COMPLEX MONGE-AMPERE EQUATION 27 Therefore, Eµ(x0) is contained in B (1+C′nσ 1 2 ) � µµ1−k 0 (0) , and contains B√ µµ1−k 0 1+C′nσ 1 2 (0) under z(k−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Therefore, working under z(k−1), one has image of 10Sµ(x0) in z(k−1) ⊂ image of 10(1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)Eµ(x0) under z(k−1) ⊂ B 10(1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)(1+C′nσ 1 2 ) � µµ1−k 0 (0) ⊂ (1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)B 11 � µµ1−k 0 (0) ⊂ image of S121µ(x0) under z(k−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' The above inequality holds as long as 10(1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)(1 + C′ nσ 1 2 ) ≤ 11(1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' On the other hand, still working under z(k−1): Image of S121µ(x0) in z(k−1) ⊂ 11(1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)B� µµ1−k 0 (0) ⊂ 11(1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)(1 + C′ nσ 1 2)×Eµ(x0) in z(k−1) ⊂ 12(1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)×Eµ(x0) in z(k−1) ⊂ 12×Sµ(x0) in z(k−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' In the above, we need to require that 11(1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)(1 + C′ nσ 1 2) ≤ 12(1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' □ As a consequence, we also have the following version of engulfing property: Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Under the assumptions of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='16, we have: Sµ1(x0) ⊂ S121µ2(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Some measure theoretic lemmas In this section we will prove a covering lemma which will be used in the W 2,p estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' In the following, m always denotes the standard Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let Sµα(xα) ⊂ Rd be a family of sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Assume that 0 < µα ≤ µ0 for all α and ∪αSµα(xα) is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Assume that the volume of Sµα(xα) is comparable to that of a standard ball with radius √µα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' there exists a uniform constant C such that 1 C m(B√µα(0)) ≤ m(Sµα(xα)) ≤ Cm(B√µα(0)) Assume that Sµα(xα) satisfies the following engulfing property: For any Sµα1(xα1) and Sµα2(xα2) with Sµα1(xα1) ∩ Sµα2(xα2) ̸= ∅, if √µα1 ≤ 2√µα2, then Sµα1(xα1) ⊂ 10Sµα2(xα2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let X be a measurable set with X ⊂ ∪αSµα(xα), then one can choose a sequence (finite or infinite) Sµi(xi), such that: (1) Sµi(xi) are all disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' (2)X ⊂ ∪i10Sµi(xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' The proof for this lemma is very similar to the standard Vitali’s covering lemma in measure theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' First, we choose a set Sµ1(x1) with √µ1 > 1 2 supα √µα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then we consider all Sµα(xα) which does not intersect Sµ1(x1), and you choose Sµ2(x2) so that √µ2 > 1 2 sup{√µα : Sµα(xα) ∩ Sµ1(x1) = ∅}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then you consider all Sµα(xα) which does not intersect with Sµ1(x1) or Sµ2(x2) and you choose Sµ3(x3) among those so that √µ3 > 1 2sup of √µα among all Sµα(xα) which don’t intersect with Sµ1(x1) or Sµ2(x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' We continue this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' 28 JINGRUI CHENG, YULUN XU This process may stop in finite steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' If this happens, then we get a finite sequence Sµ1(x1), Sµ2(x2), · · · , SµN (xN) such that they are mutually disjoint, and all Sµα(xα) must intersect with one of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let i0 be the first index such that Sµi(xi)∩Sµα(xα) ̸= ∅, then Sµi0−1(xi0−1) ∩ Sµα(xα) = ∅, and due to our inductive choice, √µα ≤ 2√µi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Using the engulfing property, we get: Sµα(xα) ⊂ 10Sµi0(xi0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Therefore, X ⊂ ∪αSµα(xα) ⊂ ∪N i=110Sµi(xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' The other possibility is that we find an infinite sequence of {Sµi(xi)}∞ i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' They are mutually disjoint because of our construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then we must have that µi → 0, since ∪αSµα(xα) is bounded and the volume of Sµα(xα) is comparable to that of a standard ball with radius √µα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' In particular, if you define di = sup{µα : Sµα(xα) ∩ Sµj(xj) = ∅, 1 ≤ j ≤ i}, then we have di → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' It follows that any Sµα(xα), there exists Sµi0(xi0) such that √µα ≤ 2√µi0 such that Sµα(xα)∩Sµi0(xi0) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Hence if you use the engulfing property, you see that Sµα(xα) ⊂ 10Sµi0(xi0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' □ Using this, we can follow the usual proof of Lebesgue differentiation theorem to con- clude that: Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let {Sµ(x)}0<µ≤µ0, x∈B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8 be a family of sets such that: (1) Sµ(x) ⊂ B1, for all 0 < µ ≤ µ0, and x ∈ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8, (2) There is C > 0, such that for all 0 < µ ≤ µ0 and all x ∈ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8, 1 C m(B√µ(0)) ≤ m(Sµ(x)) ≤ Cm(B√µ(0)), (3) For any Sµ1(x1) and Sµ2(x2), if √µ1 ≤ 2√µ2 and Sµ1(x1) ∩ Sµ2(x2) ̸= ∅, then Sµ1(x1) ⊂ 10Sµ2(x2), (4) diam Sµ(x) tends to 0 as µ → 0, uniformly for x ∈ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let f : B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8 → R be an L1 function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then for m-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' x ∈ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8, we have: lim sup x∈Sµα(xα), µα→0 1 m(Sµα(xα)) � Sµα(xα) |f(y) − f(x)|dm(y) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' In particular, for all measurable set A, we have: lim inf x∈Sµα(xα), µα→0 m(Sµα(xα) ∩ A) m(Sµα(xα)) = 1, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' The proof of this lemma follows the proof of the standard Lebesgue differentiation theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' First, we define the maximal function: given f ∈ L1(B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8), M(f)(x) = sup x∈Sµ(x′), 0<µ≤µ0, x′∈B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8 1 m(Sµ(x′)) � Sµ(x′) f(y)dm(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' As in the proof of Lebesgue differentiation theorem, the result would follow from the following estimate: (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1) m{x ∈ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8 : M(|f|)(x) > t} ≤ Cn ||f||L1 t , ∀t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Indeed, for t > 0, we can define: Ωt := {x ∈ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8 : lim sup x∈Sµα(xα), µα→0 1 m(Sµα(xα)) � Sµα(xα) |f(y) − f(x)|dm(y) > t}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' INTERIOR W 2,p ESTIMATE FOR SMALL PERTURBATIONS TO THE COMPLEX MONGE-AMPERE EQUATION 29 We only need to show that m(Ωt) = 0 for any t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' On the other hand, for any g ∈ C( ¯B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8), we have 1 m(Sµα(xα)) � Sµα(xα) |f(y) − f(x)|dm(y) ≤ M(|f − g|)(x) + 1 m(Sµα(xα)) � Sµα(xα) |g(y) − g(x)|dm(y) + |f − g|(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='2) Therefore, Ωt ⊂ {x : M(|f − g|) > t 3} ∪ {x : |f − g|(x) > t 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Here we implicitly used item (4) of the assumption, so that the middle term in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='2) tends to zero as µα → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Hence m(Ωt) ≤ m{x : M(|f − g|) > t 3}+ m{x : |f − g|(x) > t 3} ≤ Cn 3||f − g||L1 t + ||f − g||L1 t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' We can then choose a sequence gj ∈ C( ¯B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8) such that gj → f in L1, so that we may conclude that m(Ωt) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Now it only remains to show (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Denote the set on the left hand side to be E, then for any x ∈ E, we can find 0 < µx ≤ µ0 and yx ∈ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8 with x ∈ Sµx(yx), such that: tm(Sµx(yx)) ≤ � Sµx(yx) |f(y)|dm(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Hence we get a covering of E: E ⊂ ∪x∈ESµx(yx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Now we are in a position to apply Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1 to choose a countable sequence Sµi(xi), which is mutually disjoint, and E ⊂ ∪i10Sµi(xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Hence m(E) ≤ m(∪i10Sµi(xi)) ≤ � i m(10Sµi(xi)) = 102n � i m(Sµi(xi)) ≤ 102n � i 1 t � Sµi(xi) |f(y)|dm(y) ≤ 102n ||f||L1 t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' This proves (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' □ Another lemma we will need is: Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let {Sµ(x)}0<µ≤µ0, x∈B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8 satisfy the assumptions of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='2, and we assume additionally: (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='3) 10Sµ(x) ⊂ S121µ(x) ⊂ 12Sµ(x), for any x ∈ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8, 0 < µ ≤ µ0 121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let X, Y ⊂ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8 be two measurable sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let 0 < ¯ε < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Assume that: (1) For any x0 ∈ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8, m(Sµ(x0) ∩ X) < ¯εm(Sµ(x0)) for any µ0 4 ≥ µ ≥ µ0 484.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' (2) For any Sµ(x) with m(Sµ(x)∩X) ≥ ¯εm(Sµ(x)) and µ ≤ µ0 2 , one has Sµ(x) ⊂ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then m(X) ≤ 122n¯εm(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' For a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' any x0 ∈ X, we know from the previous lemma that: (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='4) lim µ→0 m(X ∩ Sµ(x)) m(Sµ(x)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' 30 JINGRUI CHENG, YULUN XU Hence, if we define µ′ x = sup{0 < µ ≤ 1 4µ0 : m(X ∩Sµ(x)) ≥ ¯εSµ(x)}, then 0 < µ′ x ≤ µ0 484 for x ∈ X satisfying (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' For such x, we may choose µx, such that 5 6µ′ x ≤ µx ≤ µ′ x, and that m(X ∩ Sµx(x)) ≥ ¯εSµx(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' We may assume without loss of generality that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='4) holds for all x ∈ X, then we get a covering of X: X ⊂ ∪x∈XSµx(x) (otherwise we get a covering of X modulo a measure zero set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then we may use Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1 to obtain a countable sequence Sµi(xi), such that Sµi(xi) are mutually disjoint, with X ⊂ ∪i10Sµi(xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Hence m(X) ≤ � i m(X ∩ 10Sµi(xi)) ≤ � i m(X ∩ S121µi(xi)) ≤ � i ¯εm(S121µi(xi)) ≤ ¯ε � i m(12Sµi(xi)) ≤ ¯ε122n � i m(Sµi(xi)) ≤ 122n¯εm(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' In the second inequality, we used (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' In the third inequality, we used that µ0 4 ≥ 121µi > µ′ xi, since our choice of µx guaran- tees µi ≥ 5 6µ′ xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Therefore m(X ∩ S121µi(xi)) < ¯εm(S121µi(xi)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' In the forth inequality, we used (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='3) again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' In the last inequality, we used that Sµi(xi) are disjoint, and contained in Y , due to assumption (2) of this lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' The W 2,p estimate Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' We define Dk to be the set of z0 ∈ ¯B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8 such that for any 0 < µ ≤ µ0, Sµ(z0) ⊂ B(z0, � 10kµ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Define Ak = ¯B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8 − Dk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' In the above, one should think of Dk to be the “good set” and Ak the “bad set”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Roughly speaking, Dk is the set on which λi(z0) ≥ 10−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' So that heuristically, we can conclude that λi(z0) ≤ 10k(n−1) since the equation is Πiλi = f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' To see this picture, we can pretend that Sµ(z0) ≈ {z : � i,j ui¯j(z0)(z − z0)i(z − z0)j ≤ µ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' The requirement that this ellipsoid is contained in B√ 10kµ(z0) implies that 1 √λi ≤ 10 k 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' This of course needs to be made rigorous since our solution u is merely a viscosity solution (hence only continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=') In order to show the W 2,p estimates, there are roughly two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' (1) For every k ≥ 1, m(Ak ∩ Brk(0)) ≤ (122n¯ε)k−1m(B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='7), for k ≥ 1, by choosing σ and ε chosen sufficiently small and rk = rk−1 − 1 102−k, r0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' (2) Show that for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' x ∈ Dk, there is a paraboloid with opening Mk(n−1) 0 touching u from above at x, and a paraboloid with opening M−k 0 touching u from below at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' This is the viscosity interpretation of D2u(x) ≤ M(k−1)n 0 and ui¯j(x) ≥ M−k 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' We will carry out steps (1) and (2) in the following two subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' For the convenience of argument, we will assume that u ∈ C2(Ω) ∩ C(¯Ω), and obtain a quantitative W 2,p bound on B 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then the general case would follow from an approximation argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Power decay of the measure of bad set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' The plan is to use Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='3 with X = Ak+1 ∩ Brk+1(0), Y = Ak ∩ Brk(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Note that the sections Sµ(x) satisfy all the assumptions of that lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Therefore, we just have to show the following two things: (1) Choosing M0 large enough depending on ¯ε and n, we have m(Sµ(x0) ∩ A1 ∩ Br1(x0)) < ¯εm(Sµ(x0)) for any µ0 2 ≥ µ ≥ µ0 242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' INTERIOR W 2,p ESTIMATE FOR SMALL PERTURBATIONS TO THE COMPLEX MONGE-AMPERE EQUATION 31 (2) For all Sµ(x0) with m(Sµ(x0) ∩ Ak+1 ∩ Brk+1(0)) ≥ ¯εm(Sµ(x0)) and µ ≤ µ0 2 , one has Sµ(x0) ⊂ Ak ∩ Brk(0), by choosing σ and ε small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' We will start with the following lemma, which is the analogue of Lemma 6 in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let u0 be a C2 solution to det(u0)i¯j = f0 in Ω with B1−γ(0) ⊂ Ω ⊂ B1+γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' We also assume that |f0 − 1| < ε on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let v0 be the solution of det(v0)i¯j = 1 on Ω and v0|∂Ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then there exists a dimensional constant C6,n such that for all γ and ε small enough (depending only on dimension), m({x ∈ B 1 2 : Γ(u0 − 1 2v0) = u0 − 1 2v0}) m(B 1 2 ) ≥ 1 − C6,nε 1 2 − C6,nγ 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' In the above, Γ(u0 − 1 2v0) is the convex envelope of u0 − 1 2v0 in B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='9, defined as: Γ(u0 − 1 2v0)(z) = sup{l(z) : l(z) is affine and l ≤ u0 − 1 2v0 on B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='9}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' The proof follows similar lines as Lemma 6 of [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' First, we may use Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='3 and Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='4 to conclude that, with γ chosen small enough: (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1) |Dm(v0 − (|z|2 − 1))|B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='9 ≤ Cnγ1− m 4 , m = 0, 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' In particular, we know that v0 is strictly convex from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1), after choosing γ small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Also by maximum principle, (1 + 3ε)v0 ≤ (1 + ε) 1 n v0 ≤ u0 ≤ (1 − ε) 1 n v0 ≤ (1 − 3ε)v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' So we get: (1 2 + 3ε)v0 ≤ Γ(u0 − 1 2v0) ≤ (1 2 − 3ε)v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' In the following, we will simply denote Γ(u0 − 1 2v0) by Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' We make the following claim: Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='2) ∇ � (1 2 − 3ε)v0 � (B 1 2− √ 96ε) ⊂ ∇Γ(B 1 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' First we use Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='3 to finish the proof and then prove the Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='3 itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Indeed, we have: m(∇Γ(B 1 2 )) ≥ m � ∇(1 2 − 3ε)v0 � (B 1 2− √ 96ε) = � B 1 2 − √ 96ε det � (1 2 − 3ε)D2v0 � dm(x) ≥ (1 2 − 3ε)2n(2 − Cnγ 1 2)2nm(B 1 2 − √ 96ε) ≥ m(B 1 2 )(1 − C5,nε 1 2 − C5,nγ 1 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='3) In the equality of the first line above, we used that v0 is strictly convex on B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='9, hence ∇v0 is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' In the first inequality of the second line, we used (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1) with m = 2 to get that |D2(v0 − |z|2)| ≤ C′ nγ 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' 32 JINGRUI CHENG, YULUN XU On the other hand, we have, using the Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='4 below: m(∇Γ(B 1 2)) ≤ � B 1 2 ∩{Γ=u0− 1 2v0} � 2(1 + ε) 1 n − det(D2(1 2v0)) 1 2n � dm(x) ≤ � B 1 2 ∩{Γ=u0− 1 2v0} � 2(1 + ε) 1 n − (1 − C′ nγ 1 2) �2ndm(x) ≤ (1 + C5,nγ 1 2 + C5,nε)2nm(B 1 2 ∩ {Γ = u0 − 1 2v0}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='4) The result follows from combing (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='3) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Now it only remains to prove Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Indeed, let p = ∇(1 2 − 3ε)v0(x0), with x0 ∈ B 1 2 − √ 96ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Denote lx0 = (1 2 − 3ε)v0(x0) + p · (x − x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' We just need to show that: {z ∈ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='9 : Γ(z) < lx0(z)} ⊂ {(z ∈ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='9 : (1 2 + 3ε)v0 ≤ lx0} ⊂ B√ 96ε(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' That is, the minimum of Γ − lx0 is achieved in the interior of B√ 96ε(x0) ⊂ B 1 2 , giving p ∈ ∇Γ(B 1 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' The first inclusion above is obvious since Γ ≥ (1 2 +3ε)v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' To see the second inclusion, we need the following calculation: First, we note that, for z ∈ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='9, we have: v0(z) ≥ v0(x0) + ∇v0(x0) · (z − x0) + (2 − Cnγ 1 2 )|z − x0|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' In the above, we used (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1) again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Therefore, {z ∈ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='9 : (1 2 + 3ε)v0 ≤ lx0} ⊂ {z ∈ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='9 : (1 2 + 3ε)(v0(x0) + ∇v0(x0) · (z − x0) + (2 − Cnγ 1 2 )|z − x0|2} ≤ (1 2 − 3ε)(v0(x0) + ∇v0(x0) · (z − x0))} So that the above implies (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='5) (1 2 + 3ε)(2 − Cnγ 1 2)|z − x0|2 ≤ −6ε(v0(x0) + ∇v0(x0) · (z − x0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' In the above, we may assume |v0(x0)| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1 and |∇v0(x0)| ≤ 2, and we get |z − x0|2 ≤ 6ε(2+2·2) ( 1 2 +3ε)(2−Cnγ 1 2 ) ≤ 96ε, and Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='3 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' □ In the above proof, we used the following lemma, which is the analogue of Lemma 5 in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let u0 and v0 be as given by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='2, with γ small enough so that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Denote Γ = Γ(u0 − 1 2v0) as defined by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then we have: M(Γ) ≤ � 2(1 + ε) 1 n − det(D2(1 2v0)) 1 2n �2nχΓ=u0− 1 2 v0 on B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Here M(Γ) is the (real) Monge-Ampere measure of a convex function, defined as M(Γ)(E) = m(∂Γ(E)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Since u0 is C2, then we know that Γ is C1,1 on B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Hence for any Borel set E ⊂ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8, one would have: M(Γ)(E) = � E det(D2Γ)(x)dm(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Moreover, it is a general INTERIOR W 2,p ESTIMATE FOR SMALL PERTURBATIONS TO THE COMPLEX MONGE-AMPERE EQUATION 33 fact that M(Γ) is concentrated on the contact set {Γ = u0 − 1 2v0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' On such a set, we would have: D2u0 ≥ D2Γ + D2(1 2v0), a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Therefore det(D2Γ) 1 2n + det(D2(1 2v0)) 1 2n ≤ det(D2u0) 1 2n ≤ � 22n det(u0)2 i¯j � 1 2n ≤ 2(1 + ε) 1 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' In the above, we used the concavity of the function A �→ det 1 d (A), restricted to positive definite d × d symmetric matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' □ As a corollary to Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='2, we get: Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let u0 be as Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='2 and Cn be from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Define E to be the subset in B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8 such that x0 ∈ E if and only if there is a paraboloid with opening 1 2(1 − Cnγ 1 2 ) touching u0 from below at x0 in B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then we have: m(E ∩ B 1 2 ) m(B 1 2 ) ≥ 1 − C6,nε 1 2 − C6,nγ 1 2, where C6,n is from Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' We just need to show that Γ(u0 − 1 2v0) = u0 − 1 2v0 has the property described in this lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let x0 ∈ B 1 2 with Γ(u0 − 1 2v0)(x0) = (u0 − 1 2v0)(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Since Γ is a convex function on B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='9, we may find p ∈ Cn which defines a supporting plane for Γ, then we have: (u0 − 1 2v0)(z) ≥ Γ(u0 − 1 2v0)(z) ≥ (u0 − 1 2v0)(x0) + p · (z − x0), z ∈ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' On the other hand, from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1), we see that: v0(z) ≥ v0(x0) + ∇v0(x0) · (z − x0) + (1 − Cnγ 1 2 )|z − x0|2, z ∈ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Hence on B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='9 : u0(z) ≥ u0(x0) + (p + 1 2∇v0(x0)) · (z − x0) + 1 2(1 − Cnγ 1 2 )|z − x0|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' The above has equality at z = x0 and the right hand side defines a paraboloid with opening 1 2(1 − Cnγ 1 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' □ Having a paraboloid touching below at a point x0 is a very strong condition, and it will imply the control of the shape of Sµ(x0) on all scales of µ, together with a control on the associated pluriharmonic function hµ,x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' More precisely Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let u be a function on B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let x0 ∈ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8 be such that there is a paraboloid with opening κ > 0, touching u from below at x0 in B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let 0 < ˜γ < 1, µ > 0, and A = ai¯j be positive Hermitian with det A = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Define E(x0) = {z : � i,j ai¯j(z − x0)i(z − x0)j ≤ µ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let h(z) be a degree 2 pluriharmonic polynomial with h(x0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' (1 − 1 2˜γ)E(x0) ⊂ {z ∈ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='9 : (u − h)(z) ≤ u(x0) + µ} ⊂⊂ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' 34 JINGRUI CHENG, YULUN XU Then we have the following estimates for A and h(z): κ(1 − ˜γ)2 ≤ λi(A) ≤ κ1−n(1 − ˜γ)2(1−n), 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' − 1 (1 − ˜γ)2n κ1−nI2n ≤ D2h ≤ n − 1 (1 − ˜γ)2n κ1−nI2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Denote S(x0) = {z ∈ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='9 : (u − h)(z) ≤ u(x0) + µ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' First we see that � i,j ai¯j(z − x0)i(z − x0)j > (1 − ˜γ)2µ on (1 − 1 2˜γ)∂E(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Hence (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='6) 1 (1 − ˜γ)2 � i,j ai¯j(z − x0)i(z − x0)j + u(x0) > (u − h)(z), z ∈ (1 − 1 2˜γ)∂E(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' On the other hand, we use that u is being touched below by a paraboloid, we get u(z) ≥ κ|z − x0|2 + lx0(z), z ∈ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Here lx0(z) is an affine function with lx0(x0) = u(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Hence (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='7) 1 (1 − ˜γ)2 � i,j ai¯j(z−x0)i(z − x0)j+u(x0) > κ|z−x0|2+lx0(z)−h(z), z ∈ (1− 1 2˜γ)∂E(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' However, the above inequality has equality with z = x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Hence LHS of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='7)−RHS of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='7) has a minimum in the interior of S(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Denote this point to be x′ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Taking the complex Hessian at x′ 0, we see that, 1 (1 − ˜γ)2 ai¯j ≥ κI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' That is λi(A) ≥ κ(1 − ˜γ)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' On the other hand, since Πiλi(A) = 1, we see that λi(A) ≤ κ1−n(1 − ˜γ)2(1−n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Next we take the full Hessian of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='7) at x′ 0, we get: 1 (1 − ˜γ)2 D2� � i,j ai¯j(z − x0)i(z − x0)j � ≥ 2κI − D2h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' So that D2h ≥ − 1 (1 − ˜γ)2n κ1−nI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' On the other hand, since h is harmonic, we get that: D2h ≤ n − 1 (1 − ˜γ)2n κ1−n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' □ As a consequence, we get that: Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let u0 be as stated in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='2, with γ and ε small enough as required by that lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Define D to be the subset of B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8 such that x0 ∈ D if and only Sµ(x0) ⊂ B√M1µ(x0) for any 0 < µ ≤ µ0, where M1 = 2(1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)2(1 − Cnγ 1 2)−1(1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='2σ)−2, INTERIOR W 2,p ESTIMATE FOR SMALL PERTURBATIONS TO THE COMPLEX MONGE-AMPERE EQUATION 35 where Sµ(x0) is the family of sections constructed in Section 2 (applied to u0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then we have: m(B 1 2 ∩ D) m(B 1 2 ) ≥ 1 − C6,nγ 1 2 − C6,nε 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' First, from Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='5, we just need to show that E ∩ B 1 2 ⊂ D ∩ B 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Indeed, let x0 ∈ E such that there is a paraboloid with opening 1 2(1 − Cnγ 1 2), with Cn coming from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' On the other hand, our construction in Section 2 gives Sµ(x0), 0 < µ ≤ µ0 with (1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)Eµ(x0) ⊂ Sµ(x0) ⊂ (1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)Eµ(x0), where Sµ(x0) = {z : u(z) ≤ hµ,x0(z) + u(x0) + µ} and Eµ(x0) = {� i,j aµ,x0,i¯j(z − x0)i(z − x0)j ≤ µ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='. Now we are in a position to use Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='6 (we could assume µ0 small enough earlier so that Sµ(x0) ⊂ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='9 with 0 < µ ≤ µ0, x0 ∈ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8), with κ = 1 2(1 − Cnγ 1 2), ˜γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='2σ, A = aµ,x0,i¯j, to get: 1 2(1 − Cnγ 1 2)(1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='2σ)2In ≤ aµ,x0,i¯j ≤ 2n−1(1 − Cnγ 1 2 )1−n(1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='2σ)2(1−n)In.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' This would imply that Eµ(x0) ⊂ B(1−Cnγ 1 2 )− 1 2 (1−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='2σ)−1√2µ(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' So that Sµ(x0) ⊂ B√M1µ(x0) for all 0 < µ ≤ µ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' □ Now we are ready to prove the first statement made in the beginning of this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' More precisely, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let 0 < ¯ε < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' If σ and ε are small enough depending only on n and ¯ε, we have: m(Sµ(x0) ∩ A1 ∩ Br1(x0)) < ¯εm(Sµ(x0)), for any µ0 484 ≤ µ ≤ µ0 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' We fix some µ between µ0 484 and µ0 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' We may also assume µ0 ≤ 1 484 so that µ2 0 < µ ≤ µ0, so that E4µ(x0), Eµ(x0), and Eµ0(x0) have the same coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Recall from Section 2 that ˜T1,x0 is the coordinate change such that z = x0 + √4µ ˜T1,x0(w) will make E4µ(x0) becomes B1 under w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Denote ˜Ω to be the image of S4µ(x0) under w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then from (1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)E4µ(x0) ⊂ S4µ(x0) ⊂ (1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)E4µ(x0), we see that: (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8) B1−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ ⊂ ˜Ω ⊂ B1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Also we define ˜u = 1 4µ(u−h4µ,x0)(x0+√4µ ˜T1,x0(w)), then ˜u, ˜Ω will fullfil the assumptions we made in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Now we are in a position to apply Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='7 to conclude that: (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='9) m(B 1 2 ∩ ˜D) m(B 1 2) ≥ 1 − C6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='nσ 1 2 − C6,nε 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Here ˜D is the subset of B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8 (under the w variable) such that w0 ∈ ˜D if and only if ˜S˜µ(w0) ⊂ B√M1˜µ(w0) for any 0 < ˜µ ≤ µ0, where M1 = 2(1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)2 (1−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1Cnσ 1 2 )(1−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='2σ)2 is given by Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='7, but with γ replaced by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ because of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Here ˜S˜µ(w0) is the section given by the construction of Section 2, but carried out for ˜u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' First, since Eµ(x0) and 36 JINGRUI CHENG, YULUN XU E4µ(x0) have the same coefficients, Eµ(x0) is now B 1 2 under w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Therefore, if we define ˜Ω1 to be the image of Sµ(x0) under w, we would get that: (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='10) m(˜Ω1 ∩ ˜D) m(˜Ω1) ≥ 1 − C7,nσ 1 2 − C7,nε 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Next we would like to translate the set ˜D back to z variable, and show that: (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='11) the image of ˜D under z variable ⊂ D1, where D1 is defined in Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' This would finish the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' In order to show (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='11), we take w0 ∈ ˜D, then we have ˜S˜µ(w0) ⊂ B√M1˜µ(w0) for 0 < ˜µ ≤ µ0, and (1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ) ˜E˜µ(w0) ⊂ ˜S˜µ(w0) ⊂ (1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ) ˜E˜µ(w0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Now we switch back to z coordinates, then the above inclusions become: S′ 4µ˜µ(z0) ⊂ x0 + � 4µ ˜T1,x0(B√M1˜µ(w0)) ⊂ B(1+Cnσ 1 2 )√4M1µ˜µ(z0), (1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)E′ 4µ˜µ(z0) ⊂ S′ 4µ˜µ(z0) ⊂ (1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)E′ 4µ˜µ(z0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' In the above, z0 is the image of w0 under z and S′ 4µ˜µ is of the form {u−h ≤ u(z0)+4µ˜µ}, and E′ 4µ˜µ is an ellipsoid having the same volume as B√4µ˜µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Now we may use Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='18 to conclude that (1−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)E4µ˜µ(z0) ⊂ (1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)E′ 4µ˜µ(z0), where E4µ˜µ is the ellipsoid given by Section 2, but constructed directly with z0 (under z coordinate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' From this we see: S4µ˜µ(z0) ⊂ (1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)E4µ˜µ(z0) ⊂ (1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)2 1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ E′ 4µ˜µ(z0) ⊂ B(1+Cnσ 1 2 ) (1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)2 (1−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)2 √4M1µ˜µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' This would give the control of Sµ′(z0) for µ′ ≤ 4µµ0, and 4µµ0 ≥ µ2 0 121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' For µ′ ≥ µ2 0 121, we may assume without loss of generality that µ3 0 < µ′ ≤ µ2 0, then Sµ′(z0) ⊂ (1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)Eµ′(z0) = z0 + (1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)T2,z0(B√µ′(0)) ⊂ B(1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)(1+Cnσ 1 2 )2√µ′(z0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Hence we see that if we have: (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='12) 10 ≥ max � (1 + Cnσ 1 2) � M1, (1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)(1 + Cnσ 1 2 )2� , we can conclude that z0 ∈ D1, thereby finishing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' This is indeed true if we choose σ small enough depending only on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Moreover, we need to take σ and ε so that in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='10), we have C7,nσ 1 2 + C7,nε 1 2 < ¯ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' □ Now we move on to prove Statement (2) made in the beginning of Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' First, we need to show that, if m(Sµ(x0) ∩ Ak+1 ∩ Brk+1(0)) ≥ ¯εm(Sµ(x0)), then Sµ(x0) ⊂ Brk(0), with σ and ε small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' For this we observe that: Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let 0 < µ ≤ µ0 4 and x0 ∈ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8 such that S4µ(x0) ⊂ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Assume that there exists Λ > 0 such that ||Tµ1,x0|| ≤ Λ for all 4µ ≤ µ1 ≤ µ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then for σ > 0, ε > 0 chosen small enough depending only on n and ¯ε, we have m(Sµ(x0) ∩ D2Λ) m(Sµ(x0)) ≥ 1 − ¯ε 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Here D2Λ is the set of z0 such that Sµ′(z0) ⊂ B2Λ√µ′(z0) for all 0 < µ′ ≤ µ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' INTERIOR W 2,p ESTIMATE FOR SMALL PERTURBATIONS TO THE COMPLEX MONGE-AMPERE EQUATION 37 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' We consider the section S4µ(x0) with µk+1 0 < 4µ ≤ µk 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' From the assumption we see that ||T4µ,x0|| ≤ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Similar to the proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8, we consider the change of coordinates z = x0 + √4µT4µ,x0(w), so that E4µ(x0) gets transformed to be a unit ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' We denote ˜Ω to be the image of S4µ(x0) under w and ˜Ω1 to be the image of Sµ(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Similar to the proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' m(˜Ω1 ∩ ˜D) m(˜Ω1) ≥ 1 − C7,nσ 1 2 − C7,nε 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Here ˜D is the subset of B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8 under the w variable such that w0 ∈ ˜D if and only if ˜S˜µ(w0) ⊂ B√M1˜µ(w0) for 0 < ˜µ ≤ µ0, where M1 = 2(1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)2 (1−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1Cnσ 1 2 )(1−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='2σ)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' We repeat the argument in the proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8 below (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='11), transform this containment back to z variable and conclude that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='13) Sµ′(z0) ⊂ B(1+Cnσ 1 2 ) (1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)2 (1−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)2 Λ√M1µ′, 0 < µ′ ≤ 4µµ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Now it only remains to control Sµ′(z0) for µ′ > 4µµ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' First, if µ′ ≥ 4µ, then we have z0 ∈ Sµ′(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='19, we know that: ||T −1 µ′,x0 ◦ Tµ′,z0|| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='13, ||T −1 µ′,z0 ◦ Tµ′,x0|| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Hence ||Tµ′,z0|| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='13Λ, µ′ ≥ 2µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Therefore, we have: (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='14) Sµ′(z0) ⊂ (1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)Eµ′(z0) ⊂ B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='13(1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)Λ√µ′(z0), µ′ ≥ 2µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Finally, for 2µµ0 < µ′ ≤ 2µ, we note that T2µ,x0 and Tµ′,x0 differ at most by ˜Tk+1,x0, whose norm is bounde by 1 + Cnσ 1 2, hence we have: (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='15) Sµ′(z0) ⊂ B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='13(1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)(1+Cnσ 1 2 )Λ√µ′(z0), 2µµ0 < µ′ < 2µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Hence if one combines (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='13)-(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='15) and define (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='16) M2 = max � (1 + Cnσ 1 2 )2 (1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)4 (1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)4 M1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='16(1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)2(1 + Cnσ 1 2)2� , then for any z0 in the image of ˜D in the z coordinate, Sµ′(z0) ⊂ B√ M2Λ2µ′(z0), 0 < µ′ ≤ µ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Also we note that the right hand side of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='16) is less than 4, if σ is small enough depending only on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' □ As a consequence, we get that Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let 0 < µ ≤ µ0 4 and x0 ∈ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8 such that S4µ(x0) ⊂ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Assume that for some k0 ≥ 1, m(Sµ(x0) ∩ Ak0+1 ∩ Brk0+1(0)) ≥ ¯εm(Sµ(x0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Assume also that σ and ε are chosen small enough depending on ¯ε and n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then we have: µ ≤ 10 − | log(µ0)| 2 log(1+Cnσ 1 2 ) k0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' 38 JINGRUI CHENG, YULUN XU Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let k1 ≥ 1 be such that µk1+1 0 < 4µ ≤ µk1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then T4µ,x0 = Tk1,x0 and we have the estimate: ||Tk,x0||, ||T −1 k,x0|| ≤ (1 + Cnσ 1 2 )k1, 1 ≤ k ≤ k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Therefore, we may use Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='9, with Λ = (1 + Cnσ 1 2)k1, to conclude that: m(Sµ(x0) ∩ D2Λ) m(Sµ(x0)) ≥ 1 − ¯ε 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' From our assumption that m(Sµ(x0) ∩ Ak0+1 ∩ Brk0+1(0)) ≥ ¯εm(Sµ(x0)), we must have Dk0+1 ⊂ D2Λ (note that we either have Dk0+1 ⊂ D2Λ or D2Λ ⊂ Dk0+1 by definition), in other words, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='17) 10k0+1 ≤ 4(1 + Cnσ 1 2 )2k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' So that µ ≤ 1 4µk1 0 ≤ 10 − | log(µ0)| 2 log(1+Cnσ 1 2 ) k0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' □ As a further corollary, we see that Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Assume that for some k0 ≥ 1, m(Sµ(x0) ∩ Ak0+1 ∩ Brk0+1(0)) ≥ ¯εm(Sµ(x0)) for some x0 ∈ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8 and µ ≤ µ0 4 , then Sµ(x0) ⊂ Brk0(0), if σ and ε are chosen small enough depending on n and ¯ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' By our assumption, Sµ(x0) ∩ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='7 ̸= ∅, hence we know that S4µ(x0) ⊂ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8 (by choosing σ, hence µ0 small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=') Also we denote k1 so that µk1+1 0 < 4µ ≤ µk1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then we may use Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='10 to obtain that: diam Sµ(x0) ≤ diam E4µ(x0) ≤ (4µ) 1 2 ||T4µ,x0|| ≤ µk1 0 (1 + Cnσ 1 2 )k1 ≤ � µ0(1 + Cnσ 1 2 ) � log 10 2 log(1+Cnσ 1 2 ) k0 ≤ 1 40 · 2−k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='18) The above is true if we choose σ small enough depending on n (according to our choice of µ0 made in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='2), µ0 ≤ σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=') On the other hand, rk0 − rk0+1 = 1 10 · 2−k0−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Since Sµ(x0) ∩ Brk0+1(0) ̸= ∅, the conclusion follows from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' □ The remaining part of Statement (2) is to show that Sµ(x0) ⊂ Ak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' This would directly follow from the following observation: Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let 0 < µ ≤ µ0 4 and x0 ∈ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8 be such that S4µ(x0) ⊂ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8 and Sµ(x0) ∩ Dk0 ̸= ∅, for some k0 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then we have: m(Sµ(x0) ∩ Dk0+1) ≥ (1 − ¯ε 2)m(Sµ(x0)), provided that σ is small enough depending only on ¯ε and n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' From the assumption, we can find x∗ ∈ Sµ(x0) ∩ Dk0, which means that Sµ′(x∗) ⊂ B√ 10k0µ′(x∗) for any 0 < µ′ ≤ µ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Since (1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)Eµ′(x∗) ⊂ Sµ′(x∗) and Tµ′,x∗(B√µ′(0)) = Eµ′(0), we see that ||Tµ′,x∗|| ≤ √ 10k0 1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ , 0 < µ′ ≤ µ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' INTERIOR W 2,p ESTIMATE FOR SMALL PERTURBATIONS TO THE COMPLEX MONGE-AMPERE EQUATION 39 Since x∗ ∈ Sµ(x0), we know that x∗ ∈ Sµ′(x0) for all µ′ ≥ 4µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Hence we may use Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='19 to conclude that: ||T −1 µ′,x∗ ◦ Tµ′,x0|| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='13, ||T −1 µ′,x0 ◦ Tµ′,x∗|| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Hence we obtain that ||Tµ′,x0|| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='13√ 10k0 1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ , 4µ ≤ µ′ ≤ µ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Now we are in a position to use Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='9 with Λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='13√ 10k0 1−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then we conclude that: m(Sµ(x0) ∩ D2Λ) m(Sµ(x0)) ≥ 1 − ¯ε 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Here D2Λ is the set of z0 such that Sµ′(z0) ⊂ B2Λ√µ′(z0) for all 0 < µ′ ≤ µ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' We will be done if we can ensure that D2Λ ⊂ Dk0+1, and we just need: 4Λ2 = 4 · 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='16 · 10k0 (1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1σ)2 ≤ 10k0+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' This is clear if σ is small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' □ Now we are ready to show the Statement (2) in the beginning of Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let 0 < µ ≤ µ0 4 , x0 ∈ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8 be such that m(Sµ(x0)∩Ak0+1∩Brk0+1) ≥ ¯εm(Sµ(x0)) for some k0 ≥ 1, then Sµ(x0) ⊂ Ak0 ∩ Brk0(0), if σ and ε are small enough depending on ¯ε and n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='11 already implies Sµ(x0) ⊂ Brk(0), and we just have to show Sµ(x0) ⊂ Ak0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' If not, namely Sµ(x0) ∩ Dk0 ̸= ∅, then Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='12 would give us: m(Sµ(x0) ∩ Dk0+1) ≥ (1 − ¯ε 2)m(Sµ(x0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' This is in contradiction with our assumption that: m(Sµ(x0) ∩ Ak0+1 ∩ Brk0+1) ≥ ¯εm(Sµ(x0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' □ Now we are ready to show that: Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let Ak be define by Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let 0 < ¯ε < 1 be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let σ > 0 and ε > 0 be small enough depending on n and ε, we have: m(Ak ∩ Brk(0)) ≤ m(B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='7)(122n¯ε)k−1, k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Here rk = rk−1 − 1 10 · 2−k, r0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' In particular, m(Ak ∩ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='6) ≤ m(B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='7)(122n¯ε)k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' For k = 1, the above estimate is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' For k ≥ 1, we have the following estimate holds: m(Ak+1 ∩ Brk+1(0)) ≤ 122n¯εm(Ak ∩ Brk(0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' This follows from taking X = Ak+1∩Brk+1(0), Y = Brk(0)∩Brk(0) in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='3, where the two assumptions of that lemma indeed hold, due to Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='8 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' □ 40 JINGRUI CHENG, YULUN XU 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Control of the second derivatives on the good set and completion of proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' In this section, we show that the derivatives are controlled on the good sets Dk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' We wish to emphasize that we are assuming u ∈ C2(Ω), only for the sake of convenience of argument, but the regularity of u does not go into the quantitative estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' We start with the following lemma: Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let u ∈ C2(B1) solving det ui¯j = f with |f − 1| < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let x0 ∈ Dk for some k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then 1 10k In ≤ ui¯j(x0) ≤ 2 · 10(n−1)kIn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' From Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1, we know that Sµ(x0) ⊂ B√ 10kµ(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let 0 < c < 1, then we have, on ∂Sµ(x0), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='19) u − hµ,x0 = u(x0) + µ ≥ u(x0) + |x − x0|2 10k > u(x0) + c|x − x0|2 10k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' In the above, we noted that |x − x0|2 ≤ 10kµ on ∂Sµ(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' On the other hand, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='19) achieves equality when x = x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Hence the function x �→ (u − hµ,x0)(x) − c|x−x0|2 10k achieves minimum in the interior of Sµ(x0), say xµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then we have: (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='20) ui¯j(xµ) ≥ c 10k In.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Since det ui¯j ≤ 1 + ε, we see that ui¯j(xµ) ≤ (1 + ε)c1−n10k(n−1)In.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Since diam Sµ(x0) → 0 and xµ ∈ Sµ(x0), we see that xµ → x0 as µ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Hence we conclude that: c 10k In ≤ ui¯j(x0) ≤ 2 · c1−n10k(n−1)In.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Since 0 < c < 1 is arbitrary, we can make c → 1 to conclude the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' □ As a consequence, we get that: Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let p > 1 be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let u be given by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='3 and is C2(B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let γ be small enough depending only on n, and ε small enough depending on p and n, then we have: � B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='6 � (∆u)p + (truωE)p� ≤ Cp,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Here ∆u = � i ui¯i, truωE = � i 1 ui¯i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Note that on Dk, we have: ∆u ≤ 2n · 10(n−1)k, truωE ≤ n · 10k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' INTERIOR W 2,p ESTIMATE FOR SMALL PERTURBATIONS TO THE COMPLEX MONGE-AMPERE EQUATION 41 Hence {∆u > 2n · 10(n−1)k} and {truωE > n · 10k} are contained in Ak for k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' � B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='6∩{∆u>2n·10n−1} (∆u)p ≤ ∞ � k=1 � B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='6∩{2n·10(n−1)k<∆u≤2n·10(n−1)(k+1)} (∆u)p ≤ ∞ � k=1 � 2n · 10(n−1)(k+1)�pm(Ak ∩ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='6) ≤ ∞ � k=1 � 2n · 10(n−1)(k+1)�p · m(B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='7)(122n¯ε)k−1 = (2n · 102(n−1))pm(B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='7) ∞ � k=1 (10(n−1)p122n¯ε)k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' In order for the above sum to be finite, we can choose ¯ε > 0 so that 10(n−1)p122n¯ε = 1 2 so that the above integral ≤ 2(2n · 102(n−1))pm(B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' In the above, we used Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' In order for the theorem to apply, we need to choose ε and σ small enough depending only on ¯ε and n, so the choice of ε and σ eventually depend on p and n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' The estimation for truωE is completely similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' □ From Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='15, we can get full second order estimate by applying the Lp- estimate for the Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' That is, we get: Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Under the assumption of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='15, we have: ||u||W 2,p(B 1 2 ) ≤ Cp,n, if γ is small enough depending only on n, and ε small enough depending only on p and n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' The result would follow from Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='15 and the classical W 2,p estimates (Gilbarg-Trudinger [8], Chapter 9): for any u ∈ C2(B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='6) and 1 < p < ∞, ||u||W 2,p(B 1 2 ) ≤ C(||u||Lp(B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='6) + ||∆u||Lp(B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='6)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' □ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Some corollaries of the main theorem First we prove Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' (of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1) First, we wish to use the following Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1 to conclude that ϕ is close to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Indeed, by taking g = 1, ψ = 0, p = 2 in the following, we find that: ||ϕ||L∞ ≤ cε 1 2(n+4) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Here c depends only on the background metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Now we take some point p0 ∈ M and take normal coordinates (z1, · · · , zn) at p0 so that gi¯j(p0) = δij and ∇g(p0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' We can choose local potential ρ(z), such that ω0 = √−1∂ ¯∂ρ near p0, say on B1(p0) (under local coordinates z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' So that on this neighborhood, the equation can be written as: (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1) det((ρ + ϕ)i¯j) = f det(gi¯j), in B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' In order to use Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='4, we need to zoom in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1) at p0 at a suitable scale so that the right hand side is close to a contant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' 42 JINGRUI CHENG, YULUN XU Denote u = ρ + ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let 0 < r0 < 1, we perform a change of variable z = r0w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Next we define ˜ur0(w) = 1 r2 0 u(r0w), ˜ρr0 = 1 r2 0 ρ(r0w), ˜ϕr0(w) = 1 r2 0 ϕ(r0w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Assume that for some C0 > 0, we have 1 C0 I ≤ ρi¯j ≤ C0I, |D3ρ| ≤ C0 on B1, then we see that with the same C0 > 1: 1 C0 I ≤ (˜ρr0)wi ¯wj ≤ C0I, |D3 w ˜ρr0| ≤ C0, for |w| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Also |˜ur0 − ˜ρr0| ≤ 1 r2 0 ||ϕ||L∞ ≤ c 1 r2 0 ε 1 2(n+4) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Also on {|w| < 1}, we have det(˜ur0)i¯j = f det(gi¯j)(r0w), and we can estimate how close the right hand side is from 1: |f det(gi¯j) − 1| ≤ |f − 1| det gi¯j(r0w) + f| det(gi¯j)(r0w) − 1| ≤ C1ε + C1r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Here C1 depends only on the background metric and n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Hence in order to apply Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='4, we need to make sure that: C1ε + C1r0 ≤ εp,n, c 1 r2 0 ε 1 2(n+4) ≤ δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Here εp,n and δ0 are determined by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Hence we need to choose r0 first so that C1r0 = 1 2εp,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then we fix this choice, and choose ε small enough so as to make sure C1ε ≤ 1 2εp,n and c 1 r2 0 ε 1 2(n+4) ≤ δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='4 gives W 2,p estimate for ˜ur0 in B 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Scaling back to u, we get W 2,p estimate for u in B 1 2 r0(p0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' □ In the above proof, we used the following stability estimate due to Kolodziej [11] Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' ([11]) Let (M, ω0) be a compact K¨ahler manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Let ϕ ∈ PSH(M, ω0) and ψ ∈ PSH(M, ω0) be the solution to the complex Monge-Ampere equations: (ω0 + √ −1∂ ¯∂ϕ)n = fωn 0 , sup M ϕ = 0, (ω0 + √ −1∂ ¯∂ψ)n = gωn 0 , sup M ψ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Assume that there is some c0 > 0, p > 1 such that ||f||Lp ≤ c0, ||g||Lp ≤ c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then we have sup M |ϕ − ψ| ≤ c(c0, p)||f − g|| 1 n+4 Lp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Now we prove Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' The argument is similar to Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' INTERIOR W 2,p ESTIMATE FOR SMALL PERTURBATIONS TO THE COMPLEX MONGE-AMPERE EQUATION 43 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' (of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='2) We first use Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='4 with q = 2 to estimate the difference between ϕ and ϕ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' We have that sup Ω |u − u0| ≤ ε + cε 1 2n ≤ c′ε 1 2n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Here c depends only on n and diam Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' The rest of the argument is similar to Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' In other words, we take any point z0 ∈ Ω′ and consider rescaling ˜ur0(w) = 1 r2 0 u(z0 + r0w), and we similarly consider ˜u0,r0 = 1 r2 0 u0(z0 + r0w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then for suitable chosen r0, we will have that ˜ur0 satisfy the hypothesis of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' □ Next, we prove the Liouville theorem Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' (of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='3) Let r > 1, we consider: ur(w) = 1 r2 u(rw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then we know that for r large enough, one has |ur(w) − |w|2| ≤ 2ε, w ∈ B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' On the other hand, we have 1 − ε ≤ det ua¯b ≤ 1 + ε, hence from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='4 we obtain: ||ur||W 2,p(B 1 2 ) ≤ Cp,n, as long as we choose ε small enough depending on n and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' On the other hand, ur will also satisfy the scalar flat equation, hence we may use the following Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='2, and choose p = pn, then we obtain that: ||ur||C2,α(B 1 4 ) ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Here C is a uniform constant independent of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Rescaling back to u, it gives: |D2u(x) − D2u(y)|rα ≤ C|x − y|α, ∀x, y ∈ B r 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' One can fix x, y and let r → ∞ and conclude that D2u is a constant, hence u is a quadratic polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' □ In the following, we used the following estimate of the scalar flat equation, which originates from Chen-Cheng [6], Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Note that this is from the preprint version on arxiv, which was deleted in the published version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' ([6]) Let u ∈ C4(B1) ∩ PSH(B1) be a bounded solution to the scalar flat equation: n � i,j=1 ui¯j∂i¯j � log det ua¯b � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Assume that for some p > 3n(n − 1), we have ∆u ∈ Lp(B1), � i 1 ui¯i ∈ Lp(B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then for any 0 < α < 1, ||u||C2,α(B 1 2 ) ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Here C depends on n, p, ||u||L∞(B1), ||∆u||Lp(B1), || � i 1 ui¯i ||Lp(B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Now let us prove the Schauder type estimate, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' 44 JINGRUI CHENG, YULUN XU Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' (of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='4) First, we just need to prove that u ∈ W 2,p for p large enough, depending only on n and α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Indeed, W 2,p embeds into C1,1− 2n p for p > 2n (keep in mind that the real dimension is 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=') In order to apply Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='3, we just need to choose p large, so that 1 − 2n p > 1 − α n(2 + α) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Now we fix this p and it only remains to show u ∈ W 2,p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' The argument is similar to Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' We need to do rescaling, so that we are in the situation of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' In other words, we take z0 ∈ B 1 2 , and consider rescaling ˜ur0(w) = 1 r2 0(f(z0)) 1 n u(z0 + r0w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Similarly, we define ˜wr0(w) = 1 r2 0(f(z0)) 1 n w0(z0 + r0w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then for |w| < 1, the closeness between ˜ur0 and ˜wr0 becomes: |˜ur0 − ˜wr0| ≤ δ0 r2 0(f(z0)) 1 n ≤ δ0 C 1 n 1 r2 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' On the other hand, det(˜ur0)i¯j = f(z0 + r0w) f(z0) , and we can estimate: |f(z0 + r0w) f(z0) − 1| ≤ 1 f(z0)rα 0 K ≤ C 1 n 1 Krα 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Hence, in order for Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='4 to apply, we just need to guarantee: δ0 C 1 n 1 r2 0 ≤ δ′ 0, C 1 n 1 Krα 0 ≤ εp,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Here δ′ 0 is the required closeness from the solution to the smooth background, given by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='4, and εp,n is the required small ε under the above choice of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Hence we should first choose r0 so that C 1 n 1 Krα 0 = 1 2εp,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then with r0 fixed, we choose δ0 small enough so as to make sure δ0 C 1 n 1 r2 0 ≤ δ′ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then we may use Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='4 to conclude that u is in W 2,p on B 1 2 r0(z0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' □ Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' ([12]) Let Ω be a domain in Cn and u ∈ PSH(Ω)∩C(Ω) be a weak solution of the complex Monge-Ampere equation in Ω with 0 < λ ≤ f ∈ Cα(Ω) for some constant λ and some α ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' If u ∈ C1,β(Ω) with β ∈ (β0, 1), where β0 = β0(n, α) = 1 − α n(2 + α) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Then u ∈ C2,α(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Furthermore the C2,α norm of u in any relatively compact subset is estimable in terms of n, α, β, λ, ||u||C1,β (Ω), ||f||Cα(Ω) and the distance of the set to ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' INTERIOR W 2,p ESTIMATE FOR SMALL PERTURBATIONS TO THE COMPLEX MONGE-AMPERE EQUATION 45 References [1] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Bedford and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Taylor: The Dirichlet problem for a complex Monge-Ampere equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Invent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' vol 37 (1976), 1-44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' [2] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Blocki: Interior regularity of the degenerate Monge-Ampere equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' of the Aust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' vol 68(2003), 81-92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' [3] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Caffarelli: Interior W 2,p estimates for solutions of the Monge-Amp`ere equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' vol 131, issue 1(1990), 135-150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' [4] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Caffarelli, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Nirenberg and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Spruck: The Dirichlet problem for nonlinear second order elliptic equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Monge-Ampere equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Pure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' vol 37(1984), 369-402.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' [5] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Caffarelli, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Kohn, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Nirenbger, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Spruck: The Dirichlet problem for nonlinear second order elliptic equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Complex Monge-Ampere and uniformly elliptic equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Pure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' vol 38(1985), 209-252.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' [6] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Chen and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Cheng: On the existence of constant scalar curvature K¨ahler metrics (I): Apriori estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Soc, vol 34, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' 4(2021), 909-936.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' (arXiv: 1712.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='06679) [7] Dinew S and Ko�lodziej S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' A priori estimates for complex Hessian equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Analysis & PDE, (2014), 7(1), 227-244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' [8] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Gilbarg and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Trudinger: Elliptic partial differential equations of second order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' 224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Berlin: springer, (1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' [9] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' He: On the regularity of complex Monge-Ampere equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' vol 140, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' 5(2012), 1719-1727.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' [10] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Ivochikina: Construction of aprioru bounds for convex solutions of the Monge-Ampere equa- tion by integral methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Ukrian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' vol 30(1978), 32-38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' [11] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Kolodziej: The Monge-Ampere equation on compact K¨ahler manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Indiana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' vol 52, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' 3(2003), 667-686.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' [12] Chao, Li, Jiayu Li, and Xi Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' A C2,α estimate of the complex Monge–Amp`ere equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Journal of Functional Analysis , vol 275, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' 1 (2018), 149-169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' [13] Pogorelov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' V: The regularity of the generalized solutions of the equation det(∂2u/∂xi∂xj) = ϕ(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=', xn) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Dokl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Akad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Nauk SSSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' (1971), 534-537.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' [14] Pogorelov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' V: The Minkowski multidimensional problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Wiley, New York, 1978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' [15] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Savin: Small Perturbation Solutions for Elliptic Equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' in PDE, vol 73, issue 4(2007), 557-578.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' [16] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Schulz: A C2 estimate for solutions of complex Monge-Ampere equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Reine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Angew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' vol 348(1984), 88-93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' [17] Yu Wang: A Liouville Theorem for the Complex Monge-Ampere Equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Preprint, arXiv:1303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content='2403 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' [18] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' -T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Yau: On the Ricci curvature if a compact K¨ahler manifold and the complex Monge-Amp`ere equation, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Pure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} +page_content=' vol 31(1978), 339-441.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfBPql/content/2301.00940v1.pdf'} diff --git a/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf b/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..e6f99916b6d40f75bc9a266615870986fe452ca3 --- /dev/null +++ b/t9E0T4oBgHgl3EQf9QLr/content/2301.02800v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:ae606ec483ab82282914315fa3ba1dbc3ee03eb3f8d2e9a426abef82870a50b0 +size 501362 diff --git a/ttE2T4oBgHgl3EQffwca/content/tmp_files/2301.03929v1.pdf.txt b/ttE2T4oBgHgl3EQffwca/content/tmp_files/2301.03929v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b2bcfba51238f841b0f359ac7092077928b67d12 --- /dev/null +++ b/ttE2T4oBgHgl3EQffwca/content/tmp_files/2301.03929v1.pdf.txt @@ -0,0 +1,1015 @@ +arXiv:2301.03929v1 [gr-qc] 10 Jan 2023 +Semi-Classical quantisation of 3-particles Toda lattice +augmented +Application to the Mixmaster anisotropy Hamiltonian +Herv´e Bergeron∗ +Univ Paris-Saclay, ISMO, UMR 8214 CNRS, 91405 Orsay, France and +APC, Universit´e Paris Diderot, Sorbonne Paris Cit´e, 75205 Paris Cedex 13, France +Ewa Czuchry† +National Centre for Nuclear Research, +Pasteura 7, 02-093 Warszawa, Poland +Abstract +Usual approaches to quantisation of a 3-Toda lead to numerical calculations requiring +many steps that can be time consuming to insure their reliability. In order to reduce as +much as possible the numerical part of the EKB quantisation procedure, and then to ease +numerical calculations, we propose a reformulation of the mathematical framework with +more adapted variables. The resulting equations and procedure might be easily imple- +mented in a short Mathematica code. +This more explicit framework will be useful for +studying quantum Toda-Bianchi IX models in quantum cosmology where the true Bianchi +IX anisotropy Hamiltonian can be approximated by a 3-particle Toda system. +∗ herve.bergeron@universite-paris-saclay.fr +† ewa.czuchry@ncbj.gov.pl +Typeset by REVTEX +1 + +CONTENTS +I. Introduction +2 +II. Classical Toda lattice and its semi-classical quantization +4 +A. Classical formulation +4 +B. EKB quantization +6 +III. New parametrisation for 3-Toda system +8 +A. New parametrisation of the roots µ+ +1 and µ+ +2 +9 +B. New parametrisation of the roots µ− +2 and µ− +3 +10 +IV. New parametrisation of the actions I1 and I2 +11 +V. Application to the semi-classical quantisation of Toda-Bianchi IX anisotropy +Hamiltonian +12 +A. Bianchi IX general framework +12 +B. The anisotropy Hamiltonian Hq +13 +VI. Summary +15 +VII. Acknowledgements +15 +A. Analysis of the equation x3 − 3x = a +15 +References +16 +I. +INTRODUCTION +In our recent papers [1, 2] we have demonstrated that the quantum behaviour of +a refined model of the earliest Universe, namely the so-called “Mixmaster universe”, +might be approximated by a 3-particle Toda system, usually arriving in solid state +physics. The classical dynamics of the “Mixmaster universe” (or Bianchi IX model) +was studied by C. Misner in the canonical formalism [3] at the end of the sixties. It +involves a Hamiltonian that is formally identical to that of a particle moving in a +3D Minkowski spacetime in a time-dependent, exponentially steep and triangle-like +potential. Many mathematical studies have been devoted to the classical evolution +of the system and have led to some important results on the asymptotic behaviour, +the non-integrability or the chaotic behaviour (see e.g. [4]–[7]). +2 + +On the other hand, the quantum behaviour of Mixmaster remains not entirely +understood despite many interesting studies [1, 8–12]. +The difficulty lies mainly +in the very different possible dynamical regimes induced by the elaborate spatial +dependence of the anisotropy potential. We (with our co-authors) have shown in +[2] that the classical Mixmaster anisotropy potential can be viewed as a difference +between two 3-particle Toda potentials, thus paving the way to a possible analytically +solvable approximation for the problem of the quantum version of the Bianchi IX +model (see Sec. +5 for more details). +This approximation should be valid in the +unexplored region in-between the known harmonic and steep-wall approximations.1 +The Toda lattice system is a one-dimensional chain consisting of equal masses +interacting via exponential forces. This system was shown to be completely inte- +grable [13] –[15] as it has as many constants of motion as the numbers of degrees +of freedom. The quantisation of a Toda lattice was first performed by Gutzwiller +[18] who formulated a systematic recursive way of constructing the eigenfunctions +and explicitly constructed them for N = 2, 3 and 4 periodic Toda systems. However +it took much time until numerical results appeared [19] and [20], as they needed +calculational power and time. Canonical quantisation was performed after direct +diagonalization of the Hamiltonian, and results were classified with respect to repre- +sentation of the permutation group S3 under which the Hamiltonian is invariant. It +was found that the numerically obtained eigenvalues fulfil Gutzwiller’s quantisation +conditions. Furthermore the semiclassical Einstein-Keller-Brillouin method of quan- +tisation (EKB quantisation) was proved to provide results in good agreement with +exact ones [19], even for the first levels. However the usual mathematical formula- +tion of the EKB quantisation in the case of 3-Toda system involved several implicit +functions that seem to impose a pure numerical approach with not simple entan- +gled steps. Therefore, at first sight, it seems difficult to export this procedure in a +straightforward way in the more complex framework of Bianchi IX. +The aim of this note is to prove that the mathematical formulation of the 3- +Toda EKB quantisation can be significantly eased by a new parametrisation of the +problem. +The outline of the paper is as follows. In Sec. II we recall the main features of the +classical 3-body Toda lattice and its known formalism for EKB semi-classical quan- +tisation. This section allows to introduce all the notations useful for the remainder. +In Sec. III we introduce a new parameterisation for the Toda system which we use +in Sec. IV for new parameterisations of the action integrals which turn to be easily +calculated. In Sec. V we discuss the possible application of the obtained procedure to +quantum Mixmaster with the anisotropy Hamiltonian where a pure Toda potential +is used but with restored necessary dependences. We conclude in Sec. VI. +1 The latter corresponding respectively to the large volume and low anisotropy excitation level or +to the small volume and high anisotropy excitation level. +3 + +II. +CLASSICAL TODA LATTICE AND ITS SEMI-CLASSICAL QUANTI- +ZATION +A. +Classical formulation +The Toda lattice system is a one-dimensional chain consisting of equal masses +interacting via exponential forces. There are two types of those, an open lattice one +and a periodic one. The main difference is that in the periodic lattice the first and the +last particles are coupled whereas in the open lattice they are not. The Hamiltonian +for the periodic Toda system is like for a system of N equal-mass particles interacting +via exponential potential: +H = 1 +2 +N +� +k=1 +p2 +k + +N +� +k=1 +e−(qk−qk+1), +(1) +with periodicity condition q0 ≡ qN and q1 ≡ qN+1, where qi are generalised positions +and pi their corresponding conjugate momenta. +The simplest nontrivial periodic crystal is the periodic 3-particle Toda system +with Hamiltonian as follows: +H = 1 +2 +� +p2 +1 + p2 +2 + p2 +3 +� ++ e−(q1−q3) + e−(q2−q1) + e−(q3−q2). +(2) +The equations of motion for this system may be written as Lax’s equation [13]: +dL +dt = [M, L] , +(3) +where matrices L and M read as follows +L := + + +b1 a1 a3 +a1 b2 a2 +a3 a2 b3 + + , +M := 1 +2 + + +0 +a1 +−a3 +−a1 +0 +a2 +a3 +−a2 +0 + + . +(4) +Elements of the symmetric matrix L and the skew one M are functions of positions +and momenta of a Toda system: +ai := 1 +2e(qi−qi+1)/2 and bi := pi +2 where i = 1, 2, 3, with q0 ≡ q3, q4 ≡ q1. +(5) +The matrices L and M form a so called Lax pair, therefore the eigenvalues of L and +also the coefficients of its characteristic polynomial Ai, are constants of motion: +det(2µI − 2L) ≡ (2µ)3 + A1(2µ)2 + A2(2µ) + A3 − 2 +(6) +4 + +In the center of mass system P := p1 + p2 + p3 ≡ 0 those coefficients simplify to: +A1 = −P = 0, A2 = 1 +2P 2 − H =: −E, +(7) +A3 = Π3 +i=1pi − +3 +� +i=1 +pieqi+1−qi+2 ≡ 8Π3 +i=1bi − 8 +3 +� +i=1 +bia2 +i+1 =: A. +(8) +Therefore, conserved quantities A1 and A2 have physical interpretations, respectively +the total momentum P and the energy of the system E. The third conserved quantity +A does not have such explicit simple physical meaning. +Hamiltonian (2) is invariant under the transformations of the dihedral group D3. +The dihedral group DN is the group of symmetry of the N-sided regular polygon. +The particular group D3 has two kinds of representations, two one-dimensional rep- +resentations A1, A2 and a two-dimensional representation E. +The Toda lattice system is integrable even if the Hamiltonian is not separable, i.e. +we cannot separate its variables into the explicit ones. However, due to integrability, +we can rewrite the Hamiltonian by means of the canonical transformation in terms +of the action-angle variables (Ii, θi). For our n = 3 Toda chain we have i = 1, 2. +Moreover, it is possible to introduce the canonical conjugate variables (µi, νi), where +i = 1, 2. The action-angle variables arise in integrable systems and can be written as +Ii := +� +νi(µi)dµi, +(9) +where integration is performed along the closed loop trajectory in phase space, here +over a period of motion. It was calculated in [21] that for a Toda lattice conjugated +momenta (such that Poisson brackets fulfil {νi, µj} = δij) are following: +ν1,2 = 2 ln |1 +2(∆(µi) ± +� +∆(µi)2 − 4|, +(10) +where +∆(µ) := det(2µI − 2L) + 2. +(11) +In the the center of mass system P := p1 + p2 + p3 ≡ 0 this function reads as: +∆(µ) = 8µ3 − 2Eµ + A, +(12) +The variables µi are defined only on a limited interval specified by the inequality +|∆(µ)| ≥ 2 and so also action integrals Ii are taken over that regions. The expression +|∆(µ)| ≥ 2 is clearly a 3rd order polynomial and thus has 2 intervals where inequality +5 + +|∆(µ)| ≥ 2 holds. Since E ≥ 0, ∆(µ) admits a relative maximum for µ = −µm and +a relative minimum for µ = +µm with µm = +� +E +12. +µ +−∞ +−µm ++µm ++∞ +∆(−µm) ++∞ +∆(µ) +ր +ց +ր +−∞ +∆(µm) +(13) +In order that the inequality |∆(µ)| ≥ 2 holds on a non-zero measure interval of µ +each of the two equations ∆(µ) = 2 and ∆(µ) = −2 must have three distinct real +roots. This is only possible if ∆(−µm) > 2 and ∆(µm) < −2. This holds for following +values of E and A: +E > 3 and |A| < +2 +3 +√ +3E3/2 − 2. +(14) +Therefore, these two conditions have to be fulfilled to develop semi-classical quanti- +sation. +We call µ+ +1 < µ+ +2 < µ+ +3 the solutions of ∆(µ) = +2 and µ− +1 < µ− +2 < µ− +3 the three +solutions of ∆(µ) = −2. The canonical conjugate momenta ν+ +i +and ν− +i +such that +{µ+ +i , ν+ +j } = δij = {µ− +i , ν− +j } are given by: +ν+ +i = 2 ln | 1 +2(∆(µ+ +i ) + +� +∆(µ+ +i )2 − 4)|, +ν− +i = 2 ln | 1 +2(∆(µ− +i ) − +� +∆(µ− +i )2 − 4)|. +(15) +B. +EKB quantization +It was shown in [22] that for a 3 particle Toda lattice the corresponding actions +I1(E, A) and I2(E, A) read as +I1(E, A) = 4 +� µ+ +2 +µ+ +1 +arcosh |∆(µ)| +2 +dµ , +I2(E, A) = 4 +� µ− +3 +µ− +2 +arcosh |∆(µ)| +2 +dµ , +(16) +where arcosh(x) = ln(x + +√ +x2 − 1) for x ≥ 1. +As we see, the energy E and +the conserved quantity A, are given implicitly in terms of the above actions Ii. +6 + +Furthermore, from (14) the range of possible values of A is dependent of the value +of E and obviously the boundaries of integrals are not defined in an explicit way as +functions of E and A. +Now the semiclassical quantisation performed through EKB (Einstein-Keller- +Brillouin) formulation can be obtained as follows: inverting (at least formally) equa- +tions (16), the classical Hamiltonian can be expressed in terms of actions Ii, and the +semiclassical quantum energies are finally given by the substitutions Ii �→ (ni+1/2)h +in the expression of the Hamiltonian. In other terms EKB quantisation consists in +finding the solutions En1,n2 and An1,n2 of the system of equations +I1(En1,n2, An1,n2) = 2πℏ +� +n1 + 1 +2 +� +, +I2(En1,n2, An1,n2) = 2πℏ +� +n2 + 1 +2 +� +. +(17) +for integer values of n1 and n2. +Equations (16) might be a bit simplified by using the definitions of arcosh and then +performing integration by parts: +I1(E, A) = 4 +� µ+ +2 +µ+ +1 +log +���� +1 +2 +� +∆(µ) + +� +∆(µ)2 − 4 +����� dµ = += −4 +� µ+ +2 +µ+ +1 +µ∆′(µ) +� +∆(µ)2 − 4 +dµ , +I2(E, A) = 4 +� µ− +3 +µ− +2 +log +���� +1 +2 +� +∆(µ) + +� +∆(µ)2 − 4 +����� dµ = += −4 +� µ− +3 +µ− +2 +µ∆′(µ) +� +∆(µ)2 − 4 +dµ , +(18) +Obviously the solutions can only be found numerically and the difficulties of the +procedure lie mainly in: +(a) the non-complete independence of parameters E and A due to conditions (14), +(b) the implicit definitions of the boundaries of integrals I1 and I2 in terms of E +and A, +(c) the dependence in E and A of the length of integration domains (lengths that +can be very large), +7 + +(d) the final numerical procedure to solve the system (17). +To obtain the sought quantities En1,n2 and An1,n2 the authors in [20] change in a +controlled manner the values of E and A, calculating the corresponding actions and +retaining those values of E and A which satisfy conditions (17). The author of [19] +used the results of a direct canonical quantisation as initial values for solving the +system (17) via the simplex algorithm. Both approaches are numerically demanding, +specially as we see in (35) the integrated function is divergent at both integral limits. +That demands a very accuracy of numerical integration! +Our approach presented in this paper consists in removing first the difficulties +(a), (b) and (c) on analytical level by a new parametrisation of the problem, and +performing numerical calculations only for the final step (d) but in much simplified +settings. +III. +NEW PARAMETRISATION FOR 3-TODA SYSTEM +The main point of our approach consists in introducing a new parametrisa- +tion of the problem in terms of two completely independent parameters (α, θ) ∈ +]0, +∞[×]0, π[ in place of the usual dynamical constants E and A constrained by +the conditions (14). These two parameters (α, θ) are defined such that: +E := 3 cosh4/3 α, +A := −2 cos θ sinh2 α. +(19) +Due to internal properties of the hyper/trygonometric functions the conditions (14) +are automatically fulfilled with a one to one correspondence, as ∀α, θ :∈ R cosh α ≥ 1 +and | cos θ| ≤ 1. This solves the first point (a) mentioned at the end of the previous +section II B. +The second problem is that we have to deal with a problematic integral with no +predefined bounded length. In order to solve that let us rescale the variable µ which +appears in ∆(µ) and defining a new variable ν as follows +µ := νµm +(20) +where µm is the value of µ where ∆(µ) reaches its relative extremal value, namely +µm = +� +E/12. With the new parametrisation (19) we have: +µm = 1 +2 cosh2/3 α. +(21) +8 + +Finally, the expression ∆(νµm) reads as: +∆(νµm) = (cosh2 α) (ν3 − 3ν) − 2 cos θ sinh2 α. +(22) +In the remainder we will prove that the maximal range of ν in integrals will be +reduced to ν ∈ [−2, +2], solving the point (c) mentioned in section II B. +A. +New parametrisation of the roots µ+ +1 and µ+ +2 +In this section we focus on the point (b) of section II B, i.e. how to obtain an +explicit mathematical expression of the boundaries of integrals (16) in terms of the +new parameters (α, θ). +Let us recall that the µ+ +i +involved in the boundaries of +integrals are solutions of ∆(µi) = 2. +Using the new parametrisation of (22) we +obtain in terms of ν the equation +ν3 − 3ν = 2 +� +1 − 2 sin2 θ +2 tanh2 α +� +. +(23) +In the Appendix A we have described a procedure of finding explicit solutions of the +3rd order polynomial of the form ν3 − 3ν = 2 cos Φ leading to the explicit solutions +in ν. In order to apply this procedure to the above equation let us introduce a new +parameter Φ+ ∈]0, π[ such that +1 − 2 sin2 θ +2 tanh2 α = cos Φ+. +(24) +This is equivalent to the equation +sin2 θ +2 tanh2 α = sin2 Φ+ +2 . +(25) +Because α > 0, θ/2, Φ+/2 ∈]0, π/2[ implying sin(θ/2), sin(Φ+/2) > 0, we can sim- +plify this equation as follows +sin θ +2 tanh α = sin Φ+ +2 , +(26) +and then +Φ+ = 2 arcsin +� +sin θ +2 tanh α +� +. +(27) +9 + +Introducing the parameters ν+ +i such that µ+ +i = ν+ +i µm and using the solutions (A1) +of appendix A, we find that the sought values ν+ +1 and ν+ +2 needed for the definition +of I1 (which is now a function of α and θ in place of E and A) are explicitly: +ν+ +1 = −2 cos π − Φ+ +3 +< +ν+ +2 = −2 cos π + Φ+ +3 +, +(28) +where Φ+ defined in (27) is an explicit function of α and θ. +B. +New parametrisation of the roots µ− +2 and µ− +3 +Similarly let us recall that the µ− +i are solutions of ∆(µi) = −2. Using again the +parametrisation of (22) we obtain in terms of ν the equation +ν3 − 3ν = −2 +� +1 − 2 cos2 θ +2 tanh2 α +� +. +(29) +Now let us introduce a new parameter Φ− similar to the one used previously, namely +we would like to find Φ− ∈]0, π[ such that +− +� +1 − 2 cos2 θ +2 tanh2 α +� += cos Φ−. +(30) +This is equivalent to +cos2 θ +2 tanh2 α = cos2 Φ− +2 . +(31) +Because α > 0, θ/2, Φ−/2 ∈]0, π/2[ and then cos(θ/2), cos(Φ−/2) > 0, we can +simplify this equation as +cos θ +2 tanh α = cos Φ− +2 , +(32) +and then we have +Φ− = 2 arccos +� +cos θ +2 tanh α +� +. +(33) +Introducing the parameters ν− +i such that µ− +i = ν− +i µm and using the solutions (A1) +of appendix A, we find that the sought values ν− +2 and ν− +3 needed for the definition +of I2 (which is now a function of α and θ in place of E and A) are explicitly: +ν− +2 = −2 cos π + Φ− +3 +< +ν− +3 = 2 cos Φ− +3 , +(34) +where Φ− defined in (33) is an explicit function of α and θ. +10 + +IV. +NEW PARAMETRISATION OF THE ACTIONS I1 AND I2 +Using the definitions (16) of I1 and I2 and using the change of variable µ = νµm +in the integrals, we end with the formula +I1(α, θ) = 4µm +� ν+ +2 +ν+ +1 arcosh |∆(µmν)| +2 +dν +I2(α, θ) = 4µm +� ν− +3 +ν− +2 arcosh |∆(µmν)| +2 +dν +, +(35) +where µm given in (21); ν+ +1 , ν+ +2 given in (27), (28); ν− +2 , ν− +3 given in (33) (34) and +the function ν �→ ∆(νµm) given in (22) are explicit functions of α and θ. We may +observe that in our new parameterization the integrated functions in the expressions +for I1 and I2 do not exhibit any singular behavior! +For the final step (i.e. solving the EKB system (17) for a given pair (n1, n2)), +these integrals (35) must be computed numerically. +But since the function ν �→ +arcosh ∆(νµm) of Eq.(22) has no singular behaviour on each interval and the intervals +[ν+ +1 , ν+ +2 ], [ν− +2 , ν− +3 ] have a maximal length of 4, the numerical estimates are easy and +reliable. +The search of semi-classical quantised energies can now be done numerically with- +out much effort. We used a program Mathematica running on a laptop. After quick +numerical calculations using build-in functions we obtained solutions in terms of α +and θ. Finally, we needed only to use the definition of E and A in terms of α and θ +in (19) to go back to the sought quantities. +As a proof of efficiency of our procedure, we can compare the values obtained with +our method (with few seconds of computation on a laptop) with the ones calculated +in [19] (where ℏ = 1). The results are summarised in the table (36): the agreement +is perfect. +Symmetry +Matsuyama [19] +Our code +A or E +n1 n2 +E +A +E +A +A +0 +0 +4.7748 +0 +4.7748 +0 +A +1 +1 +8.5854 +0 +8.5854 +0 +A +3 +0 +10.8558 +9.2294 +10.8558 +9.2293 +A +4 +4 +21.9378 +0 +21.9378 +0 +A +7 +1 +22.6452 +29.0562 +22.6452 29.0562 +E +1 +0 +6.6686 +2.4110 +6.6686 +2.4110 +E +0 +2 +8.7002 +-5.4897 +8.7002 +-5.4897 +E +1 +3 +12.8280 +-6.9356 +12.8279 -6.9356 +E +5 +1 +17.5336 +16.6194 +17.5336 16.6194 +(36) +11 + +Remark on a particular situation: for n1 = n2 (which belongs to the symmetry “A”) +it is possible to prove directly that A = 0, which means in terms of our variables +that θ = π/2. Therefore, only the energy E (or the parameter α) is unknown. Then +it is better from a numerical point of view to solve a unique equation either the one +involving I1 or the one in I2, imposing by hand θ = π/2. +V. +APPLICATION TO THE SEMI-CLASSICAL QUANTISATION OF TODA- +BIANCHI IX ANISOTROPY HAMILTONIAN +A. +Bianchi IX general framework +Let us first recall Hamiltonian formulation of the Bianchi type IX model. The +respective Hamiltonian constraint in the Misner variables reads [3]: +HB9 = Ne−3Ω +24 +�2κ +V0 +�2 � +−p2 +Ω + p2 + 36 +�V0 +2κ +�3 +n2e4Ω[V (β) − 1] +� +, (Ω, pΩ, β, p) ∈ R6, +(37) +where β := (β+, β−), p := (p+, p−), V0 = 16π2 +n3 +is the fiducial volume, κ = 8πG is the +gravitational constant, N is the non-vanishing and otherwise arbitrary lapse function. +The variable Ω describes the isotropic geometry, whereas β± describe distortions to +the isotropic geometry and are referred to as the anisotropic variables. +In what follows we set n = 1 and 2κ = V0. The spacetime variables used in eq. (37) +have the following metric interpretation: +Ω = 1 +3 ln a1a2a3, +β+ = 1 +6 ln a1a2 +a2 +3 +, +β− = +1 +2 +√ +3 +ln a1 +a2 +. +(38) +The Hamiltonian constraint (37) is a sum of the isotropic and anisotropic parts, +C = −Ciso + Cani, where (up to a factor) +Ciso = p2 +Ω + 36e4Ω, +(39) +Cani = p2 + 36e4ΩV (β) , +(40) +and the anisotropy potential V (β) reads as: +V (β) = e4β+ +3 +�� +2 cosh(2 +√ +3β−) − e−6β+�2 +− 4 +� ++ 1 . +(41) +For the purpose of quantisation, we redefined partially the phase space variables +[23, 24] of the isotropic geometry (scale factor a = eΩ) by introducing the canonical +12 + +pair (q, p) := (a3/2, 2pa/(3√a). This leads to a more convenient form of Hamiltonian +(37): +C = 3 +16p2 + 3 +4q2/3 − Hq, +(42) +where +Hq = +1 +12q2(p2 ++ + p2 +−) + 3 +4q2/3V (β) +(43) +is the anisotropy Hamiltonian. +B. +The anisotropy Hamiltonian Hq +We proved in [1] the discreteness of the spectrum of the quantum Hamiltonian +ˆHq originated by the “exact” Bianchi IX anisotropic potential V (β), despite the +existence of three non-confining canyons of this potential. +These canyons could +suggest the existence of some continuum spectrum, but it is not the case. Moreover +it was also shown on the classical level [4, 5] that those canyons do not contribute +either to the chaotic behaviour of the system. This validates all possible implemen- +tations of approximations of the potential removing the three non-confining canyons. +Furthermore, we (with our co-authors) have shown in [2] that the anisotropy po- +tential V (β) in (43) can be decomposed into two parts, each corresponding to a +different Toda potential. Therefore, approximating Mixmaster with Toda system +should preserve key properties of the model. In a supplementary step, we showed +in [2] that after applying a Weyl-Heisenberg integral quantisation procedure to that +classical potential (instead of a canonical quantisation), one of the parts of the po- +tential become dominant whereas the other one becomes negligible. In other words, +we showed that in the first order of approximation: +V 0(β) ≈ D16 +3 +� +e4 +√ +3β−+4β+ + e−4 +√ +3β−+4β+ + e−8β+� +, +(44) +where D = e2/σ2 is reminiscent of the applied quantisation procedure, where σ stands +for weight of the applied Gaussian distribution coming from Weyl-Heisenberg quan- +tisation. One may introduce new variables q1, q2, q3 such that: +q1 − q2 = 4 +√ +3β− + 4β+, +q2 − q3 = −4 +√ +3β− + 4β+, +(45) +13 + +that lead to +V 0(q1, q2, q3) ≈ D16 +3 +� +eq1−q2 + eq2−q3 + eq3−q1� +≡ D16 +3 VT(q1, q2, q3), +(46) +where VT is 3-body Toda lattice potential VT(q1, q2, q3) = eq1−q2 + eq2−q3 + eq3−q1. +Therefore, after suitable position, momenta and time reparameterisation (see [2]), +the Bianchi IX quantum anisotropy Hamiltonian (43) may be written in first order +of approximation as +ˆH0 +q = p2 +q2 + Kq2/3VT(β), +(47) +where K is a fixed constant, p = −iℏ∇β and VT(⃗β) = e4 +√ +3β−+4β+ + e−4 +√ +3β−+4β+ + +e−8β+. +Therefore, we have +ˆH0 +q = Kq2/3 + +−1 +2 +� +ℏ +q4/3� +K/2 +�2 +∇2 +β + VT(⃗β) + + . +(48) +On the other hand, a periodic 3-body Toda lattice is actually a system with 2 de- +grees of freedom, due to periodicity condition. In the paper [15] it was presented how +to reduce the classical 3-Toda Hamiltonian (2) to a Hamiltonian for a 2 dimensional +system via a suitable change of position and corresponding momenta variables and +also time reparameterisation similar to transformation (45). Therefore, the 3-Toda +classical Hamiltonian (2) might be written as well at the quantum level (canonical +quantisation) in a two dimensional form as follows +ˆHT = −ℏ2 +2 ∇2 +β + VT(β), +(49) +where we denoted by β the new configurations variables (45). Therefore, we obtain +that the quantum Hamiltonian ˆHq(q, ℏ) formally fulfils +ˆHq(q, ℏ) = Kq2/3 ˆHT(ℏ/ηq) +with +ηq = q4/3� +K/2, +(50) +Thus, the semi-classical EKB quantisation of Toda can be easily extended to Toda- +Bianchi IX, by replacing ℏ by ℏ/η in (17) and then by renormalising the obtained +energies by Kq2/3. +14 + +VI. +SUMMARY +We proposed a novel parameterisation of the Toda lattice, the new varable descrip- +tion simplified significantly calculation action integrals coming from semi-classical +EKB quantisation method. The new variables helped to ged rid of a large part of +the initial numerical problem, replacing it by explicit formula (even if a final nu- +merical step is unavoidable). This reduces significantly the process of finding energy +levels of a 3-particle Toda system. +Regarding the energy levels of the “exact” quantum anisotropy Hamiltonian of +Mixmaster a semi-classical quantization like EKB seems interesting since this po- +tential is “not so far” from Toda potential, and it is known that the semi-classical +approach is reliable for Toda system[19]. +On a side note, the explored similarity between the anisotropy potential and the +Toda potential was already used on the classical level, to introduce the so-called dis- +turbed Toda lattices [16]. Nevertheless, due to the integrability of the Toda latices +this similarity seems to be too small to be useful in the context of the classical dy- +namics (although see [17]). However, quantum mechanics and classical mechanics are +very different. For example, the Helium atom made of a nucleus and two electrons is +a chaotic system at the classical level, but reliable quantum approximations of this +system can be obtained without making reference of the chaotic classical behaviour. +So it is not unreasonable to look at quantum Toda approximations of quantum Mix- +master. Furthermore, the Weyl-Heisenberg quantization procedure that we used in +[2] amplifies the relative contribution of the positive Toda potential that is shown +dominate over the negative one. Thus, the latter can be neglected in the first approx- +imation and the approximate evolution of the anisotropic variables for a fixed volume +of the universe becomes integrable. The negative Toda potential can be treated as a +quantum perturbation to the integrable dynamics. +VII. +ACKNOWLEDGEMENTS +The project is cofinanced by the Polish National Agency for Academic Exchange +and PHC POLONIUM 2019 (Grant No. 42657QJ). +Appendix A: Analysis of the equation x3 − 3x = a +This 3rd order polynomial admits three solutions only if |a| < 2, therefore let us +define a as +a = 2 cos φ , +15 + +where we can assume without loss 0 < φ < π. +The three solutions of the title equation are now explicit in terms of φ and can be +written as x1 < x2 < x3: +x1 = −2 cos π − φ +3 +, +x2 = −2 cos π + φ +3 +, +x3 = 2 cos φ +3 . +(A1) +Furthermore: +for: +0 < φ < π, +− cos π − φ +3 +< − cos π + φ +3 +< cos φ +3 +and +x1 < x2 < x3, +which comes from the properties of the cosine function. +[1] H. Bergeron, E. Czuchry, J-P. Gazeau, P. Ma�lkiewicz, Spectral properties of the quan- +tum Mixmaster universe, Phys. Rev. D 96, 043521 (2017). +[2] H. Bergeron, E. Czuchry, J.-P. Gazeau, P. Ma�lkiewicz, +Integrable Toda system as +a quantum approximation to the anisotropy of Mixmaster, Phys. Rev. D 98, 083512 +(2018). +[3] C. W. Misner, Mixmaster Universe, Phys. Rev. Lett. 22, 1071 (1969); Quantum Cos- +mology, Phys. Rev. 186, 1319 (1969). +[4] J. M. Heinzle, C. Uggla and N. Rohr, The Cosmological billiard attractor, Adv. Theor. +Math. Phys. 13, no. 2, 293 (2009). +[5] J. M. Heinzle and C. Uggla, Mixmaster: Fact and Belief, Class. Quant. Grav. 26, +075016 (2009). +[6] N. J. Cornish and J. J. Levin, The Mixmaster universe is chaotic, Phys. Rev. Lett. +78, 998 (1997). +[7] A. Latifi, M. Musette, R. Conte, The Bianchi IX (mixmaster) cosmological model is +not integrable, Phys. Lett. A 194 83 (1994). +[8] G. Montani, M. V. Battisti, R. Benini, and G. Imponente, Classical and quantum +features of the mixmaster singularity, Int. J. Mod. Phys. A 23, 2353 (2008), +[9] D. Craig and J. B. Hartle, Generalized quantum theory of recollapsing homogeneous +cosmologies, Phys. Rev. D 69, 123525 (2004). +[10] T. Damour and P. Spindel, Quantum supersymmetric Bianchi IX cosmology, Phys. +Rev. D 90, 103509 (2014). +[11] H. Bergeron, E. Czuchry, J-P. Gazeau, P. Ma�lkiewicz, Vibronic framework for quantum +mixmaster universe, PRD 93, 064080 (2016) +[12] H. Bergeron, E. Czuchry, J-P. Gazeau, P. Ma�lkiewicz, Quantum Mixmaster as a Model +of the Primordial Universe, Universe 6(1), 7 (2020) +16 + +[13] M. Flaschka, The Toda lattice. II. Existence of integrals, Phys. Rev. B 9, 1924 (1974). +[14] M. H´enon, Integrals of the Toda lattice, Phys. Rev. B 9, 1926 (1974). +[15] J. Ford, S.D. Stoddard and J.S. Turner, On the Integrability of the Toda Lattice, +Progress of Theoretical Physics 50, 1547 (1973). +[16] O. I. Bogoyavlensky, On perturbations of the periodic Toda lattice, Comm. Math. Phys. +51, no. 3, 201 (1976). +[17] M. Biesiada, M. Szyd�lowski, Mixmaster cosmological models as disturbed Toda lattices, +Phys. Lett. A 160, 123 (1991). +[18] M. C. Gutzwiller, The Quantum mechanical Toda lattice, Ann. Phys. 124, 347 (1980); +The Quantum mechanical Toda lattice II, Ann. Phys. 133, 304 (1981). +[19] A. Matsuyama, Numerical study of the quantum mechanical Toda lattice, Phys. Lett. +A 161, 121 (1991); Periodic Toda lattice in quantum mechanics, Annals of Phys. 220, +300 (1992). +[20] S. Isola, H. Kantz and R. Livi, On the quantization of the three-particle Toda lattice, +J. Phys. A 24, 3061 (1991). +[21] M. Toda, Theory of Nonlinear lattices, J. Phys. Soc. Japan 22, 431 (1967). +[22] W. E. Ferguson Jr., H. Flaschka and D. W. Mclaughlin, Nonlinear Normal Modes for +the Toda Chain, J. Comp. Phys. 45, 157 (1982). +[23] H. Bergeron, E. Czuchry, J.-P. Gazeau, P, Ma�lkiewicz, W Piechocki, Smooth quantum +dynamics of the mixmaster universe Phys. Rev. D 92, 061302(R) (2015). +[24] H. Bergeron, E. Czuchry, J.-P. Gazeau, P. Ma�lkiewicz and W. Piechocki, Singularity +avoidance in a quantum model for mixmaster universe, Phys. Rev. D. 92, 124018 +(2015). +17 + diff --git a/ttE2T4oBgHgl3EQffwca/content/tmp_files/load_file.txt b/ttE2T4oBgHgl3EQffwca/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6eb9292d45b09655616dae767cff584529dbd75b --- /dev/null +++ b/ttE2T4oBgHgl3EQffwca/content/tmp_files/load_file.txt @@ -0,0 +1,410 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf,len=409 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content='03929v1 [gr-qc] 10 Jan 2023 Semi-Classical quantisation of 3-particles Toda lattice augmented Application to the Mixmaster anisotropy Hamiltonian Herv´e Bergeron∗ Univ Paris-Saclay,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' ISMO,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' UMR 8214 CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' 91405 Orsay,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' France and APC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Universit´e Paris Diderot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Sorbonne Paris Cit´e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' 75205 Paris Cedex 13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' France Ewa Czuchry† National Centre for Nuclear Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Pasteura 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' 02-093 Warszawa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Poland Abstract Usual approaches to quantisation of a 3-Toda lead to numerical calculations requiring many steps that can be time consuming to insure their reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' In order to reduce as much as possible the numerical part of the EKB quantisation procedure, and then to ease numerical calculations, we propose a reformulation of the mathematical framework with more adapted variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' The resulting equations and procedure might be easily imple- mented in a short Mathematica code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' This more explicit framework will be useful for studying quantum Toda-Bianchi IX models in quantum cosmology where the true Bianchi IX anisotropy Hamiltonian can be approximated by a 3-particle Toda system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' ∗ herve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content='bergeron@universite-paris-saclay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content='fr † ewa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content='czuchry@ncbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content='gov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content='pl Typeset by REVTEX 1 CONTENTS I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Introduction 2 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Classical Toda lattice and its semi-classical quantization 4 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Classical formulation 4 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' EKB quantization 6 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' New parametrisation for 3-Toda system 8 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' New parametrisation of the roots µ+ 1 and µ+ 2 9 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' New parametrisation of the roots µ− 2 and µ− 3 10 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' New parametrisation of the actions I1 and I2 11 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Application to the semi-classical quantisation of Toda-Bianchi IX anisotropy Hamiltonian 12 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Bianchi IX general framework 12 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' The anisotropy Hamiltonian Hq 13 VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Summary 15 VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Acknowledgements 15 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Analysis of the equation x3 − 3x = a 15 References 16 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' INTRODUCTION In our recent papers [1, 2] we have demonstrated that the quantum behaviour of a refined model of the earliest Universe, namely the so-called “Mixmaster universe”, might be approximated by a 3-particle Toda system, usually arriving in solid state physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' The classical dynamics of the “Mixmaster universe” (or Bianchi IX model) was studied by C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Misner in the canonical formalism [3] at the end of the sixties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' It involves a Hamiltonian that is formally identical to that of a particle moving in a 3D Minkowski spacetime in a time-dependent, exponentially steep and triangle-like potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Many mathematical studies have been devoted to the classical evolution of the system and have led to some important results on the asymptotic behaviour, the non-integrability or the chaotic behaviour (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' [4]–[7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' 2 On the other hand, the quantum behaviour of Mixmaster remains not entirely understood despite many interesting studies [1, 8–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' The difficulty lies mainly in the very different possible dynamical regimes induced by the elaborate spatial dependence of the anisotropy potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' We (with our co-authors) have shown in [2] that the classical Mixmaster anisotropy potential can be viewed as a difference between two 3-particle Toda potentials, thus paving the way to a possible analytically solvable approximation for the problem of the quantum version of the Bianchi IX model (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' 5 for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' This approximation should be valid in the unexplored region in-between the known harmonic and steep-wall approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content='1 The Toda lattice system is a one-dimensional chain consisting of equal masses interacting via exponential forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' This system was shown to be completely inte- grable [13] –[15] as it has as many constants of motion as the numbers of degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' The quantisation of a Toda lattice was first performed by Gutzwiller [18] who formulated a systematic recursive way of constructing the eigenfunctions and explicitly constructed them for N = 2, 3 and 4 periodic Toda systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' However it took much time until numerical results appeared [19] and [20], as they needed calculational power and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Canonical quantisation was performed after direct diagonalization of the Hamiltonian, and results were classified with respect to repre- sentation of the permutation group S3 under which the Hamiltonian is invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' It was found that the numerically obtained eigenvalues fulfil Gutzwiller’s quantisation conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Furthermore the semiclassical Einstein-Keller-Brillouin method of quan- tisation (EKB quantisation) was proved to provide results in good agreement with exact ones [19], even for the first levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' However the usual mathematical formula- tion of the EKB quantisation in the case of 3-Toda system involved several implicit functions that seem to impose a pure numerical approach with not simple entan- gled steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Therefore, at first sight, it seems difficult to export this procedure in a straightforward way in the more complex framework of Bianchi IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' The aim of this note is to prove that the mathematical formulation of the 3- Toda EKB quantisation can be significantly eased by a new parametrisation of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' The outline of the paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' II we recall the main features of the classical 3-body Toda lattice and its known formalism for EKB semi-classical quan- tisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' This section allows to introduce all the notations useful for the remainder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' III we introduce a new parameterisation for the Toda system which we use in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' IV for new parameterisations of the action integrals which turn to be easily calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' V we discuss the possible application of the obtained procedure to quantum Mixmaster with the anisotropy Hamiltonian where a pure Toda potential is used but with restored necessary dependences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' We conclude in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' 1 The latter corresponding respectively to the large volume and low anisotropy excitation level or to the small volume and high anisotropy excitation level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' 3 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' CLASSICAL TODA LATTICE AND ITS SEMI-CLASSICAL QUANTI- ZATION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Classical formulation The Toda lattice system is a one-dimensional chain consisting of equal masses interacting via exponential forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' There are two types of those, an open lattice one and a periodic one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' The main difference is that in the periodic lattice the first and the last particles are coupled whereas in the open lattice they are not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' The Hamiltonian for the periodic Toda system is like for a system of N equal-mass particles interacting via exponential potential: H = 1 2 N � k=1 p2 k + N � k=1 e−(qk−qk+1), (1) with periodicity condition q0 ≡ qN and q1 ≡ qN+1, where qi are generalised positions and pi their corresponding conjugate momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' The simplest nontrivial periodic crystal is the periodic 3-particle Toda system with Hamiltonian as follows: H = 1 2 � p2 1 + p2 2 + p2 3 � + e−(q1−q3) + e−(q2−q1) + e−(q3−q2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' (2) The equations of motion for this system may be written as Lax’s equation [13]: dL dt = [M, L] , (3) where matrices L and M read as follows L := \uf8ee \uf8f0 b1 a1 a3 a1 b2 a2 a3 a2 b3 \uf8f9 \uf8fb , M := 1 2 \uf8ee \uf8f0 0 a1 −a3 −a1 0 a2 a3 −a2 0 \uf8f9 \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' (4) Elements of the symmetric matrix L and the skew one M are functions of positions and momenta of a Toda system: ai := 1 2e(qi−qi+1)/2 and bi := pi 2 where i = 1, 2, 3, with q0 ≡ q3, q4 ≡ q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' (5) The matrices L and M form a so called Lax pair, therefore the eigenvalues of L and also the coefficients of its characteristic polynomial Ai, are constants of motion: det(2µI − 2L) ≡ (2µ)3 + A1(2µ)2 + A2(2µ) + A3 − 2 (6) 4 In the center of mass system P := p1 + p2 + p3 ≡ 0 those coefficients simplify to: A1 = −P = 0, A2 = 1 2P 2 − H =: −E, (7) A3 = Π3 i=1pi − 3 � i=1 pieqi+1−qi+2 ≡ 8Π3 i=1bi − 8 3 � i=1 bia2 i+1 =: A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' (8) Therefore, conserved quantities A1 and A2 have physical interpretations, respectively the total momentum P and the energy of the system E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' The third conserved quantity A does not have such explicit simple physical meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Hamiltonian (2) is invariant under the transformations of the dihedral group D3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' The dihedral group DN is the group of symmetry of the N-sided regular polygon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' The particular group D3 has two kinds of representations, two one-dimensional rep- resentations A1, A2 and a two-dimensional representation E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' The Toda lattice system is integrable even if the Hamiltonian is not separable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' we cannot separate its variables into the explicit ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' However, due to integrability, we can rewrite the Hamiltonian by means of the canonical transformation in terms of the action-angle variables (Ii, θi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' For our n = 3 Toda chain we have i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Moreover, it is possible to introduce the canonical conjugate variables (µi, νi), where i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' The action-angle variables arise in integrable systems and can be written as Ii := � νi(µi)dµi, (9) where integration is performed along the closed loop trajectory in phase space, here over a period of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' It was calculated in [21] that for a Toda lattice conjugated momenta (such that Poisson brackets fulfil {νi, µj} = δij) are following: ν1,2 = 2 ln |1 2(∆(µi) ± � ∆(µi)2 − 4|, (10) where ∆(µ) := det(2µI − 2L) + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' (11) In the the center of mass system P := p1 + p2 + p3 ≡ 0 this function reads as: ∆(µ) = 8µ3 − 2Eµ + A, (12) The variables µi are defined only on a limited interval specified by the inequality |∆(µ)| ≥ 2 and so also action integrals Ii are taken over that regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' The expression |∆(µ)| ≥ 2 is clearly a 3rd order polynomial and thus has 2 intervals where inequality 5 |∆(µ)| ≥ 2 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Since E ≥ 0, ∆(µ) admits a relative maximum for µ = −µm and a relative minimum for µ = +µm with µm = � E 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' µ −∞ −µm +µm +∞ ∆(−µm) +∞ ∆(µ) ր ց ր −∞ ∆(µm) (13) In order that the inequality |∆(µ)| ≥ 2 holds on a non-zero measure interval of µ each of the two equations ∆(µ) = 2 and ∆(µ) = −2 must have three distinct real roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' This is only possible if ∆(−µm) > 2 and ∆(µm) < −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' This holds for following values of E and A: E > 3 and |A| < 2 3 √ 3E3/2 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' (14) Therefore, these two conditions have to be fulfilled to develop semi-classical quanti- sation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' We call µ+ 1 < µ+ 2 < µ+ 3 the solutions of ∆(µ) = +2 and µ− 1 < µ− 2 < µ− 3 the three solutions of ∆(µ) = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' The canonical conjugate momenta ν+ i and ν− i such that {µ+ i , ν+ j } = δij = {µ− i , ν− j } are given by: ν+ i = 2 ln | 1 2(∆(µ+ i ) + � ∆(µ+ i )2 − 4)|, ν− i = 2 ln | 1 2(∆(µ− i ) − � ∆(µ− i )2 − 4)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' (15) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' EKB quantization It was shown in [22] that for a 3 particle Toda lattice the corresponding actions I1(E, A) and I2(E, A) read as I1(E, A) = 4 � µ+ 2 µ+ 1 arcosh |∆(µ)| 2 dµ , I2(E, A) = 4 � µ− 3 µ− 2 arcosh |∆(µ)| 2 dµ , (16) where arcosh(x) = ln(x + √ x2 − 1) for x ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' As we see, the energy E and the conserved quantity A, are given implicitly in terms of the above actions Ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' 6 Furthermore, from (14) the range of possible values of A is dependent of the value of E and obviously the boundaries of integrals are not defined in an explicit way as functions of E and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Now the semiclassical quantisation performed through EKB (Einstein-Keller- Brillouin) formulation can be obtained as follows: inverting (at least formally) equa- tions (16), the classical Hamiltonian can be expressed in terms of actions Ii, and the semiclassical quantum energies are finally given by the substitutions Ii �→ (ni+1/2)h in the expression of the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' In other terms EKB quantisation consists in finding the solutions En1,n2 and An1,n2 of the system of equations I1(En1,n2, An1,n2) = 2πℏ � n1 + 1 2 � , I2(En1,n2, An1,n2) = 2πℏ � n2 + 1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' (17) for integer values of n1 and n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Equations (16) might be a bit simplified by using the definitions of arcosh and then performing integration by parts: I1(E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' A) = 4 � µ+ 2 µ+ 1 log ���� 1 2 � ∆(µ) + � ∆(µ)2 − 4 ����� dµ = = −4 � µ+ 2 µ+ 1 µ∆′(µ) � ∆(µ)2 − 4 dµ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' I2(E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' A) = 4 � µ− 3 µ− 2 log ���� 1 2 � ∆(µ) + � ∆(µ)2 − 4 ����� dµ = = −4 � µ− 3 µ− 2 µ∆′(µ) � ∆(µ)2 − 4 dµ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' (18) Obviously the solutions can only be found numerically and the difficulties of the procedure lie mainly in: (a) the non-complete independence of parameters E and A due to conditions (14),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' (b) the implicit definitions of the boundaries of integrals I1 and I2 in terms of E and A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' (c) the dependence in E and A of the length of integration domains (lengths that can be very large),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' 7 (d) the final numerical procedure to solve the system (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' To obtain the sought quantities En1,n2 and An1,n2 the authors in [20] change in a controlled manner the values of E and A, calculating the corresponding actions and retaining those values of E and A which satisfy conditions (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' The author of [19] used the results of a direct canonical quantisation as initial values for solving the system (17) via the simplex algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Both approaches are numerically demanding, specially as we see in (35) the integrated function is divergent at both integral limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' That demands a very accuracy of numerical integration!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Our approach presented in this paper consists in removing first the difficulties (a), (b) and (c) on analytical level by a new parametrisation of the problem, and performing numerical calculations only for the final step (d) but in much simplified settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' NEW PARAMETRISATION FOR 3-TODA SYSTEM The main point of our approach consists in introducing a new parametrisa- tion of the problem in terms of two completely independent parameters (α, θ) ∈ ]0, +∞[×]0, π[ in place of the usual dynamical constants E and A constrained by the conditions (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' These two parameters (α, θ) are defined such that: E := 3 cosh4/3 α, A := −2 cos θ sinh2 α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' (19) Due to internal properties of the hyper/trygonometric functions the conditions (14) are automatically fulfilled with a one to one correspondence, as ∀α, θ :∈ R cosh α ≥ 1 and | cos θ| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' This solves the first point (a) mentioned at the end of the previous section II B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' The second problem is that we have to deal with a problematic integral with no predefined bounded length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' In order to solve that let us rescale the variable µ which appears in ∆(µ) and defining a new variable ν as follows µ := νµm (20) where µm is the value of µ where ∆(µ) reaches its relative extremal value, namely µm = � E/12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' With the new parametrisation (19) we have: µm = 1 2 cosh2/3 α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' (21) 8 Finally, the expression ∆(νµm) reads as: ∆(νµm) = (cosh2 α) (ν3 − 3ν) − 2 cos θ sinh2 α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' (22) In the remainder we will prove that the maximal range of ν in integrals will be reduced to ν ∈ [−2, +2], solving the point (c) mentioned in section II B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' New parametrisation of the roots µ+ 1 and µ+ 2 In this section we focus on the point (b) of section II B, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' how to obtain an explicit mathematical expression of the boundaries of integrals (16) in terms of the new parameters (α, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Let us recall that the µ+ i involved in the boundaries of integrals are solutions of ∆(µi) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Using the new parametrisation of (22) we obtain in terms of ν the equation ν3 − 3ν = 2 � 1 − 2 sin2 θ 2 tanh2 α � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' (23) In the Appendix A we have described a procedure of finding explicit solutions of the 3rd order polynomial of the form ν3 − 3ν = 2 cos Φ leading to the explicit solutions in ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' In order to apply this procedure to the above equation let us introduce a new parameter Φ+ ∈]0, π[ such that 1 − 2 sin2 θ 2 tanh2 α = cos Φ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' (24) This is equivalent to the equation sin2 θ 2 tanh2 α = sin2 Φ+ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' (25) Because α > 0, θ/2, Φ+/2 ∈]0, π/2[ implying sin(θ/2), sin(Φ+/2) > 0, we can sim- plify this equation as follows sin θ 2 tanh α = sin Φ+ 2 , (26) and then Φ+ = 2 arcsin � sin θ 2 tanh α � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' (27) 9 Introducing the parameters ν+ i such that µ+ i = ν+ i µm and using the solutions (A1) of appendix A, we find that the sought values ν+ 1 and ν+ 2 needed for the definition of I1 (which is now a function of α and θ in place of E and A) are explicitly: ν+ 1 = −2 cos π − Φ+ 3 < ν+ 2 = −2 cos π + Φ+ 3 , (28) where Φ+ defined in (27) is an explicit function of α and θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' New parametrisation of the roots µ− 2 and µ− 3 Similarly let us recall that the µ− i are solutions of ∆(µi) = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Using again the parametrisation of (22) we obtain in terms of ν the equation ν3 − 3ν = −2 � 1 − 2 cos2 θ 2 tanh2 α � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' (29) Now let us introduce a new parameter Φ− similar to the one used previously, namely we would like to find Φ− ∈]0, π[ such that − � 1 − 2 cos2 θ 2 tanh2 α � = cos Φ−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' (30) This is equivalent to cos2 θ 2 tanh2 α = cos2 Φ− 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' (31) Because α > 0, θ/2, Φ−/2 ∈]0, π/2[ and then cos(θ/2), cos(Φ−/2) > 0, we can simplify this equation as cos θ 2 tanh α = cos Φ− 2 , (32) and then we have Φ− = 2 arccos � cos θ 2 tanh α � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' (33) Introducing the parameters ν− i such that µ− i = ν− i µm and using the solutions (A1) of appendix A, we find that the sought values ν− 2 and ν− 3 needed for the definition of I2 (which is now a function of α and θ in place of E and A) are explicitly: ν− 2 = −2 cos π + Φ− 3 < ν− 3 = 2 cos Φ− 3 , (34) where Φ− defined in (33) is an explicit function of α and θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' 10 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' NEW PARAMETRISATION OF THE ACTIONS I1 AND I2 Using the definitions (16) of I1 and I2 and using the change of variable µ = νµm in the integrals, we end with the formula I1(α, θ) = 4µm � ν+ 2 ν+ 1 arcosh |∆(µmν)| 2 dν I2(α, θ) = 4µm � ν− 3 ν− 2 arcosh |∆(µmν)| 2 dν , (35) where µm given in (21);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' ν+ 1 , ν+ 2 given in (27), (28);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' ν− 2 , ν− 3 given in (33) (34) and the function ν �→ ∆(νµm) given in (22) are explicit functions of α and θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' We may observe that in our new parameterization the integrated functions in the expressions for I1 and I2 do not exhibit any singular behavior!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' For the final step (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' solving the EKB system (17) for a given pair (n1, n2)), these integrals (35) must be computed numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' But since the function ν �→ arcosh ∆(νµm) of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' (22) has no singular behaviour on each interval and the intervals [ν+ 1 , ν+ 2 ], [ν− 2 , ν− 3 ] have a maximal length of 4, the numerical estimates are easy and reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' The search of semi-classical quantised energies can now be done numerically with- out much effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' We used a program Mathematica running on a laptop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' After quick numerical calculations using build-in functions we obtained solutions in terms of α and θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Finally, we needed only to use the definition of E and A in terms of α and θ in (19) to go back to the sought quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' As a proof of efficiency of our procedure, we can compare the values obtained with our method (with few seconds of computation on a laptop) with the ones calculated in [19] (where ℏ = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' The results are summarised in the table (36): the agreement is perfect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Symmetry Matsuyama [19] Our code A or E n1 n2 E A E A A 0 0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content='7748 0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content='7748 0 A 1 1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content='5854 0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content='5854 0 A 3 0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content='8558 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content='2294 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content='8558 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content='2293 A 4 4 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content='9378 0 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content='9378 0 A 7 1 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content='6452 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content='0562 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content='6452 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content='0562 E 1 0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content='6686 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content='4110 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content='6686 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content='4110 E 0 2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content='7002 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content='4897 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content='7002 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content='4897 E 1 3 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content='8280 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content='9356 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content='8279 -6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content='9356 E 5 1 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content='5336 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content='6194 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content='5336 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content='6194 (36) 11 Remark on a particular situation: for n1 = n2 (which belongs to the symmetry “A”) it is possible to prove directly that A = 0, which means in terms of our variables that θ = π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Therefore, only the energy E (or the parameter α) is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Then it is better from a numerical point of view to solve a unique equation either the one involving I1 or the one in I2, imposing by hand θ = π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' APPLICATION TO THE SEMI-CLASSICAL QUANTISATION OF TODA- BIANCHI IX ANISOTROPY HAMILTONIAN A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Bianchi IX general framework Let us first recall Hamiltonian formulation of the Bianchi type IX model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' The respective Hamiltonian constraint in the Misner variables reads [3]: HB9 = Ne−3Ω 24 �2κ V0 �2 � −p2 Ω + p2 + 36 �V0 2κ �3 n2e4Ω[V (β) − 1] � , (Ω, pΩ, β, p) ∈ R6, (37) where β := (β+, β−), p := (p+, p−), V0 = 16π2 n3 is the fiducial volume, κ = 8πG is the gravitational constant, N is the non-vanishing and otherwise arbitrary lapse function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' The variable Ω describes the isotropic geometry, whereas β± describe distortions to the isotropic geometry and are referred to as the anisotropic variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' In what follows we set n = 1 and 2κ = V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' The spacetime variables used in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' (37) have the following metric interpretation: Ω = 1 3 ln a1a2a3, β+ = 1 6 ln a1a2 a2 3 , β− = 1 2 √ 3 ln a1 a2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' (38) The Hamiltonian constraint (37) is a sum of the isotropic and anisotropic parts, C = −Ciso + Cani, where (up to a factor) Ciso = p2 Ω + 36e4Ω, (39) Cani = p2 + 36e4ΩV (β) , (40) and the anisotropy potential V (β) reads as: V (β) = e4β+ 3 �� 2 cosh(2 √ 3β−) − e−6β+�2 − 4 � + 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' (41) For the purpose of quantisation, we redefined partially the phase space variables [23, 24] of the isotropic geometry (scale factor a = eΩ) by introducing the canonical 12 pair (q, p) := (a3/2, 2pa/(3√a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' This leads to a more convenient form of Hamiltonian (37): C = 3 16p2 + 3 4q2/3 − Hq, (42) where Hq = 1 12q2(p2 + + p2 −) + 3 4q2/3V (β) (43) is the anisotropy Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' The anisotropy Hamiltonian Hq We proved in [1] the discreteness of the spectrum of the quantum Hamiltonian ˆHq originated by the “exact” Bianchi IX anisotropic potential V (β), despite the existence of three non-confining canyons of this potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' These canyons could suggest the existence of some continuum spectrum, but it is not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Moreover it was also shown on the classical level [4, 5] that those canyons do not contribute either to the chaotic behaviour of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' This validates all possible implemen- tations of approximations of the potential removing the three non-confining canyons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Furthermore, we (with our co-authors) have shown in [2] that the anisotropy po- tential V (β) in (43) can be decomposed into two parts, each corresponding to a different Toda potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Therefore, approximating Mixmaster with Toda system should preserve key properties of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' In a supplementary step, we showed in [2] that after applying a Weyl-Heisenberg integral quantisation procedure to that classical potential (instead of a canonical quantisation), one of the parts of the po- tential become dominant whereas the other one becomes negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' In other words, we showed that in the first order of approximation: V 0(β) ≈ D16 3 � e4 √ 3β−+4β+ + e−4 √ 3β−+4β+ + e−8β+� , (44) where D = e2/σ2 is reminiscent of the applied quantisation procedure, where σ stands for weight of the applied Gaussian distribution coming from Weyl-Heisenberg quan- tisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' One may introduce new variables q1, q2, q3 such that: q1 − q2 = 4 √ 3β− + 4β+, q2 − q3 = −4 √ 3β− + 4β+, (45) 13 that lead to V 0(q1, q2, q3) ≈ D16 3 � eq1−q2 + eq2−q3 + eq3−q1� ≡ D16 3 VT(q1, q2, q3), (46) where VT is 3-body Toda lattice potential VT(q1, q2, q3) = eq1−q2 + eq2−q3 + eq3−q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Therefore, after suitable position, momenta and time reparameterisation (see [2]), the Bianchi IX quantum anisotropy Hamiltonian (43) may be written in first order of approximation as ˆH0 q = p2 q2 + Kq2/3VT(β), (47) where K is a fixed constant, p = −iℏ∇β and VT(⃗β) = e4 √ 3β−+4β+ + e−4 √ 3β−+4β+ + e−8β+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Therefore, we have ˆH0 q = Kq2/3 \uf8ee \uf8f0−1 2 � ℏ q4/3� K/2 �2 ∇2 β + VT(⃗β) \uf8f9 \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' (48) On the other hand, a periodic 3-body Toda lattice is actually a system with 2 de- grees of freedom, due to periodicity condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' In the paper [15] it was presented how to reduce the classical 3-Toda Hamiltonian (2) to a Hamiltonian for a 2 dimensional system via a suitable change of position and corresponding momenta variables and also time reparameterisation similar to transformation (45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Therefore, the 3-Toda classical Hamiltonian (2) might be written as well at the quantum level (canonical quantisation) in a two dimensional form as follows ˆHT = −ℏ2 2 ∇2 β + VT(β), (49) where we denoted by β the new configurations variables (45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Therefore, we obtain that the quantum Hamiltonian ˆHq(q, ℏ) formally fulfils ˆHq(q, ℏ) = Kq2/3 ˆHT(ℏ/ηq) with ηq = q4/3� K/2, (50) Thus, the semi-classical EKB quantisation of Toda can be easily extended to Toda- Bianchi IX, by replacing ℏ by ℏ/η in (17) and then by renormalising the obtained energies by Kq2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' 14 VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' SUMMARY We proposed a novel parameterisation of the Toda lattice, the new varable descrip- tion simplified significantly calculation action integrals coming from semi-classical EKB quantisation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' The new variables helped to ged rid of a large part of the initial numerical problem, replacing it by explicit formula (even if a final nu- merical step is unavoidable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' This reduces significantly the process of finding energy levels of a 3-particle Toda system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Regarding the energy levels of the “exact” quantum anisotropy Hamiltonian of Mixmaster a semi-classical quantization like EKB seems interesting since this po- tential is “not so far” from Toda potential, and it is known that the semi-classical approach is reliable for Toda system[19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' On a side note, the explored similarity between the anisotropy potential and the Toda potential was already used on the classical level, to introduce the so-called dis- turbed Toda lattices [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Nevertheless, due to the integrability of the Toda latices this similarity seems to be too small to be useful in the context of the classical dy- namics (although see [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' However, quantum mechanics and classical mechanics are very different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' For example, the Helium atom made of a nucleus and two electrons is a chaotic system at the classical level, but reliable quantum approximations of this system can be obtained without making reference of the chaotic classical behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' So it is not unreasonable to look at quantum Toda approximations of quantum Mix- master.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Furthermore, the Weyl-Heisenberg quantization procedure that we used in [2] amplifies the relative contribution of the positive Toda potential that is shown dominate over the negative one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Thus, the latter can be neglected in the first approx- imation and the approximate evolution of the anisotropic variables for a fixed volume of the universe becomes integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' The negative Toda potential can be treated as a quantum perturbation to the integrable dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' ACKNOWLEDGEMENTS The project is cofinanced by the Polish National Agency for Academic Exchange and PHC POLONIUM 2019 (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' 42657QJ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Appendix A: Analysis of the equation x3 − 3x = a This 3rd order polynomial admits three solutions only if |a| < 2, therefore let us define a as a = 2 cos φ , 15 where we can assume without loss 0 < φ < π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' The three solutions of the title equation are now explicit in terms of φ and can be written as x1 < x2 < x3: x1 = −2 cos π − φ 3 , x2 = −2 cos π + φ 3 , x3 = 2 cos φ 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' (A1) Furthermore: for: 0 < φ < π, − cos π − φ 3 < − cos π + φ 3 < cos φ 3 and x1 < x2 < x3, which comes from the properties of the cosine function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' [1] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Bergeron, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Czuchry, J-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Gazeau, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Ma�lkiewicz, Spectral properties of the quan- tum Mixmaster universe, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' D 96, 043521 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' [2] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Bergeron, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Czuchry, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Gazeau, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Ma�lkiewicz, Integrable Toda system as a quantum approximation to the anisotropy of Mixmaster, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' D 98, 083512 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' [3] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Misner, Mixmaster Universe, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' 22, 1071 (1969);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Quantum Cos- mology, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' 186, 1319 (1969).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' [4] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Heinzle, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Uggla and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Rohr, The Cosmological billiard attractor, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' 13, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' 2, 293 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' [5] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Heinzle and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Uggla, Mixmaster: Fact and Belief, Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' 26, 075016 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' [6] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Cornish and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Levin, The Mixmaster universe is chaotic, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' 78, 998 (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' [7] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Latifi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Musette, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Conte, The Bianchi IX (mixmaster) cosmological model is not integrable, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' A 194 83 (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' [8] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Montani, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Battisti, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Benini, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Imponente, Classical and quantum features of the mixmaster singularity, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' A 23, 2353 (2008), [9] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Craig and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Hartle, Generalized quantum theory of recollapsing homogeneous cosmologies, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' D 69, 123525 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' [10] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Damour and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Spindel, Quantum supersymmetric Bianchi IX cosmology, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' D 90, 103509 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' [11] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Bergeron, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Czuchry, J-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Gazeau, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Ma�lkiewicz, Vibronic framework for quantum mixmaster universe, PRD 93, 064080 (2016) [12] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Bergeron, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Czuchry, J-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Gazeau, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Ma�lkiewicz, Quantum Mixmaster as a Model of the Primordial Universe, Universe 6(1), 7 (2020) 16 [13] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Flaschka, The Toda lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Existence of integrals, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' B 9, 1924 (1974).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' [14] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' H´enon, Integrals of the Toda lattice, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' B 9, 1926 (1974).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' [15] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Ford, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Stoddard and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Turner, On the Integrability of the Toda Lattice, Progress of Theoretical Physics 50, 1547 (1973).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' [16] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Bogoyavlensky, On perturbations of the periodic Toda lattice, Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' 51, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' 3, 201 (1976).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' [17] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Biesiada, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Szyd�lowski, Mixmaster cosmological models as disturbed Toda lattices, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' A 160, 123 (1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' [18] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Gutzwiller, The Quantum mechanical Toda lattice, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' 124, 347 (1980);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' The Quantum mechanical Toda lattice II, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' 133, 304 (1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' [19] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Matsuyama, Numerical study of the quantum mechanical Toda lattice, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' A 161, 121 (1991);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Periodic Toda lattice in quantum mechanics, Annals of Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' 220, 300 (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' [20] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Isola, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Kantz and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Livi, On the quantization of the three-particle Toda lattice, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' A 24, 3061 (1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' [21] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Toda, Theory of Nonlinear lattices, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Japan 22, 431 (1967).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' [22] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Ferguson Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=', H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Flaschka and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Mclaughlin, Nonlinear Normal Modes for the Toda Chain, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' 45, 157 (1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' [23] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Bergeron, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Czuchry, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Gazeau, P, Ma�lkiewicz, W Piechocki, Smooth quantum dynamics of the mixmaster universe Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' D 92, 061302(R) (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' [24] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Bergeron, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Czuchry, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Gazeau, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Ma�lkiewicz and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Piechocki, Singularity avoidance in a quantum model for mixmaster universe, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' 92, 124018 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} +page_content=' 17' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQffwca/content/2301.03929v1.pdf'} diff --git a/vdE3T4oBgHgl3EQfkwp-/vector_store/index.pkl b/vdE3T4oBgHgl3EQfkwp-/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..35beb6a60ecdfffff9f3eded669ea9624978c681 --- /dev/null +++ b/vdE3T4oBgHgl3EQfkwp-/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:ea1485e2e31da4d9a887beed0fad3f0d60bbec96c5dfe0af60695dc2064bd3ef +size 60000 diff --git a/wNAyT4oBgHgl3EQfOfYw/content/tmp_files/2301.00005v1.pdf.txt b/wNAyT4oBgHgl3EQfOfYw/content/tmp_files/2301.00005v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2771e5cbc886620e8aac69766ccd374c5b610c0e --- /dev/null +++ b/wNAyT4oBgHgl3EQfOfYw/content/tmp_files/2301.00005v1.pdf.txt @@ -0,0 +1,1025 @@ +Intrinsic Motivation in Dynamical Control Systems +Stas Tiomkina, Ilya Nemenmanb,c,d, Daniel Polanie, and Naftali Tishby∗f,g +aComputer Engineering Department, Charles W. Davidson College of Engineering San Jose State University, CA, 95192, +bDepartment of Physics, cDepartment of Biology, dInitiative in Theory and Modeling of Living Systems, Emory University, +Atlanta, GA 30322, USA , eAdaptive Systems Research Group, University of Hertfordshire, Hatfield, UK, fThe Rachel and +Selim Benin School of Computer Science and Engineering, gEdmond and Lilly Safra Center for Brain Sciences (ELSC), +Hebrew University of Jerusalem, 96906 Israel +Abstract +Biological systems often choose actions without an ex- +plicit reward signal, a phenomenon known as intrinsic +motivation. +The computational principles underlying +this behavior remain poorly understood. In this study, +we investigate an information-theoretic approach to in- +trinsic motivation, based on maximizing an agent’s em- +powerment (the mutual information between its past ac- +tions and future states). +We show that this approach +generalizes previous attempts to formalize intrinsic moti- +vation, and we provide a computationally efficient algo- +rithm for computing the necessary quantities. We test +our approach on several benchmark control problems, +and we explain its success in guiding intrinsically mo- +tivated behaviors by relating our information-theoretic +control function to fundamental properties of the dynam- +ical system representing the combined agent-environment +system. This opens the door for designing practical ar- +tificial, intrinsically motivated controllers and for linking +animal behaviors to their dynamical properties. +Keywords— information capacity | sensitivity gain | sta- +bilization | predictive information +Introduction +Living organisms are able to generate behaviors that solve +novel challenges without prior experience. +Can this ability +be explained by a single, generic mechanism? One proposal +is that novel, useful behaviors can be generated through in- +trinsic motivation [1], which is defined informally as a set +of computational algorithms that are derived directly from +the intrinsic properties of the organism-environment dynam- +ics and not specifically learned. +Increasingly, there is a move away from reinforcement learn- +ing and its extrinsically specified reward structure [2,3] in the +theory and practice of artificial agents, robots, and machine +learning more generally [4–20]. A specific class of such intrin- +sic motivation algorithms for artificial systems is known as +∗Prof. Naftali Tishby passed away when this work was in de- +velopment. +This project began under his leadership when Stas +Tiomkin was a PhD student in his group. +The rest of the au- +thors agree that he should be a senior author on this manuscript, +but his consent for this was not obtained. +empowerment maximization. It proposes that agents should +maximize the mutual information [21] between their poten- +tial actions and a subsequent future state of the world [22]. +This corresponds to maximizing the diversity of future world +states achievable as a result of the chosen actions, potentiating +a broader set of behavior options in the future. Intrinsically +motivated synthetic agents develop behaviors that are atypi- +cal for inanimate engineered systems and often resemble those +of simple living systems. Interestingly, potentiating future ac- +tions is also a key part of the success of modern reward-based +training algorithms [8,23,24]. +Despite the successes of empowerment maximization, it re- +mains unclear how well it can be used as a general intrinsic +motivation principle. There are many different versions of in- +trinsic motivation related to empowerment, and their relation +to each other is unknown [20,23,25]. Additionally, most work +on empowerment maximization has relied on simulational case +studies and ad hoc approximations, and analytical results are +scarce. In order to gain insight, it is important to link em- +powerment to other, better-understood characterizations of +the systems in question. Finally, calculating the mutual in- +formation between two interlinked processes in the general +case is a challenging task [26,27], which has so far limited the +use of empowerment maximization to simple cases. +In this work, we unify different versions of intrinsic moti- +vation related to the empowerment maximization paradigm. +Here our main contribution is in showing analytically that +empowerment-like quantities are linked to the sensitivity +of the agent-environment dynamics to the agent’s actions. +This connects empowerment maximization to well-understood +properties of dynamical systems. +Since highly sensitive re- +gions of the dynamics potentiate many diverse future behav- +iors, the connection to dynamical systems also explains why +empowerment-based intrinsic motivations succeed in generat- +ing behaviors that resemble those of living systems. +The analytical results allow us to develop a practical com- +putational algorithm for calculating empowerment for com- +plex scenarios in the continuous time limit, which is the sec- +ond major contribution of the paper. +We apply the algo- +rithm to standard benchmarks used in intrinsic motivation +research [14, 28, 29]. +Specifically, a controller based on the +efficient calculation of empowerment manages to balance an +inverted pendula without extrinsic rewards. This opens the +door for designing complex robotic intrinsically motivated +agents with systematically computed — rather than heuristi- +cally estimated — empowerment. +1 +arXiv:2301.00005v1 [cs.AI] 29 Dec 2022 + +CTe,Ta,∆T � +x0 ≡ x(t) +� += max +p(⃗a|x0) I[{XTe, XTe−1, . . . , XTa+∆T } +� +�� +� +⃗ +X−future states +; {ATa−1, ATa−2, . . . , A0} +� +�� +� +⃗A−possible actions +| x0 ≡ x(t)] +Empowerment +Ta = Te, ∆T = 0 +· · · controlled Lyapunov expt. +Ta = 1, ∆T = Te − 1 +· · · +kicked CEF +Ta = 1, ∆T = 0 +Figure 1: Unified view on information theoretical intrinsic motivation, for a discretized process sequence. Starting at time +x0 (i.e. x(t)), potential actions are applied for Ta times, following that, after waiting for ∆T time steps, the future system +trajectory is considered until Te. A controlled Lyapunov exponent is a Lyapunov exponent, but only in directions controlled +by the agent, cf., (11). “Kicked CEF” refers to a variant of Causal Entropic Forcing [30], with the addition that an action +kicks the system at the beginning of a trajectory. For more details see Generalized Empowerment. +Results +A +Preliminaries +Notation We consider an agent that takes on states x(t) ∈ +X := Rdx, evolving in time under the dynamics f with (small) +stochastic perturbations η(t) ∈ Rdx. Via its (small) actions, +a(t) ∈ A := Rda filtered through the control gain g, the agent +can affect the dynamics of the system: +dx(t) = f(x(t))dt + g(x(t))da(t) + dη . +(1) +Here dη denotes the system noise, modeled as a Wiener pro- +cess. +The agent’s actions a(t) are modeled by a stochastic +control process with variance σ2 +t controlled by the agent and +with a mean of zero. This models potential effect of actions +centered around the null action. +To compute various quantities of interest, we will consider +a discretized versions of this system, for which we adopt a +modified notation. +To distinguish it from the continuous +version, we replace the continuous time in parentheses by +an integer index, xk := x(t + k · ∆t). +Here ∆t denotes +the physical time step, and we adopted the convention that +x0 = x(t), so that the index corresponding to the current +physical time, t, is chosen as 0. We will consider trajectories +of a fixed duration, and the agent will apply actions over a +part of that trajectory. We denote by Te the time index of +the very last state of the trajectory, which we also refer to +as the time horizon. We further use Ta to denote the (dis- +cretized) duration of the action sequence. Then state, con- +trol and perturbation trajectories at finite equidistant times, +{t+k·∆t}T +k=0, are denoted by xTe +0 +≡ {xk}Te +k=0, aTa +0 +≡ {ak}Ta +k=0, +and ηTe +0 +≡ {ηk}Te +k=0, respectively. +For consistency with the +control theory literature, we write a trajectory in the reverse +order, e.g., xTe +0 += (xTe, . . . x0). When we wish to emphasize +the continuous nature of the underlying process, we will write +te ≡ t + Te · ∆t and ta ≡ t + Ta · ∆t for explicitly continuous +times. +Reinforcement Learning vs. Intrinsic Motivation To elicit +a desired behavior in an agent, one typically uses reinforce- +ment learning (RL). RL is task-specific, and an agent needs an +extrinsic feedback about its performance from a reward func- +tion to learn the behavior. The precise construction of this +reward function is critical to achieve a desired performance in +a short training time [2]. Some of the complications include +a significant degree of arbitrariness when choosing amongst +reward functions with equivalent performance [31] and the +difficulty of translating an often vague desired behavior into +a concrete reward function. Furthermore, complex behaviors +consist of combinations of shorter sequences. Designing a re- +ward function capable of partitioning the solution into such +parts and hence learning it in a realistic time is hard [32]. +In contrast to this, in living systems, acquisition of skills +often starts with task-unspecific learning. +This endows or- +ganisms with potentiating skills, which are not rewarding on +their own. This is then followed by task-oriented specializa- +tion, which combines task-unspecific behaviors into complex +and explicitly rewarding tasks [1,33]. While specific tasks are +often refined with the help of an extrinsic reinforcement, the +potentiating tasks usually are intrinsically motivated [9]. +Empowerment +The type of intrinsic motivation we focus +on is empowerment. Empowerment is based on information- +theoretic quantities [4,23,30,34–40]. It defines a pseudo-utility +function on the state space, based on the system dynamics +only, without resorting to a reward. Formally, we express the +dynamics of the system by the conditional probability dis- +tribution p(xTe | aTe−1 +0 +, x0) of the resulting state when one +starts in a state x0 and subsequently carries out an action se- +quence aTe−1 +0 +. Then the empowerment C(x0) is a function of +the starting state, x0. It is given by the maximally achievable +mutual information (the channel capacity [21]) between the +control action sequence of length Te and the final state when +starting in the state x0: +C(x0) := +max +p(aTe−1 +0 +|x0) +I(XTe; ATe−1 +0 +|x0). +(2) +Here p(·) denotes a probability density or a probability distri- +bution function, and I is the mutual information [21] +I(XTe; ATe−1 +0 +|x0) = H(XTe|x0) − H(XTe | ATe−1 +0 +x0). +(3) +H is the entropy, and conditioning an entropy on a random +variable means the entropy of the conditional distribution, +averaged over the conditioning variable. The empowerment +C(x0) depends on both the state, x0, and the time horizon, Te. +However, for notational convenience, we omit all parameters +from the notation except for the dependency on x0. +Locally maximizing empowerment (e.g., by following its +gradient over x0) guides an agent to perform actions atypical +within the natural dynamics of the system. Indeed, since em- +powerment measures the diversity of achievable future states, +maximizing it increases this diversity (“empowers” the agent – +hence the name). Thus it is expected to be particularly useful +for learning potentiating tasks [9]. Crucially, empowerment +quantifies the relation between the final state and the inten- +tional control, rather than the diversity of states due to the +stochasticity of the system. In particular, it is not just the en- +2 + +tropy of a passive diffusion process in the state variables, but +of the subprocess that the agent can actively generate. Fur- +thermore, it quantifies diversity due to potential future action +sequences, which are not then necessarily carried out. +Empowerment is typically used in the form of the empow- +erment maximization principle [17], treats C(x0) as a pseudo- +utility function. At each time step, an agent chooses an action +to greedily optimize its empowerment at the next time step. +That is, the agent climbs up in its empowerment landscape, +eventually achieving a local maximum of C: +a∗� +x(t) +� += argmax +a∈A +Eη +� +C +� +f(x(t)) + g(x(t))a∆t′ + dη +�� +. (4) +Here A is the set of permitted actions, ∆t′ is a small time +step used to simulate the actual behavior of the system (and +which is selected independently from the time step ∆t used to +discretize (1)). An empowerment-maximizing agent generates +its behavior by repeating this action selection procedure for +each decision step it takes. +Crucially, no general analytical solutions or efficient algo- +rithms for numerical estimation of empowerment for arbitrary +dynamical systems are known, limiting adoption of the em- +powerment maximization principle. Our goal is to provide a +method to calculate it under specific approximations. +B +Empowerment in dynamical systems +The linear response approximation To relate empowerment +to traditional quantities used to describe dynamical systems, +we assume that the control signal a in (1) is small. This is +true in some of the most interesting cases, where the chal- +lenge is to solve a problem with only weak controls that can- +not easily “force” a solution. Under this assumption, (1) is +approximated by a linear time-variant dynamics around the +trajectories of the autonomous dynamics (i.e., for a = 0). To +proceed, we now introduce the following notation. +We de- +fine ¯xs as the s-th step of the trajectory in the discretized +deterministic approximation of the dynamics (1), given by +¯xs = f(¯xs−1) + g(¯xs−1)∆as−1 +(5) +with ¯x0 = x0 ≡ x(t). +For example, ¯x3 = f(f(f(¯x0) + +g(¯x0)∆a0) + g(¯x1)∆a1) + g(¯x2)∆a2. We denote this recur- +sive mapping from ¯x0 to ¯xs by F, ¯xs = F(¯x0; ∆as−1 +0 +). Then +the sensitivity of the state at the time step s to the action at +the time step r can be calculated via the iterated differentia- +tion chain rule applied to the state derivative of the dynamics +F: +∂¯xs +∂ar = +s +� +τ=r+2 +∇¯xf(¯xτ−1) g(¯xr), +(6) +where ∇¯xf(¯xτ) is the dx ×dx Jacobian matrix, which approx- +imates f up to the linear order in the state and the control. +Specifically, the (i, j)-th entry of ∇¯xf(¯xτ) is +∂fi(¯xτ ) +∂¯xτ,j , where +indices i, j stand for components of the vectors x and f. For +s = r + 1, the expression in (6) evaluates to +∂¯xr+1 +∂ar += g(xr). +Now we define the linear response of the sequence of the +system’s states xs2 +s1 to the sequence of the agent’s actions ∆ar2 +r1 +F s1,s2 +r1,r2 (x0) = +∂¯xs2 +∂ar2 +∂¯xs2 +∂ar2−1 +. . . +∂¯xs2 +∂ar1 +∂¯xs2−1 +∂ar2 +∂¯xs2−1 +∂ar2−1 +. . . +∂¯xs2−1 +∂ar1 +... +... +. . . +... +∂¯xs1 +∂ar2 +∂¯xs1 +∂ar2−1 +. . . +∂¯xs1 +∂ar1 +� +������������ +� +������������ +dx·s×da·r +, +(7) +where s = s2 − s1 + 1, r = r2 − r1 + 1, s + ∆T + r − 1 = Te, +and the entries are computed via (6). +Usually we consider +situations where the agent applies its controls for r time steps, +and then after a gap observes the state for s steps. That is, +s1 = r2 + 1 + ∆T, where ∆T ≥ 0 is the gap between the end +of the control sequence and the start of the observations. +Notice that traditional definitions of sensitivity of a dy- +namical system to its controls are blocks F +s′ +1,s′ +2 +r′ +1,r′ +2 in this over- +all sensitivity matrix, F s1,s2 +r1,r2 . For example, if r′ +1 = r′ +2 = 0, +∆T ′ = Te − 1, and s′ +1 = Te, then s′ +2 = Te, and the sensitivity +matrix collapses to just the entries that measure the sensitiv- +ity of the current state to the controls during the immediately +preceding time step, F Te,Te +0,0 +(x0) = +∂¯xTe +∂a0 . This is also the blue +block of the overall sensitivity matrix, (7). +With the definitions above, in the linear response regime, +the effect of a sequence of (small) actions on a sequence of +states, (1), becomes +∆xs2 +s1 = F s1,s2 +r1,r2 (x0)∆ar2 +r1 + ˜η, +(8) +where ∆a and ∆x are the reverse-time-ordered vectors of +small actions and the induced deviations of states (which +themselves can be vectors).Here ˜η models both the total noise +resulting from the integration of the process noise dη from (1) +and the noise of the subsequent observation of the state per- +turbation ∆xs2 +s1, which we assume as Gaussian. +Generalized Empowerment +Since the entire dynamics is +now linear, cf. (8), we can consider formally effects of arbitrary +length sequences of actions on arbitrary length sequences of +future states. In other words, we can define the generalized +empowerment, +CTe,Ta,∆T (x0) := max +p(⃗a|x0) I(XTe +Ta+∆T ; ATa−1 +0 +|x0) . +(9) +Here, Ta denotes the number of time steps at which actions +are performed, ∆T is the time gap between the action se- +quence and the beginning of the observation of the resulting +states, and Te is the last step in that observed sequence. That +is, CTe,Ta,∆T measures the maximum mutual information be- +tween a sequence of actions and a later sequence of states, +rather than just one final state, like empowerment does. +Plugging in (8) into (4), we observe that computing the gen- +eralized empowerment in discretized time with an arbitrary +discretization step and an arbitrary time horizon Te reduces +to a traditional calculation of the channel capacity of a linear +Gaussian channel, though with a large number of dimensions +reflecting both the duration of the signal and the duration of +the response. Specifically, +CTe,Ta,∆T (x0) = max +σi≥0 +� +i +σi=P +1 +2 +dx +� +i=1 +ln(1 + ρi(x0)σi). +(10) +3 + +Here ρi(x0) are the singular values of the appropriate sub- +matrix F +s′ +1,s′ +2 +r′ +1,r′ +2 (x0); for example, the traditional empowerment +corresponds to the red-dashed submatrix in (7). Further, P +is the power of the control signal ∆a over the whole control +period, and σi ≥ 0 is that part of the overall power of the +control signal which is associated with the i-th singular value +(called channel power). The channel power can be computed +by the usual water-filling procedure [21]. Note that here we +denote P as power, as per control-theoretic convention, but +since we fix the time interval over which it is applied, the units +of P are those of energy. As per our weak control assumption, +we assume P to be suitably small. +With (10), calculation of any generalized empowerment be- +comes tractable, at least in principle. This also shows explic- +itly that the (generalized) empowerment is a function of the +sensitivity matrix F, and with it of quantities used to char- +acterize dynamics, such as the Lyapunov exponents. +To compute CTe,Ta,∆T (x0) efficiently for an arbitrary dy- +namical system (1) and arbitrary long time horizons and ar- +bitrary small discretization steps, we start by discretizing the +time and calculating the linear response matrix F. While in +this paper we do this by analytical differentiation, numerical +differentiation can be used whenever f is unknown. We then +calculate the singular values of F; this is straightforward on +modern computers for dimensionalities of up to a few hundred. +Finally, we apply the “water filling” procedure to find the set +of channel powers σi to match the available total power P in +(10), and from there we calculate the (generalized) empower- +ment value. We will employ this approach for all examples in +this paper. +Connecting Generalized Empowerment to Related Quan- +tities Generalized empowerment with different durations of +action and observation sequences is related to various quan- +tities describing dynamical systems, including those defining +intrinsic motivation [8, 20, 23, 41]. For example, Causal En- +tropic Forcing (CEF) [20] is defined as actions that maximize +the entropy of future trajectories of a system. With Ta = 1 +and ∆T = 0, CTe,Ta,∆T in (9) measures the immediate con- +sequences of a single action on a trajectory with a fixed time +horizon Te. Maximizing CTe,Ta,∆T is then equivalent to choos- +ing actions that maximize susceptibility, and not the entropy +of trajectories with a given time horizon. In other words, one +can interpret CTe,1,0 as a “kicked”, or agent-controllable, ver- +sion of CEF, where just the first action can be selected by +the agent at any time, and uncontrolled future variability is +discarded in action planning (see Fig. 1 for an illustration). +Such kicked CEF corresponds to the green submatrix in (7). +Now consider the top right corner (blue) of (7) with Te = +Ta = 1, or, equivalently, s′ +2 = s2 and s′ +1 = s′ +2 − 1. In the limit +of a very long horizon, s2 → ∞, the appropriate submatrix of +F is +Λ ≡ lim +s2→∞ +��∂¯xs2 +∂ar1 +��∂¯xs2 +∂ar1 +�† +� 1 +s2 +, +(11) +where † is the transpose, and +∂¯xs2 +∂ar1 is given by (6). In the +special case that the control gain is the identity, g(x) = x, +the logarithm of the eigenvalues of Λ reduces to the usual +characteristic Lyapunov exponents of the dynamical system +[42]. However, once a more general control gain is applied, +the action-controlled perturbation, ar1 may be able to affect +only a part of the state space. This means that Λ not only +is a generalized empowerment with specific indices, but it is +also a specialization of the concept of Lyapunov exponents to +the controllable subspace. Thus we refer to the log-spectrum +of Λ as the control Lyapunov exponents, cf. Fig. 1. +In summary, (9) and the linearization, (7), provide a unified +view of various sensitivties of the dynamics to the controls, +and hence on various versions of intrinsic motivation. +C +Intrinsic motivation in power-constrained agents +An agent controlling a system with unconstrained actions +can trivially reach any state in a controllable dynamical sys- +tem [43] by simply forcing their desired outcome without so- +phisticated control. Thus to render the setup interesting, we +consider only power-constrained, or weak agents. +To show +that empowerment maximization, in the linearized regime, is +an efficient control principle, we use it to stabilize a family +of inverted pendula (single pole, double pole, and cart-pole), +which are simple, paradigmatic models of important phenom- +ena, such as human walking [44]. +Solutions for the stabilization problem are known. They +require to accumulate energy by swinging the pendulum back +and forth into resonance without overshooting and then to +keep the pendulum upright. When details of the system are +not specified a priori, this solution needs to be learned by +the agent. +Finding such an indirect control policy by tra- +ditional reinforcement learning is nontrivial [3], since the in- +creasing oscillations require a long time for the balancing to +take place, and the acquisition of informative rewards indi- +cating success is significantly delayed. As we will show, it is +precisely in such situations that intrinsic motivation based on +empowerment is especially useful, since it is determined from +only comparatively local properties of the dynamics along the +present trajectory and its potential future variations. +Inverted pendulum We start with a relatively simple task +of swinging up and stabilizing an inverted pendulum without +an external reward. With an angle of θ (in radians) from the +upright vertical, the equations of motion of the pendulum are +�dθ(t) +d ˙θ(t) +� += +� +˙θ(t)dt +g +l sin(θ(t)) dt + da(t) +ml2 + dW (t) +ml2 +� +, +(12) +where ˙θ is the angular velocity of the pendulum, m is its mass, +l is the length, a is the torque applied by the agent, g is the +free fall acceleration, and dW(t) is a Wiener process. +We apply a (stochastically chosen) control signal a(t) for +the duration Te and observe the final state ˜θ = θ + ˜ηobs, +where ˜ηobs is the standard Gaussian observation noise at the +final state. +Empowerment is then given by the maximally +achievable mutual information between a(t) and ˜θ at a given +power level for a(t), i.e., the channel capacity between the +two. +The observation noise effectively determines the resolution, +at which the end state is considered. Note that in our linear +approximation the process noise dW(t) undergoes the same +gain sequence as the control signal, and thus it rescales the +empowerment landscape and changes the behavior of the sys- +tem. Thus to compare empowerment values in different states, +it is essential to include the observation noise. +We now apply our empowerment-based control protocol, +(4), to the inverted pendulum. We calculate the empowerment +landscape by using the time-discretized version of Eqs. (1, 12). +For this, we map the deterministic part of the dynamics (f, g +in (1)) onto discrete time as per (5). We then compute the +4 + +Figure 2: Intrinsic motivation based control in the power-constrained regime. Top row: generalized empowerment landscapes +in the linear approximation for empowerment (left), controlled Lyapunov exponent (middle), and kicked CEF (right) versions +of the problem, plotted against θ (horizontal axis) and ˙θ (vertical axis), measured in rad and rad/s, respectively. Black dots +in each panel are the final state, and white lines are the trajectories of the pendulum, starting at the bottom denoted by the +red dots. Bottom row: the control signals chosen from the generalized empowerment maximization as a function of time. +Here the time horizon is te = 0.5s. +channel capacity by applying (10) using the singular values +from (8), where states are given by (θ, ˙θ) ∈ Rdx, and actions +consist of applying a torque a. The landscapes for the orig- +inal empowerment, the controlled Lyapunov exponent, and +the kicked CEF versions of the problem, all with the time +horizons of te = 0.5 s and the discretization ∆t = 10−3 are +shown in Fig. 2. Then, from each state, we choose the con- +trol action to greedily optimize the generalized empowerment. +The panels in the upper row in this Figure also show trajec- +tories obtained this way. +The lower row shows time traces +of the control signal derived from the generalized empower- +ment maximization. +In all cases, initially, the agent drives +the pendulum at the maximum allowable torque, which we +set to be power-constrained to ±1 N m. Around 13, 10, and +10 seconds after the start (for the three versions of the em- +powerment, respectively), the pendulum accumulates enough +energy to reach the vertical, and the agents reduce the torques +to very small values, a ≪ 1 N m, which are now sufficient to +keep the pendulum in the upright position and prevent it from +falling. It is striking that the generalized empowerment land- +Figure 3: +Convergence of the method for ∆t → 0 and +te = 0.5s. As time resolution is refined fourfold at every stage, +one arrives at a well-defined value for the empowerment es- +timation as ∆t → 0. +The numerical stability of this limit +approximation is consistent throughout the landscape. +scapes and their induced trajectories are qualitatively similar +to those that would be generated by an optimal value func- +tion, derived by standard optimal control techniques based on +a reward specifically designed to achieve the top position [3]. +In our analysis, we chose a particular discretization ∆t = +10−3 s, and we need to show that our results depend only +weekly on this choice. For this, we repeat our analysis at dif- +ferent ∆t. Figure 3 shows the dependence of the maximum +value of the original empowerment (black dot in left panel of +Fig. 2) on ∆t. To the extent that the estimate converges to a +well-defined number linearly as ∆t → 0, the discrete time dy- +namics provides a consistent approximation to the continuous +time dynamics. +Double Pendulum +Now we show that the empowerment +maximization formalism is capable of dealing with more chal- +lenging problems, such as a power-constrained control of a +(potentially chaotic) double pendulum [16], Fig. 4, with equa- +tions of motion: +d¨θ1(t) = − +1 +d1(t) +� +d2(t)¨θ2(t) + φ1(t) +� +, +(13) +d¨θ2(t) = +1 +m2ℓ2c2 + I2 − +d2 +2(t) +d1(t) +� +da(t) + dW(t) + d2 +2(t) +d1(t)φ1(t) +− m2ℓ1ℓc2 ˙θ1(t)2 sin θ2(t) − φ2(t) +� +, +with +d1(t) =m1ℓ2 +c1 + m2(ℓ2 +1 + ℓ2 +c2 + 2ℓ1ℓc2 cos θ2(t)) + I1 + I2, +d2(t) =m2(ℓ2 +c2 + ℓ1ℓc2 cos θ2(t)) + I2, +φ1(t) = − m2ℓ1ℓc2 ˙θ(t)2 sin θ2(t)−2m2ℓ1ℓc2 ˙θ2(t) ˙θ1(t) sin θ2(t) ++ (m1ℓc1 + m2ℓ1)g cos θ1(t) + φ2(t), +φ2(t) =m2ℓc2g cos(θ1(t) + θ2(t)). +We add Wiener noise, dW(t), and permit the controller to +apply a scalar control signal |a(t)| ≤ 1, at the joint between +the two links. In the equations of motion, mi, ℓi, ℓci, and Ii +stand for the mass, the length, the length to center of mass, +and the moment of inertia of the i-th link, i ∈ [1, 2], respec- +5 + +Figure 4: Top left: Double pendulum with control torque on the joint between the links with dynamics given by (13) Top +right: Slices through the empowerment landscape of a double pendulum. Each subplot shows a particular slice in the 4D +landscape, when two other coordinates are zero. For example, the plot with axes ˙θ2, ˙θ1 is shown for θ2 = 0 rad and θ1 = 0 rad. +Bottom: Traversing the state space of the double pendulum according to (4). The first and the second 15s are shown with +different scale for the instantaneous empowerment. The initial and the final positions are both links down and both links up, +respectively. Torque is applied to the middle joint only. +tively. Figure 4 shows the landscape for the original empow- +erment for selected slices of the phase space. This landscape +is more complex than for the single-pendulum. Nonetheless it +retains the property that, following the local gradient in the +state space directly, one ultimately reaches the state of the +maximum empowerment, which is precisely where both links +of the pendulum are balanced upright. The vertical position, +however, is a priori not sufficient to guarantee the balanc- +ing since the control only applies torque at the joint linking +the pendulum halves. +That is, the controller cannot move +the pendulum in arbitrary directions through the state space. +Surprisingly, this concern notwithstanding, the algorithm still +balances the pendulum, cf. Fig. 4. +Cart-Pole We have additionally verified that the empower- +ment maximization also balances an inverted pendulum on a +moving cart, cf. Fig.5. Here the control signal (force) is ap- +plied to the cart. Thus the pendulum is now affected only +indirectly. The dynamics of this system is: +d¨x(t) =m sin θ(t)(ℓ ˙θ2(t) + g cos θ(t)) + da(t) + dW(t) +M + m sin2 θ(t) +, (14) +d¨θ(t) = − da(t) cos θ(t) − mℓ ˙θ2(t) cos θ(t) sin θ(t) +− (M + m)g sin θ(t), +where x(t), θ(t), m, M, ℓ, g, |a(t)| ≤ 1 are the x coordinate of +the center of mass of the cart, the angle of the pole, the pole +mass, the cart mass, the pole length, the free fall acceleration, +and the force applied to the cart. +Discussion +In this study, we focused on a class of intrinsic motivation +models that mimic decision-making abilities of biological or- +ganisms in various situations without explicit reward signals. +We used an information-theoretic formulation in which the +controller starts with knowledge of the (stochastic) dynami- +cal equations describing the agent and the environment, and +then selects actions that “empower” the agent. That is, the +controller improves its ability to affect the system in the fu- +ture, as measured by the mutual information between the ac- +tion sequence and the subsequent responses. This leads the +system to the most sensitive points in the state space, which +we showed solves a problem known to be difficult for simple +reinforcement learning algorithms: balancing inverted pen- +dula. +Depending on which subsets of the past actions and +future responses are used to drive the intrinsic motivation, +our approach interpolates between the original formulation +of empowerment maximization, maximization of the “kicked” +version of Causal Entropic Forcing, and maximization of the +“controlled” subset of the Lyapunov exponents of the agent- +environment pair. This provides insight into which properties +of the dynamical system are responsible for the behaviors pro- +6 + +upright balance +01 +01 +2 02 +Motor +02)0.912 +1.6 +C(α*) in nats +0.910 +1.4 +0.908 +1.2 +0.906 +1.0 +0.904 +0.8 +10 +15 +15 +20 +25 +30 +seconds +secondsFigure 5: Left: Cart-Pole system with control force, ⃗a(t), applied to the cart only, which moves on the rail (or on the edge +of a table), allowing the pole to rotate in the x-y plane. Its dynamics is given by (14). Right: Traversing the state space of +the pendulum on a cart according to empowerment maximization. The initial and the final state of the pole are down and +up, respectively. The horizontal axis is time in seconds t ∈ [0, 20]s. +duced by these different motivation functions. +One big challenge in using information-theoretic quantities +is computing them, which can be difficult to do either analyt- +ically or from data. Our paper makes a significant contribu- +tion to solving this problem in the context of empowerment by +providing an explicit algorithm for computing various versions +of empowerment, for arbitrary lengths of pasts and futures, +using the small noise/small control approximation to the dy- +namics, while still treating the dynamics as nonlinear. This +is often the most interesting regime, modeling weak, power- +constrained controllers. Crucially, our algorithm is local, so +that climbing up the empowerment gradient only requires es- +timation of the dynamics in the vicinity of the current state +of the system. This should be possible in real control appli- +cations by using the data directly, possibly with the help of +deep neural networks to approximate the relevant dynamical +landscapes [45–47]. Therefore, knowing the exact form of the +dynamical system, which could be a potential limitation of +our approach, is not strictly required. This opens up oppor- +tunities for scaling our method to more complex scenarios. +Our work suggests that, in addition to the Lyapunov spec- +trum, defined via the trajectory divergence in time due to a +small arbitrary perturbation, one may want to consider the +optimal Lyapunov spectrum, where the initial perturbation is +optimally aligned with the controllable directions in the dy- +namics. +We defer a systematic study of optimal Lyapunov +spectra to future work. +A potential extension of our analysis relates to social in- +teractions. Interacting agents have their own intrinsic moti- +vations and affect each other’s ability to achieve their goals. +Understanding how multiple agents interact, each trying to +empower itself in the presence of others, and whether and +when this leads to cooperation or conflict is a promising area +for future research. Crucially, the ability to affect someone +else’s empowerment may provide insight into what distin- +guishes social interactions from purely physical interactions +among nearby individuals. +Acknowledgements ST was supported in part by Califor- +nia State University, and the College of Engineering at SJSU. +IN was supported in part by the Simons Foundation Investi- +gator award, the Simons-Emory Consortium on Motor Con- +trol, and NIH grant 2R01NS084844. DP acknowledges partial +support by the EC H2020-641321 socSMCs FET Proactive +project and the Pazy Foundation. +References +[1] P.-Y. Oudeyer and F. Kaplan. What is intrinsic motiva- +tion? a typology of computational approaches. Frontiers +in neurorobotics, 1:6, 2009. +[2] R. S. Sutton and A. G. Barto. Reinforcement learning: +An introduction. MIT press, 2018. +[3] K. Doya. Reinforcement learning in continuous time and +space. Neural computation, 12(1):219–245, 2000. +[4] S. Mohamed and D. J. Rezende. +Variational informa- +tion maximisation for intrinsically motivated reinforce- +ment learning. In Advances in neural information pro- +cessing systems, pages 2125–2133, 2015. +[5] K. Gregor, D J. Rezende, and D. Wierstra. Variational +intrinsic control. arXiv preprint arXiv:1611.07507, 2016. +[6] K. Baumli, D. Warde-Farley, S. Hansen, and V. Mnih. +Relative variational intrinsic control. Proceedings of the +AAAI Conference on Artificial Intelligence, 35(8):6732– +6740, 2021. +[7] T. Kwon. Variational intrinsic control revisited. Interna- +tional Conference on Learning Representations, (ICLR), +2021, 2021. +[8] A. Sharma, S. Gu, S Levine, V. Kumar, and K. Hausman. +Dynamics-aware unsupervised discovery of skills. arXiv +preprint arXiv:1907.01657, 2019. +[9] A. Sharma, M Ahn, S. Levine, V. Kumar, K. Haus- +man, and S. Gu. +Emergent real-world robotic skills +via unsupervised off-policy reinforcement learning. RSS, +Robotics: Science and Systems 2020, 2020. +[10] B. Eysenbach, A Gupta, J. Ibarz, and S. Levine. +Di- +versity is all you need: Learning skills without a reward +function. International Conference on Learning Repre- +sentations, (ICLR), 2019, 2018. +[11] R. +Houthooft, +X. +Chen, +Y. +Duan, +J. +Schulman, +F. De Turck, and P. Abbeel. Vime: Variational infor- +mation maximizing exploration. Advances in neural in- +formation processing systems, 29, 2016. +[12] J. Achiam, H. Edwards, D. Amodei, and P. Abbeel. +Variational option discovery algorithms. arXiv preprint +arXiv:1807.10299, 2018. +[13] J. Choi, A. Sharma, H. Lee, S. Levine, and S. S. Gu. Vari- +ational empowerment as representation learning for goal- +conditioned reinforcement learning. International Con- +ference on Machine Learning, pages 1953–1963, 2021. +[14] C. Salge, C. Glackin, and D. Polani. +Empowerment– +an introduction. In Guided Self-Organization: Inception, +pages 67–114. Springer, 2014. +[15] A. S. Klyubin, D Polani, and C. L. Nehaniv. +Em- +powerment: A universal agent-centric measure of con- +trol. +IEEE Congress on Evolutionary Computation, +2005, 1:128–135, 2005. +[16] T. Jung, D. Polani, and P. Stone. Empowerment for con- +tinuous agent—environment systems. Adaptive Behavior, +19(1):16–39, 2011. +7 + +m +0 +TY +X +()p +M[17] A. S. Klyubin, D. Polani, and C. L. Nehaniv. +Keep +your options open: An information-based driving prin- +ciple for sensorimotor systems. PLoS ONE, 3(12):e4018, +Dec 2008. +[18] R. Zhao, P. Abbeel, and S. Tiomkin. Efficient empower- +ment estimation for unsupervised stabilization. Interna- +tional Conference Learning Representations, 2020. +[19] R. Zhao, S. Tiomkin, and P. Abbeel. Dynamical system +embedding for efficient intrinsically motivated artificial +agents. Advances in Neural Information Processing Sys- +tems, (NeurIPS), DeepRL, 2019. +[20] A. D. Wissner-Gross and C. E. Freer. Causal entropic +forces. Physical review letters, 110(16):168702, 2013. +[21] T. M. Cover and J. A. Thomas. Elements of information +theory. John Wiley & Sons, 2012. +[22] A. S. Klyubin, D. Polani, and C. L. Nehaniv. Empow- +erment: A universal agent-centric measure of control. In +2005 IEEE Congress on Evolutionary Computation, vol- +ume 1, pages 128–135. IEEE, 2005. +[23] B. Eysenbach, A. Gupta, J. Ibarz, and S. Levine. Di- +versity is all you need: Learning skills without a reward +function. In ICLR, 2019. +[24] Y. Du, S. Tiomkin, E. Kiciman, D. Polani, P. Abbeel, +and A. D. Dragan. Ave: Assistance via empowerment. +Neural Information Processing Systems, NeurIPS 2020, +2020. +[25] C. Salge and D. Polani. Empowerment as replacement +for the three laws of robotics. Frontiers in Robotics and +AI, 4:25, 2017. +[26] W. Bialek, I. Nemenman, and N. Tishby. Predictabil- +ity, complexity, and learning. +Neural computation, +13(11):2409–2463, 2001. +[27] C. Holmes and I. Nemenman. Estimation of mutual infor- +mation for real-valued data with error bars and controlled +bias. Phys Rev E, 100:022404, 2019. +[28] T. Jung, D. Polani, and P. Stone. Empowerment for con- +tinuous agent—environment systems. Adaptive Behavior, +19(1):16–39, 2011. +[29] R. Zhao, K. Lu, P. Abbeel, and S. Tiomkin. Efficient em- +powerment estimation for unsupervised stabilization. In +International Conference on Learning Representations, +ICLR, 2021. +[30] A. D. Wissner-Gross and C. E. Freer. Causal entropic +forces. Physical review letters, 110(16):168702, 2013. +[31] A. Y. Ng, D. Harada, and S. Russell. Policy invariance +under reward transformations: Theory and application +to reward shaping. +[32] P. Dayan and G. E. Hinton. Feudal reinforcement learn- +ing. In Advances in neural information processing sys- +tems, pages 271–278, 1993. +[33] T. D. Kulkarni, K. Narasimhan, A. Saeedi, and J. Tenen- +baum. +Hierarchical deep reinforcement learning: Inte- +grating temporal abstraction and intrinsic motivation. +In Advances in Neural Information Processing Systems, +NeurIPS, 2016. +[34] Y. Burda, H. Edwards, D. Pathak, A. Storkey, T. Darrell, +and A. A. Efros. +Large-scale study of curiosity-driven +learning. In ICLR, 2019. +[35] D. Pathak, P. Agrawal, A. A. Efros, and T. Darrell. +Curiosity-driven exploration by self-supervised predic- +tion. In ICML, 2017. +[36] M. Karl, M. Soelch, P. Becker-Ehmck, D. Benbouzid, +P. van der Smagt, and J. Bayer. Unsupervised real-time +control through variational empowerment. 2018. +[37] C. Salge, C. Glackin, and D. Polani. Approximation of +empowerment in the continuous domain. +Advances in +Complex Systems, 16(02n03):1250079, 2013. +[38] C. Salge, C. Glackin, and D. Polani. +Empowerment– +an introduction. In Guided Self-Organization: Inception, +pages 67–114. Springer, 2014. +[39] T. Anthony, D. Polani, and C. L. Nehaniv. +General +self-motivation and strategy identification: Case studies +based on sokoban and pac-man. IEEE Transactions on +Computational Intelligence and AI in Games, 6(1):1–17, +2014. +[40] H. J. Charlesworth and M. S. Turner. Intrinsically mo- +tivated collective motion. +Proceedings of the National +Academy of Sciences, 116(31):15362–15367, 2019. +[41] J. Schmidhuber. Formal theory of creativity, fun, and +intrinsic motivation (1990–2010). IEEE Transactions on +Autonomous Mental Development, 2(3):230–247, 2010. +[42] H. D. Abarbanel, R. Brown, and M. B. Kennel. Local lya- +punov exponents computed from observed data. Journal +of Nonlinear Science, 2(3):343–365, 1992. +[43] L. C. Evans. An introduction to mathematical optimal +control theory version 0.2. 1983. +[44] A. D. Kuo. The six determinants of gait and the inverted +pendulum analogy: A dynamic walking perspective. Hu- +man movement science, 26(4):617–656, 2007. +[45] B. C. Daniels and I. Nemenman. Automated adaptive +inference of phenomenological dynamical models. Nature +communications, 6(1):1–8, 2015. +[46] S. L. Brunton, J. L. Proctor, and J. N. Kutz. Discovering +governing equations from data by sparse identification of +nonlinear dynamical systems. Proceedings of the national +academy of sciences, 113(15):3932–3937, 2016. +[47] B. Chen, K. Huang, S. Raghupathi, I. Chandratreya, +Q. Du, and H. Lipson. Automated discovery of funda- +mental variables hidden in experimental data. +Nature +Computational Science, 2(7):433–442, 2022. +8 + diff --git a/wNAyT4oBgHgl3EQfOfYw/content/tmp_files/load_file.txt b/wNAyT4oBgHgl3EQfOfYw/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..23a30e83cbd6de4d7d1d5c338c80975833ed6c6a --- /dev/null +++ b/wNAyT4oBgHgl3EQfOfYw/content/tmp_files/load_file.txt @@ -0,0 +1,640 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf,len=639 +page_content='Intrinsic Motivation in Dynamical Control Systems Stas Tiomkina, Ilya Nemenmanb,c,d, Daniel Polanie, and Naftali Tishby∗f,g aComputer Engineering Department, Charles W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Davidson College of Engineering San Jose State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' CA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' 95192,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' bDepartment of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' cDepartment of Biology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' dInitiative in Theory and Modeling of Living Systems,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Emory University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Atlanta,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' GA 30322,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' USA ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' eAdaptive Systems Research Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' University of Hertfordshire,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Hatfield,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' UK,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' fThe Rachel and Selim Benin School of Computer Science and Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' gEdmond and Lilly Safra Center for Brain Sciences (ELSC),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Hebrew University of Jerusalem,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' 96906 Israel Abstract Biological systems often choose actions without an ex- plicit reward signal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' a phenomenon known as intrinsic motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' The computational principles underlying this behavior remain poorly understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' In this study, we investigate an information-theoretic approach to in- trinsic motivation, based on maximizing an agent’s em- powerment (the mutual information between its past ac- tions and future states).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' We show that this approach generalizes previous attempts to formalize intrinsic moti- vation, and we provide a computationally efficient algo- rithm for computing the necessary quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' We test our approach on several benchmark control problems, and we explain its success in guiding intrinsically mo- tivated behaviors by relating our information-theoretic control function to fundamental properties of the dynam- ical system representing the combined agent-environment system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' This opens the door for designing practical ar- tificial, intrinsically motivated controllers and for linking animal behaviors to their dynamical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Keywords— information capacity | sensitivity gain | sta- bilization | predictive information Introduction Living organisms are able to generate behaviors that solve novel challenges without prior experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Can this ability be explained by a single, generic mechanism?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' One proposal is that novel, useful behaviors can be generated through in- trinsic motivation [1], which is defined informally as a set of computational algorithms that are derived directly from the intrinsic properties of the organism-environment dynam- ics and not specifically learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Increasingly, there is a move away from reinforcement learn- ing and its extrinsically specified reward structure [2,3] in the theory and practice of artificial agents, robots, and machine learning more generally [4–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' A specific class of such intrin- sic motivation algorithms for artificial systems is known as ∗Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Naftali Tishby passed away when this work was in de- velopment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' This project began under his leadership when Stas Tiomkin was a PhD student in his group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' The rest of the au- thors agree that he should be a senior author on this manuscript, but his consent for this was not obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' empowerment maximization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' It proposes that agents should maximize the mutual information [21] between their poten- tial actions and a subsequent future state of the world [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' This corresponds to maximizing the diversity of future world states achievable as a result of the chosen actions, potentiating a broader set of behavior options in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Intrinsically motivated synthetic agents develop behaviors that are atypi- cal for inanimate engineered systems and often resemble those of simple living systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Interestingly, potentiating future ac- tions is also a key part of the success of modern reward-based training algorithms [8,23,24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Despite the successes of empowerment maximization, it re- mains unclear how well it can be used as a general intrinsic motivation principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' There are many different versions of in- trinsic motivation related to empowerment, and their relation to each other is unknown [20,23,25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Additionally, most work on empowerment maximization has relied on simulational case studies and ad hoc approximations, and analytical results are scarce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' In order to gain insight, it is important to link em- powerment to other, better-understood characterizations of the systems in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Finally, calculating the mutual in- formation between two interlinked processes in the general case is a challenging task [26,27], which has so far limited the use of empowerment maximization to simple cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' In this work, we unify different versions of intrinsic moti- vation related to the empowerment maximization paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Here our main contribution is in showing analytically that empowerment-like quantities are linked to the sensitivity of the agent-environment dynamics to the agent’s actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' This connects empowerment maximization to well-understood properties of dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Since highly sensitive re- gions of the dynamics potentiate many diverse future behav- iors, the connection to dynamical systems also explains why empowerment-based intrinsic motivations succeed in generat- ing behaviors that resemble those of living systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' The analytical results allow us to develop a practical com- putational algorithm for calculating empowerment for com- plex scenarios in the continuous time limit, which is the sec- ond major contribution of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' We apply the algo- rithm to standard benchmarks used in intrinsic motivation research [14, 28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Specifically, a controller based on the efficient calculation of empowerment manages to balance an inverted pendula without extrinsic rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' This opens the door for designing complex robotic intrinsically motivated agents with systematically computed — rather than heuristi- cally estimated — empowerment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content='00005v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content='AI] 29 Dec 2022 CTe,Ta,∆T � x0 ≡ x(t) � = max p(⃗a|x0) I[{XTe, XTe−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' , XTa+∆T } � �� � ⃗ X−future states ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' {ATa−1, ATa−2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' , A0} � �� � ⃗A−possible actions | x0 ≡ x(t)] Empowerment Ta = Te, ∆T = 0 · · controlled Lyapunov expt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Ta = 1, ∆T = Te − 1 · · kicked CEF Ta = 1, ∆T = 0 Figure 1: Unified view on information theoretical intrinsic motivation, for a discretized process sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Starting at time x0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' x(t)), potential actions are applied for Ta times, following that, after waiting for ∆T time steps, the future system trajectory is considered until Te.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' A controlled Lyapunov exponent is a Lyapunov exponent, but only in directions controlled by the agent, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=', (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' “Kicked CEF” refers to a variant of Causal Entropic Forcing [30], with the addition that an action kicks the system at the beginning of a trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' For more details see Generalized Empowerment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Results A Preliminaries Notation We consider an agent that takes on states x(t) ∈ X := Rdx, evolving in time under the dynamics f with (small) stochastic perturbations η(t) ∈ Rdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Via its (small) actions, a(t) ∈ A := Rda filtered through the control gain g, the agent can affect the dynamics of the system: dx(t) = f(x(t))dt + g(x(t))da(t) + dη .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' (1) Here dη denotes the system noise, modeled as a Wiener pro- cess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' The agent’s actions a(t) are modeled by a stochastic control process with variance σ2 t controlled by the agent and with a mean of zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' This models potential effect of actions centered around the null action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' To compute various quantities of interest, we will consider a discretized versions of this system, for which we adopt a modified notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' To distinguish it from the continuous version, we replace the continuous time in parentheses by an integer index, xk := x(t + k · ∆t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Here ∆t denotes the physical time step, and we adopted the convention that x0 = x(t), so that the index corresponding to the current physical time, t, is chosen as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' We will consider trajectories of a fixed duration, and the agent will apply actions over a part of that trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' We denote by Te the time index of the very last state of the trajectory, which we also refer to as the time horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' We further use Ta to denote the (dis- cretized) duration of the action sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Then state, con- trol and perturbation trajectories at finite equidistant times, {t+k·∆t}T k=0, are denoted by xTe 0 ≡ {xk}Te k=0, aTa 0 ≡ {ak}Ta k=0, and ηTe 0 ≡ {ηk}Te k=0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' For consistency with the control theory literature, we write a trajectory in the reverse order, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=', xTe 0 = (xTe, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' When we wish to emphasize the continuous nature of the underlying process, we will write te ≡ t + Te · ∆t and ta ≡ t + Ta · ∆t for explicitly continuous times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Reinforcement Learning vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Intrinsic Motivation To elicit a desired behavior in an agent, one typically uses reinforce- ment learning (RL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' RL is task-specific, and an agent needs an extrinsic feedback about its performance from a reward func- tion to learn the behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' The precise construction of this reward function is critical to achieve a desired performance in a short training time [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Some of the complications include a significant degree of arbitrariness when choosing amongst reward functions with equivalent performance [31] and the difficulty of translating an often vague desired behavior into a concrete reward function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Furthermore, complex behaviors consist of combinations of shorter sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Designing a re- ward function capable of partitioning the solution into such parts and hence learning it in a realistic time is hard [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' In contrast to this, in living systems, acquisition of skills often starts with task-unspecific learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' This endows or- ganisms with potentiating skills, which are not rewarding on their own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' This is then followed by task-oriented specializa- tion, which combines task-unspecific behaviors into complex and explicitly rewarding tasks [1,33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' While specific tasks are often refined with the help of an extrinsic reinforcement, the potentiating tasks usually are intrinsically motivated [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Empowerment The type of intrinsic motivation we focus on is empowerment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Empowerment is based on information- theoretic quantities [4,23,30,34–40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' It defines a pseudo-utility function on the state space, based on the system dynamics only, without resorting to a reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Formally, we express the dynamics of the system by the conditional probability dis- tribution p(xTe | aTe−1 0 , x0) of the resulting state when one starts in a state x0 and subsequently carries out an action se- quence aTe−1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Then the empowerment C(x0) is a function of the starting state, x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' It is given by the maximally achievable mutual information (the channel capacity [21]) between the control action sequence of length Te and the final state when starting in the state x0: C(x0) := max p(aTe−1 0 |x0) I(XTe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' ATe−1 0 |x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' (2) Here p(·) denotes a probability density or a probability distri- bution function, and I is the mutual information [21] I(XTe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' ATe−1 0 |x0) = H(XTe|x0) − H(XTe | ATe−1 0 x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' (3) H is the entropy, and conditioning an entropy on a random variable means the entropy of the conditional distribution, averaged over the conditioning variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' The empowerment C(x0) depends on both the state, x0, and the time horizon, Te.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' However, for notational convenience, we omit all parameters from the notation except for the dependency on x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Locally maximizing empowerment (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=', by following its gradient over x0) guides an agent to perform actions atypical within the natural dynamics of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Indeed, since em- powerment measures the diversity of achievable future states, maximizing it increases this diversity (“empowers” the agent – hence the name).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Thus it is expected to be particularly useful for learning potentiating tasks [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Crucially, empowerment quantifies the relation between the final state and the inten- tional control, rather than the diversity of states due to the stochasticity of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' In particular, it is not just the en- 2 tropy of a passive diffusion process in the state variables, but of the subprocess that the agent can actively generate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Fur- thermore, it quantifies diversity due to potential future action sequences, which are not then necessarily carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Empowerment is typically used in the form of the empow- erment maximization principle [17], treats C(x0) as a pseudo- utility function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' At each time step, an agent chooses an action to greedily optimize its empowerment at the next time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' That is, the agent climbs up in its empowerment landscape, eventually achieving a local maximum of C: a∗� x(t) � = argmax a∈A Eη � C � f(x(t)) + g(x(t))a∆t′ + dη �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' (4) Here A is the set of permitted actions, ∆t′ is a small time step used to simulate the actual behavior of the system (and which is selected independently from the time step ∆t used to discretize (1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' An empowerment-maximizing agent generates its behavior by repeating this action selection procedure for each decision step it takes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Crucially, no general analytical solutions or efficient algo- rithms for numerical estimation of empowerment for arbitrary dynamical systems are known, limiting adoption of the em- powerment maximization principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Our goal is to provide a method to calculate it under specific approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' B Empowerment in dynamical systems The linear response approximation To relate empowerment to traditional quantities used to describe dynamical systems, we assume that the control signal a in (1) is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' This is true in some of the most interesting cases, where the chal- lenge is to solve a problem with only weak controls that can- not easily “force” a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Under this assumption, (1) is approximated by a linear time-variant dynamics around the trajectories of the autonomous dynamics (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=', for a = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' To proceed, we now introduce the following notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' We de- fine ¯xs as the s-th step of the trajectory in the discretized deterministic approximation of the dynamics (1), given by ¯xs = f(¯xs−1) + g(¯xs−1)∆as−1 (5) with ¯x0 = x0 ≡ x(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' For example, ¯x3 = f(f(f(¯x0) + g(¯x0)∆a0) + g(¯x1)∆a1) + g(¯x2)∆a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' We denote this recur- sive mapping from ¯x0 to ¯xs by F, ¯xs = F(¯x0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' ∆as−1 0 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Then the sensitivity of the state at the time step s to the action at the time step r can be calculated via the iterated differentia- tion chain rule applied to the state derivative of the dynamics F: ∂¯xs ∂ar = s � τ=r+2 ∇¯xf(¯xτ−1) g(¯xr), (6) where ∇¯xf(¯xτ) is the dx ×dx Jacobian matrix, which approx- imates f up to the linear order in the state and the control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Specifically, the (i, j)-th entry of ∇¯xf(¯xτ) is ∂fi(¯xτ ) ∂¯xτ,j , where indices i, j stand for components of the vectors x and f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' For s = r + 1, the expression in (6) evaluates to ∂¯xr+1 ∂ar = g(xr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Now we define the linear response of the sequence of the system’s states xs2 s1 to the sequence of the agent’s actions ∆ar2 r1 F s1,s2 r1,r2 (x0) = ∂¯xs2 ∂ar2 ∂¯xs2 ∂ar2−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' ∂¯xs2 ∂ar1 ∂¯xs2−1 ∂ar2 ∂¯xs2−1 ∂ar2−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' ∂¯xs2−1 ∂ar1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' ∂¯xs1 ∂ar2 ∂¯xs1 ∂ar2−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' ∂¯xs1 ∂ar1 � ������������ � ������������ dx·s×da·r , (7) where s = s2 − s1 + 1, r = r2 − r1 + 1, s + ∆T + r − 1 = Te, and the entries are computed via (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Usually we consider situations where the agent applies its controls for r time steps, and then after a gap observes the state for s steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' That is, s1 = r2 + 1 + ∆T, where ∆T ≥ 0 is the gap between the end of the control sequence and the start of the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Notice that traditional definitions of sensitivity of a dy- namical system to its controls are blocks F s′ 1,s′ 2 r′ 1,r′ 2 in this over- all sensitivity matrix, F s1,s2 r1,r2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' For example, if r′ 1 = r′ 2 = 0, ∆T ′ = Te − 1, and s′ 1 = Te, then s′ 2 = Te, and the sensitivity matrix collapses to just the entries that measure the sensitiv- ity of the current state to the controls during the immediately preceding time step, F Te,Te 0,0 (x0) = ∂¯xTe ∂a0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' This is also the blue block of the overall sensitivity matrix, (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' With the definitions above, in the linear response regime, the effect of a sequence of (small) actions on a sequence of states, (1), becomes ∆xs2 s1 = F s1,s2 r1,r2 (x0)∆ar2 r1 + ˜η, (8) where ∆a and ∆x are the reverse-time-ordered vectors of small actions and the induced deviations of states (which themselves can be vectors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content='Here ˜η models both the total noise resulting from the integration of the process noise dη from (1) and the noise of the subsequent observation of the state per- turbation ∆xs2 s1, which we assume as Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Generalized Empowerment Since the entire dynamics is now linear, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' (8), we can consider formally effects of arbitrary length sequences of actions on arbitrary length sequences of future states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' In other words, we can define the generalized empowerment, CTe,Ta,∆T (x0) := max p(⃗a|x0) I(XTe Ta+∆T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' ATa−1 0 |x0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' (9) Here, Ta denotes the number of time steps at which actions are performed, ∆T is the time gap between the action se- quence and the beginning of the observation of the resulting states, and Te is the last step in that observed sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' That is, CTe,Ta,∆T measures the maximum mutual information be- tween a sequence of actions and a later sequence of states, rather than just one final state, like empowerment does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Plugging in (8) into (4), we observe that computing the gen- eralized empowerment in discretized time with an arbitrary discretization step and an arbitrary time horizon Te reduces to a traditional calculation of the channel capacity of a linear Gaussian channel, though with a large number of dimensions reflecting both the duration of the signal and the duration of the response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Specifically, CTe,Ta,∆T (x0) = max σi≥0 � i σi=P 1 2 dx � i=1 ln(1 + ρi(x0)σi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' (10) 3 Here ρi(x0) are the singular values of the appropriate sub- matrix F s′ 1,s′ 2 r′ 1,r′ 2 (x0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' for example, the traditional empowerment corresponds to the red-dashed submatrix in (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Further, P is the power of the control signal ∆a over the whole control period, and σi ≥ 0 is that part of the overall power of the control signal which is associated with the i-th singular value (called channel power).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' The channel power can be computed by the usual water-filling procedure [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Note that here we denote P as power, as per control-theoretic convention, but since we fix the time interval over which it is applied, the units of P are those of energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' As per our weak control assumption, we assume P to be suitably small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' With (10), calculation of any generalized empowerment be- comes tractable, at least in principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' This also shows explic- itly that the (generalized) empowerment is a function of the sensitivity matrix F, and with it of quantities used to char- acterize dynamics, such as the Lyapunov exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' To compute CTe,Ta,∆T (x0) efficiently for an arbitrary dy- namical system (1) and arbitrary long time horizons and ar- bitrary small discretization steps, we start by discretizing the time and calculating the linear response matrix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' While in this paper we do this by analytical differentiation, numerical differentiation can be used whenever f is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' We then calculate the singular values of F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' this is straightforward on modern computers for dimensionalities of up to a few hundred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Finally, we apply the “water filling” procedure to find the set of channel powers σi to match the available total power P in (10), and from there we calculate the (generalized) empower- ment value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' We will employ this approach for all examples in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Connecting Generalized Empowerment to Related Quan- tities Generalized empowerment with different durations of action and observation sequences is related to various quan- tities describing dynamical systems, including those defining intrinsic motivation [8, 20, 23, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' For example, Causal En- tropic Forcing (CEF) [20] is defined as actions that maximize the entropy of future trajectories of a system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' With Ta = 1 and ∆T = 0, CTe,Ta,∆T in (9) measures the immediate con- sequences of a single action on a trajectory with a fixed time horizon Te.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Maximizing CTe,Ta,∆T is then equivalent to choos- ing actions that maximize susceptibility, and not the entropy of trajectories with a given time horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' In other words, one can interpret CTe,1,0 as a “kicked”, or agent-controllable, ver- sion of CEF, where just the first action can be selected by the agent at any time, and uncontrolled future variability is discarded in action planning (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' 1 for an illustration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Such kicked CEF corresponds to the green submatrix in (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Now consider the top right corner (blue) of (7) with Te = Ta = 1, or, equivalently, s′ 2 = s2 and s′ 1 = s′ 2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' In the limit of a very long horizon, s2 → ∞, the appropriate submatrix of F is Λ ≡ lim s2→∞ ��∂¯xs2 ∂ar1 ��∂¯xs2 ∂ar1 �† � 1 s2 , (11) where † is the transpose, and ∂¯xs2 ∂ar1 is given by (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' In the special case that the control gain is the identity, g(x) = x, the logarithm of the eigenvalues of Λ reduces to the usual characteristic Lyapunov exponents of the dynamical system [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' However, once a more general control gain is applied, the action-controlled perturbation, ar1 may be able to affect only a part of the state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' This means that Λ not only is a generalized empowerment with specific indices, but it is also a specialization of the concept of Lyapunov exponents to the controllable subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Thus we refer to the log-spectrum of Λ as the control Lyapunov exponents, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' In summary, (9) and the linearization, (7), provide a unified view of various sensitivties of the dynamics to the controls, and hence on various versions of intrinsic motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' C Intrinsic motivation in power-constrained agents An agent controlling a system with unconstrained actions can trivially reach any state in a controllable dynamical sys- tem [43] by simply forcing their desired outcome without so- phisticated control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Thus to render the setup interesting, we consider only power-constrained, or weak agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' To show that empowerment maximization, in the linearized regime, is an efficient control principle, we use it to stabilize a family of inverted pendula (single pole, double pole, and cart-pole), which are simple, paradigmatic models of important phenom- ena, such as human walking [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Solutions for the stabilization problem are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' They require to accumulate energy by swinging the pendulum back and forth into resonance without overshooting and then to keep the pendulum upright.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' When details of the system are not specified a priori, this solution needs to be learned by the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Finding such an indirect control policy by tra- ditional reinforcement learning is nontrivial [3], since the in- creasing oscillations require a long time for the balancing to take place, and the acquisition of informative rewards indi- cating success is significantly delayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' As we will show, it is precisely in such situations that intrinsic motivation based on empowerment is especially useful, since it is determined from only comparatively local properties of the dynamics along the present trajectory and its potential future variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Inverted pendulum We start with a relatively simple task of swinging up and stabilizing an inverted pendulum without an external reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' With an angle of θ (in radians) from the upright vertical, the equations of motion of the pendulum are �dθ(t) d ˙θ(t) � = � ˙θ(t)dt g l sin(θ(t)) dt + da(t) ml2 + dW (t) ml2 � , (12) where ˙θ is the angular velocity of the pendulum, m is its mass, l is the length, a is the torque applied by the agent, g is the free fall acceleration, and dW(t) is a Wiener process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' We apply a (stochastically chosen) control signal a(t) for the duration Te and observe the final state ˜θ = θ + ˜ηobs, where ˜ηobs is the standard Gaussian observation noise at the final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Empowerment is then given by the maximally achievable mutual information between a(t) and ˜θ at a given power level for a(t), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=', the channel capacity between the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' The observation noise effectively determines the resolution, at which the end state is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Note that in our linear approximation the process noise dW(t) undergoes the same gain sequence as the control signal, and thus it rescales the empowerment landscape and changes the behavior of the sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Thus to compare empowerment values in different states, it is essential to include the observation noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' We now apply our empowerment-based control protocol, (4), to the inverted pendulum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' We calculate the empowerment landscape by using the time-discretized version of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' (1, 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' For this, we map the deterministic part of the dynamics (f, g in (1)) onto discrete time as per (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' We then compute the 4 Figure 2: Intrinsic motivation based control in the power-constrained regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Top row: generalized empowerment landscapes in the linear approximation for empowerment (left), controlled Lyapunov exponent (middle), and kicked CEF (right) versions of the problem, plotted against θ (horizontal axis) and ˙θ (vertical axis), measured in rad and rad/s, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Black dots in each panel are the final state, and white lines are the trajectories of the pendulum, starting at the bottom denoted by the red dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Bottom row: the control signals chosen from the generalized empowerment maximization as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Here the time horizon is te = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content='5s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' channel capacity by applying (10) using the singular values from (8), where states are given by (θ, ˙θ) ∈ Rdx, and actions consist of applying a torque a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' The landscapes for the orig- inal empowerment, the controlled Lyapunov exponent, and the kicked CEF versions of the problem, all with the time horizons of te = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content='5 s and the discretization ∆t = 10−3 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Then, from each state, we choose the con- trol action to greedily optimize the generalized empowerment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' The panels in the upper row in this Figure also show trajec- tories obtained this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' The lower row shows time traces of the control signal derived from the generalized empower- ment maximization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' In all cases, initially, the agent drives the pendulum at the maximum allowable torque, which we set to be power-constrained to ±1 N m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Around 13, 10, and 10 seconds after the start (for the three versions of the em- powerment, respectively), the pendulum accumulates enough energy to reach the vertical, and the agents reduce the torques to very small values, a ≪ 1 N m, which are now sufficient to keep the pendulum in the upright position and prevent it from falling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' It is striking that the generalized empowerment land- Figure 3: Convergence of the method for ∆t → 0 and te = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content='5s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' As time resolution is refined fourfold at every stage, one arrives at a well-defined value for the empowerment es- timation as ∆t → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' The numerical stability of this limit approximation is consistent throughout the landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' scapes and their induced trajectories are qualitatively similar to those that would be generated by an optimal value func- tion, derived by standard optimal control techniques based on a reward specifically designed to achieve the top position [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' In our analysis, we chose a particular discretization ∆t = 10−3 s, and we need to show that our results depend only weekly on this choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' For this, we repeat our analysis at dif- ferent ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Figure 3 shows the dependence of the maximum value of the original empowerment (black dot in left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' 2) on ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' To the extent that the estimate converges to a well-defined number linearly as ∆t → 0, the discrete time dy- namics provides a consistent approximation to the continuous time dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Double Pendulum Now we show that the empowerment maximization formalism is capable of dealing with more chal- lenging problems, such as a power-constrained control of a (potentially chaotic) double pendulum [16], Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' 4, with equa- tions of motion: d¨θ1(t) = − 1 d1(t) � d2(t)¨θ2(t) + φ1(t) � , (13) d¨θ2(t) = 1 m2ℓ2c2 + I2 − d2 2(t) d1(t) � da(t) + dW(t) + d2 2(t) d1(t)φ1(t) − m2ℓ1ℓc2 ˙θ1(t)2 sin θ2(t) − φ2(t) � , with d1(t) =m1ℓ2 c1 + m2(ℓ2 1 + ℓ2 c2 + 2ℓ1ℓc2 cos θ2(t)) + I1 + I2, d2(t) =m2(ℓ2 c2 + ℓ1ℓc2 cos θ2(t)) + I2, φ1(t) = − m2ℓ1ℓc2 ˙θ(t)2 sin θ2(t)−2m2ℓ1ℓc2 ˙θ2(t) ˙θ1(t) sin θ2(t) + (m1ℓc1 + m2ℓ1)g cos θ1(t) + φ2(t), φ2(t) =m2ℓc2g cos(θ1(t) + θ2(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' We add Wiener noise, dW(t), and permit the controller to apply a scalar control signal |a(t)| ≤ 1, at the joint between the two links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' In the equations of motion, mi, ℓi, ℓci, and Ii stand for the mass, the length, the length to center of mass, and the moment of inertia of the i-th link, i ∈ [1, 2], respec- 5 Figure 4: Top left: Double pendulum with control torque on the joint between the links with dynamics given by (13) Top right: Slices through the empowerment landscape of a double pendulum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Each subplot shows a particular slice in the 4D landscape, when two other coordinates are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' For example, the plot with axes ˙θ2, ˙θ1 is shown for θ2 = 0 rad and θ1 = 0 rad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Bottom: Traversing the state space of the double pendulum according to (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' The first and the second 15s are shown with different scale for the instantaneous empowerment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' The initial and the final positions are both links down and both links up, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Torque is applied to the middle joint only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Figure 4 shows the landscape for the original empow- erment for selected slices of the phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' This landscape is more complex than for the single-pendulum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Nonetheless it retains the property that, following the local gradient in the state space directly, one ultimately reaches the state of the maximum empowerment, which is precisely where both links of the pendulum are balanced upright.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' The vertical position, however, is a priori not sufficient to guarantee the balanc- ing since the control only applies torque at the joint linking the pendulum halves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' That is, the controller cannot move the pendulum in arbitrary directions through the state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Surprisingly, this concern notwithstanding, the algorithm still balances the pendulum, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Cart-Pole We have additionally verified that the empower- ment maximization also balances an inverted pendulum on a moving cart, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Here the control signal (force) is ap- plied to the cart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Thus the pendulum is now affected only indirectly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' The dynamics of this system is: d¨x(t) =m sin θ(t)(ℓ ˙θ2(t) + g cos θ(t)) + da(t) + dW(t) M + m sin2 θ(t) , (14) d¨θ(t) = − da(t) cos θ(t) − mℓ ˙θ2(t) cos θ(t) sin θ(t) − (M + m)g sin θ(t), where x(t), θ(t), m, M, ℓ, g, |a(t)| ≤ 1 are the x coordinate of the center of mass of the cart, the angle of the pole, the pole mass, the cart mass, the pole length, the free fall acceleration, and the force applied to the cart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Discussion In this study, we focused on a class of intrinsic motivation models that mimic decision-making abilities of biological or- ganisms in various situations without explicit reward signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' We used an information-theoretic formulation in which the controller starts with knowledge of the (stochastic) dynami- cal equations describing the agent and the environment, and then selects actions that “empower” the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' That is, the controller improves its ability to affect the system in the fu- ture, as measured by the mutual information between the ac- tion sequence and the subsequent responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' This leads the system to the most sensitive points in the state space, which we showed solves a problem known to be difficult for simple reinforcement learning algorithms: balancing inverted pen- dula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Depending on which subsets of the past actions and future responses are used to drive the intrinsic motivation, our approach interpolates between the original formulation of empowerment maximization, maximization of the “kicked” version of Causal Entropic Forcing, and maximization of the “controlled” subset of the Lyapunov exponents of the agent- environment pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' This provides insight into which properties of the dynamical system are responsible for the behaviors pro- 6 upright balance 01 01 2 02 Motor 02)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content='912 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content='6 C(α*) in nats 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content='910 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content='908 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content='906 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content='904 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content='8 10 15 15 20 25 30 seconds secondsFigure 5: Left: Cart-Pole system with control force, ⃗a(t), applied to the cart only, which moves on the rail (or on the edge of a table), allowing the pole to rotate in the x-y plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Its dynamics is given by (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Right: Traversing the state space of the pendulum on a cart according to empowerment maximization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' The initial and the final state of the pole are down and up, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' The horizontal axis is time in seconds t ∈ [0, 20]s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' duced by these different motivation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' One big challenge in using information-theoretic quantities is computing them, which can be difficult to do either analyt- ically or from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Our paper makes a significant contribu- tion to solving this problem in the context of empowerment by providing an explicit algorithm for computing various versions of empowerment, for arbitrary lengths of pasts and futures, using the small noise/small control approximation to the dy- namics, while still treating the dynamics as nonlinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' This is often the most interesting regime, modeling weak, power- constrained controllers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Crucially, our algorithm is local, so that climbing up the empowerment gradient only requires es- timation of the dynamics in the vicinity of the current state of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' This should be possible in real control appli- cations by using the data directly, possibly with the help of deep neural networks to approximate the relevant dynamical landscapes [45–47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Therefore, knowing the exact form of the dynamical system, which could be a potential limitation of our approach, is not strictly required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' This opens up oppor- tunities for scaling our method to more complex scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Our work suggests that, in addition to the Lyapunov spec- trum, defined via the trajectory divergence in time due to a small arbitrary perturbation, one may want to consider the optimal Lyapunov spectrum, where the initial perturbation is optimally aligned with the controllable directions in the dy- namics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' We defer a systematic study of optimal Lyapunov spectra to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' A potential extension of our analysis relates to social in- teractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Interacting agents have their own intrinsic moti- vations and affect each other’s ability to achieve their goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Understanding how multiple agents interact, each trying to empower itself in the presence of others, and whether and when this leads to cooperation or conflict is a promising area for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Crucially, the ability to affect someone else’s empowerment may provide insight into what distin- guishes social interactions from purely physical interactions among nearby individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Acknowledgements ST was supported in part by Califor- nia State University, and the College of Engineering at SJSU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' IN was supported in part by the Simons Foundation Investi- gator award, the Simons-Emory Consortium on Motor Con- trol, and NIH grant 2R01NS084844.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' DP acknowledges partial support by the EC H2020-641321 socSMCs FET Proactive project and the Pazy Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' References [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Oudeyer and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Kaplan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' What is intrinsic motiva- tion?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' a typology of computational approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Frontiers in neurorobotics, 1:6, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' [2] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Sutton and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Barto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Reinforcement learning: An introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' MIT press, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' [3] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Doya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Reinforcement learning in continuous time and space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Neural computation, 12(1):219–245, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' [4] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Mohamed and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Rezende.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Variational informa- tion maximisation for intrinsically motivated reinforce- ment learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' In Advances in neural information pro- cessing systems, pages 2125–2133, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' [5] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Gregor, D J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Rezende, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Wierstra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Variational intrinsic control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' arXiv preprint arXiv:1611.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content='07507, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' [6] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Baumli, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Warde-Farley, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Hansen, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Mnih.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Relative variational intrinsic control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Proceedings of the AAAI Conference on Artificial Intelligence, 35(8):6732– 6740, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' [7] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Kwon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Variational intrinsic control revisited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Interna- tional Conference on Learning Representations, (ICLR), 2021, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' [8] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Sharma, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Gu, S Levine, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Kumar, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Hausman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Dynamics-aware unsupervised discovery of skills.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' arXiv preprint arXiv:1907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content='01657, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' [9] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Sharma, M Ahn, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Levine, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Kumar, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Haus- man, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Gu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Emergent real-world robotic skills via unsupervised off-policy reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' RSS, Robotics: Science and Systems 2020, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' [10] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Eysenbach, A Gupta, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Ibarz, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Levine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Di- versity is all you need: Learning skills without a reward function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' International Conference on Learning Repre- sentations, (ICLR), 2019, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' [11] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Houthooft, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Duan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Schulman, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' De Turck, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Abbeel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Vime: Variational infor- mation maximizing exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Advances in neural in- formation processing systems, 29, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' [12] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Achiam, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Edwards, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Amodei, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Abbeel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Variational option discovery algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' arXiv preprint arXiv:1807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content='10299, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' [13] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Choi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Sharma, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Lee, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Levine, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Gu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Vari- ational empowerment as representation learning for goal- conditioned reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' International Con- ference on Machine Learning, pages 1953–1963, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' [14] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Salge, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Glackin, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Polani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Empowerment– an introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' In Guided Self-Organization: Inception, pages 67–114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Springer, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' [15] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Klyubin, D Polani, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Nehaniv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Em- powerment: A universal agent-centric measure of con- trol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' IEEE Congress on Evolutionary Computation, 2005, 1:128–135, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' [16] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Jung, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Polani, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Stone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Empowerment for con- tinuous agent—environment systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Adaptive Behavior, 19(1):16–39, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' 7 m 0 TY X ()p M[17] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Klyubin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Polani, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Nehaniv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Keep your options open: An information-based driving prin- ciple for sensorimotor systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' PLoS ONE, 3(12):e4018, Dec 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' [18] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Zhao, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Abbeel, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Tiomkin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Efficient empower- ment estimation for unsupervised stabilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Interna- tional Conference Learning Representations, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' [19] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Zhao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Tiomkin, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Abbeel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Dynamical system embedding for efficient intrinsically motivated artificial agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Advances in Neural Information Processing Sys- tems, (NeurIPS), DeepRL, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' [20] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Wissner-Gross and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Freer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Causal entropic forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Physical review letters, 110(16):168702, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' [21] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Cover and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Thomas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Elements of information theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' John Wiley & Sons, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' [22] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Klyubin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Polani, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Nehaniv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Empow- erment: A universal agent-centric measure of control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' In 2005 IEEE Congress on Evolutionary Computation, vol- ume 1, pages 128–135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' IEEE, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' [23] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Eysenbach, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Gupta, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Ibarz, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Levine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Di- versity is all you need: Learning skills without a reward function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' In ICLR, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' [24] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Du, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Tiomkin, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Kiciman, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Polani, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Abbeel, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Dragan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Ave: Assistance via empowerment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Neural Information Processing Systems, NeurIPS 2020, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' [25] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Salge and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Polani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Empowerment as replacement for the three laws of robotics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Frontiers in Robotics and AI, 4:25, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' [26] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Bialek, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Nemenman, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Tishby.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Predictabil- ity, complexity, and learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Neural computation, 13(11):2409–2463, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' [27] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Holmes and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Nemenman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Estimation of mutual infor- mation for real-valued data with error bars and controlled bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Phys Rev E, 100:022404, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' [28] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Jung, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Polani, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Stone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Empowerment for con- tinuous agent—environment systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Adaptive Behavior, 19(1):16–39, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' [29] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Zhao, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Lu, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Abbeel, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Tiomkin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Efficient em- powerment estimation for unsupervised stabilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' In International Conference on Learning Representations, ICLR, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' [30] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Wissner-Gross and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Freer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Causal entropic forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Physical review letters, 110(16):168702, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' [31] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Ng, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Harada, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Russell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Policy invariance under reward transformations: Theory and application to reward shaping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' [32] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Dayan and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Hinton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Feudal reinforcement learn- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' In Advances in neural information processing sys- tems, pages 271–278, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' [33] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Kulkarni, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Narasimhan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Saeedi, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Tenen- baum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Hierarchical deep reinforcement learning: Inte- grating temporal abstraction and intrinsic motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' In Advances in Neural Information Processing Systems, NeurIPS, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' [34] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Burda, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Edwards, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Pathak, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Storkey, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Darrell, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Efros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Large-scale study of curiosity-driven learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' In ICLR, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' [35] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Pathak, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Agrawal, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Efros, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Darrell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Curiosity-driven exploration by self-supervised predic- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' In ICML, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' [36] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Karl, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Soelch, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Becker-Ehmck, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Benbouzid, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' van der Smagt, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Bayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Unsupervised real-time control through variational empowerment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' [37] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Salge, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Glackin, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Polani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Approximation of empowerment in the continuous domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Advances in Complex Systems, 16(02n03):1250079, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' [38] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Salge, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Glackin, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Polani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Empowerment– an introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' In Guided Self-Organization: Inception, pages 67–114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Springer, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' [39] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Anthony, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Polani, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Nehaniv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' General self-motivation and strategy identification: Case studies based on sokoban and pac-man.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' IEEE Transactions on Computational Intelligence and AI in Games, 6(1):1–17, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' [40] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Charlesworth and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Turner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Intrinsically mo- tivated collective motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Proceedings of the National Academy of Sciences, 116(31):15362–15367, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' [41] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Schmidhuber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Formal theory of creativity, fun, and intrinsic motivation (1990–2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' IEEE Transactions on Autonomous Mental Development, 2(3):230–247, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' [42] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Abarbanel, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Brown, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Kennel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Local lya- punov exponents computed from observed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Journal of Nonlinear Science, 2(3):343–365, 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' [43] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Evans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' An introduction to mathematical optimal control theory version 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' 1983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' [44] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Kuo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' The six determinants of gait and the inverted pendulum analogy: A dynamic walking perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Hu- man movement science, 26(4):617–656, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' [45] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Daniels and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Nemenman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Automated adaptive inference of phenomenological dynamical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Nature communications, 6(1):1–8, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' [46] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Brunton, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Proctor, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Kutz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Discovering governing equations from data by sparse identification of nonlinear dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Proceedings of the national academy of sciences, 113(15):3932–3937, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' [47] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Chen, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Huang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Raghupathi, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Chandratreya, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Du, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Lipson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Automated discovery of funda- mental variables hidden in experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' Nature Computational Science, 2(7):433–442, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} +page_content=' 8' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAyT4oBgHgl3EQfOfYw/content/2301.00005v1.pdf'} diff --git a/x9FKT4oBgHgl3EQfLS3X/content/tmp_files/2301.11746v1.pdf.txt b/x9FKT4oBgHgl3EQfLS3X/content/tmp_files/2301.11746v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c3292df64d295cd349c3c02e2bfdc70e094b3501 --- /dev/null +++ b/x9FKT4oBgHgl3EQfLS3X/content/tmp_files/2301.11746v1.pdf.txt @@ -0,0 +1,1156 @@ +Emergence of collapsed snaking related dark and bright Kerr dissipative +solitons with quartic-quadratic dispersion +Edem Kossi Akakpo,1 Marc Haelterman,1 Francois Leo,1 and Pedro Parra-Rivas2, 1 +1)OPERA-photonics, Université libre de Bruxelles, 50 Avenue F. D. Roosevelt, CP 194/5, B-1050 Bruxelles, +Belgium +2)Dipartimento di Ingegneria dell’Informazione, Elettronica e Telecomunicazioni, Sapienza Universitá di Roma, via Eudossiana 18, +00184 Rome +(*Electronic mail: pedro.parra-rivas@uinroma1.it) +We theoretically investigate the dynamics, bifurcation structure and stability of dark localized states emerging in Kerr +cavities in the presence of second- and fourth-order dispersion. These states form through the locking of uniform wave +fronts, or domain walls, connecting two coexisting stable uniform states. They undergo a generic bifurcation structure +known as collapsed homoclinic snaking. We characterize the robustness of these states by computing their stability +and bifurcation structure as a function of the main control parameter of the system. Furthermore, we show that by +increasing the dispersion of fourth order, bright localized states can be also stabilized. +I. +INTRODUCTION +The generation of dissipative solitons, also referred to as +localized states (LSs), in externally driven dispersive Kerr op- +tical resonators hinges in a double balance condition between +four different factors which compensate pairwise: Kerr non- +linearity counteracts chromatic dispersion (similarly to sin- +gle pass conservative systems), while energy dissipation is +compensated through external driving1. +The dynamics of +the cavity can be basically controlled by tuning the external +energy source. However, in most of the applications, it is +extremely important the careful management of dispersion, +which has become a fundamental engineering problem. In mi- +croresonators, for example, the dispersion can be engineered +through the resonator geometry2. Recently, a new accurate +control of dispersion, based on an intracavity pulse shaper, +was proposed by Runge et al. in soliton lasers, opening new +avenues regarding LSs formation and control3. +Generally, the formation of dissipative solitons in Kerr res- +onators is based on the concept of bistability and front lock- +ing: when two different stable states coexist for the same +range of parameters, front waves may form, interact and +lock, leading to the formation of a plethora of LSs of dif- +ferent extensions4,5. This mechanism has been useful to ex- +plain the formation of LSs based on second-order dispersion +(SOD)6–10, but also to understand the implications that higher- +order dispersion effect may have on the LS dynamics and +stability11. One of the first studies in this topic, showed that +fourth-order dispersion (FOD) was able to stabilize dark lo- +calized pattern in a regime where otherwise such states were +absent12. Later, different studies have tackled the effect of +third-order dispersion (TOD) on different bistable configura- +tions, showing that, in general, new type of bright solitons can +be formed11,13. +In the last few years, dissipative solitons have been also +studied in Kerr cavities driven at the pure FOD point14,15, and +their bifurcation structure and stability analyzed in the anoma- +lous and normal regimes16. In the first case, single and multi- +peak LSs persist and are stable over a wider parameter region +than those in the pure SOD case. Moreover, in the second +scenario, pure FOD is able to stabilize bright LSs. Thus, in +(a) +(b) +Figure 1. (a) Sketch of a externally driven Kerr resonator of length +L. θ represents the transmission coefficient of the coupler between +the Ein and the cavity. β2 and β4 are the second- and fourth-order +dispersion coefficients (b) Coexistence of bright and dark LSs for +large β4. +general, higher-dispersion effects seem to have a positive im- +pact on soliton formation and stability. +In this work, we analyze a externally driven Kerr resonator +[see sketch in Fig. 1(a)] in a operation regime not tackled pre- +viously, were both normal SOD and FOD effects contribute to +the cavity dynamics. We show that in this regime, the lock- +ing of fronts connecting two uniform bistable states leads to +the formation of collapsed snaking related dark LSs17. This +configuration shows morphological similarities with the nor- +mal SOD scenario6, despite the different dynamical instabili- +ties thresholds and the LSs existence domains. Increasing the +FOD strength, we find that different types of bright LSs are +stabilized and coexist with dark ones [see Fig. 1(b)], in a fash- +ion similar to the one described in the context of TOD11. In +what follows, we unveil the bifurcation structure organization +associated with such coexistence, showing the main dynami- +cal regimes of the system. +This paper is organized as follows. In Sec. II we introduce +the model and give some preliminary results. We present the +time-independent problem (Sec. II A), show the homogenous +steady state solution (Sec. II B), and compute its linear stabil- +ity (Sec. II C). Section III focuses on the bifurcation structure +and stability of dark LSs and in Sec. IV we study the spatio- +temporal dynamics of some of the LSs. The effect that the +variation of FOD may have on the dynamics and stability of +the dark LSs and the stabilization of bright LSs is analyzed in +Sec. V. Finally, we discuss the results and draw some conclu- +arXiv:2301.11746v1 [nlin.PS] 27 Jan 2023 + +2 +sions in Sec. VI. +II. +THE LUGIATO-LEFEVER MODEL WITH QUARTIC +DISPERSION +In the mean-field approximation, passive Kerr cavities can +be described by the Lugiato-Lefever (LL) equation18. Consid- +ering chromatic dispersion up to fourth-order, and neglecting +the contribution of TOD, the normalized LL equation reads +∂tA = −(1+i∆)A−id2∂ 2 +x A+id4∂ 4 +x A+i|A|2A+S, +(1) +where A is the complex field amplitude, t represents the time +coordinate, and x the fast time in fiber cavities or angular vari- +able in microresonators. The losses are normalized to 1, ∆ +is the phase detuning from the closest cavity resonance and +S in the driving field amplitude. +With this normalization, +d2 = sign(β2) = ±1 and d4 ≡ β4α/(6L|β2|2), where α repre- +sent the losses, L is the cavity length, and β2 and β4 represent +the SOD and FOD coefficients, respectively. In this work we +focus on the regime defined by d2 = 1 and d4 > 0. +A. +The time-independent problem: Spatial dynamics +LSs and any other type of time-independent states (∂tA = 0) +satisfy the complex ordinary differential equation +id4∂ 4 +x A = id2∂ 2 +x A+(1+i∆)A−i|A|2A−S. +(2) +Considering A = U + iV, Eq. (2) can be recast into the 8D +spatial dynamical system +dY +dx = F(Y,d2,d4,∆,S), +(3) +where the new variables read +Y = (Y1,Y2,··· ,Y8) = (U,V,∂xU,∂xV,∂ 2 +x U,∂ 2 +x V,∂ 3 +x U,∂ 3 +x V), +and the vector field F is defined as +� +� +� +� +� +� +� +� +� +� +� +Fm = Ym+2, +m = 1,··· ,6 +F7 = 1 +d4 +[∆Y1 −(Y 2 +1 +Y 2 +2 )Y1 +Y2 +d2Y5] +F8 = − 1 +d4 +[Y1 +∆Y2 −(Y 2 +1 +Y 2 +2 )Y2 +d2Y6 +S] +By using the dynamical system (3), we can apply well +known results of dynamical systems and bifurcation theory +to study our problem. Furthermore, this approach allows us +to apply a correspondence between the time-independent so- +lutions of the system [i.e., solutions of Eq. (2)] and different +solutions of Eq. (3). In this context, a uniform front corre- +sponds to a heteroclinic orbit, a LS is a homoclinic orbit, a +spatially periodic pattern a limit cycle, and the homogeneous +state of the system a fixed point19. +Equation 3 will be used for computing LSs through nu- +merical path-continuation algorithms based on a predictor- +corrector method20,21 using the free software package AUTO- +07p22. With this procedure, we are able to compute not only +t +m +b +t +m +b +C +Figure 2. (a) Homogeneous steady state in the bistable regime for +∆ = 5. (b) Marginal instability curve corresponding to (a). Panel (c) +shows the (∆,S)-parameter space. The vertical line corresponds to +the situation shown in (a). The horizontal line to the nonlinear cavity +resonance shown in (d) for S = 2.8. (e) Shows the formation of a +dark LS for ∆ = 5 and S = 2.8. The blue curve in (e) represents the +initial condition. +the stable but also the unstable state solutions, unveiling their +connection. The stability is determined after continuation by +solving the eigenvalue problem +L ψ = σψ, +(4) +where L is the linear operator associated with the right hand +side of Eq. (1) evaluated a given steady state, and ψ is the +eigenmode corresponding to the eigenvalue σ. +The time- +independent state is stable if Re[σ] < 0, and unstable oth- +erwise. +If by varying a control parameter of the system, +let’s say p, this transition occurs at the value p = pc (i.e., +Re[σ(pc)] = 0), we say that a local bifurcation takes place +at pc23,24. + +3 +B. +Homogeneous steady state +The simplest time-independent solution is the uniform or +homogeneous steady state (HSS) solution Ah (i.e., ∂xA = 0), +which satisfies the algebraic equation +(1+i∆)Ah −i|Ah|2Ah −S = 0. +(5) +This equation can be rewritten in the form +S2 = I3 +h −2∆I2 +h +(1+∆2)Ih, +(6) +with Ih ≡ |Ah|2. For a fixed value of ∆, this expression defines +a nonlinear dependence between the intracavity intensity Ih +and the pump S. An example of such dependence is depicted +in Fig. 2(a) for ∆ = 5. +For this value of ∆, the system shows multistability, i.e., +for the same value of S three HSSs, namely Ab +h, Am +h , and At +h, +coexist. These states are connected at two folds, or turning +points, located at the positions +Il,r +h ≡ 1 +3 +� +2∆± +� +∆2 −3 +� +. +(7) +As a function of ∆, these folds points define the two solid lines +plotted in the (∆,S)-parameter diagram shown in Fig. 2(c). +Decreasing ∆, eventually, these two folds meet and disappear +in a cusp bifurcation Ch occurring exactly at ∆ = +√ +3. The +vertical dashed line shown in Fig. 2(c) corresponds to the dia- +gram plotted in Fig. 2(a). +For a fixed value of S, Eq. (6) is solved by the expression +∆ = Ih ± +� +(S/Ih)2 −1. +(8) +An example of these solution branches is plotted in Fig. 2(d) +for S = 2.8, and represents the nonlinear resonance of the cav- +ity. +C. +Linear stability analysis of the uniform state +The linear stability of HSS against perturbations ∼ eσtψk +can be determined analitically. +In this case, eigenmodes +read ψk = eikx + c.c., while the eigenvalues depend on the +wavenumber k through the dispersion relation +σ(k) = −1± +� +−K(k)2 −2(∆−2Ih)K(k)−C, +(9) +with +K(k) = −d2k2 −d4k4, +(10a) +and +C = (∆−Ih)2 −2Ih(∆−Ih) +(10b) +For non-uniform perturbations (k ̸= 0), the transition sta- +ble/unstable takes place at the Turing or modulational in- +stability (MI), which satisfies simultaneously the conditions +∂kσ(k) = 0 and σ(k) = 0 for a critical wavenumber k = kc. +From this condition, MI occurs at Ih = 1, and the growing +perturbation at this point has the wavenumber +kc = ± +� +� +� +�−d2 ± +� +d2 +2 −4d4(2−∆) +2d4 +, +(11) +with d2 = 1 for our regime. +The condition σ(k) = 0 leads also to the marginal instabil- +ity curve +Ih = 1 +3 +�� +K2 +2K∆+∆2 −3±2(∆+K) +� +, +(12) +which separates stable from unstable regions. +Figure 2(b) shows the marginal instability curve corre- +sponding to the the HSS shown in Fig. 2(a). The minimum +of this curve corresponds to the MI, and the area inside it to +the unstable HSSs. In correspondence, stable (unstable) HSSs +are marked using solid (dashed) lines in Fig. 2(a). At k = 0, +two homogeneous instabilities [i.e., saddle-node (SN) bifurca- +tions] occur at the fold positions [see Eq. (7)]. In what follows +we label these points SNl,r +h . +III. +BIFURCATION ANALYSIS FOR DARK LOCALIZED +STATES +A. +Bistability and plane-front locking +In the bistability region shown in Fig. 2, fronts connecting +At +h and Ab +h forwards and backwards can form. These fronts +drift at a constant speed which depends on the parameters +of the system. The speed increases (decreases) as the sys- +tem parameters approach (separate) from the Maxwell point +of the system, where it cancels out. Around this zero-speed +point, fronts can lock if oscillatory tails exist in the front pro- +files, leading to the formation of LSs of different widths5,25. +Asymptotically, these tails can be described through the ex- +pression A(x) − Ah ∼ eλx, where λ is the spatial eigenvalue +of the system evaluated at Ah. The spatial eigenvalues can +be computed through the Jacobian associated with Eq. (3), or +equivalently, by solving the equation σ(−iλ) = 0, namely +d2 +4λ 8 −2d2d4λ 6 +(4Ihd4 −2∆d4 +d2 +2)λ 4 +−(4Ihd2 +2∆d2)λ 2 +(∆2 −4Ih∆+3I2 +h +1) = 0. +In our regime (d2 = 1 and d4 = 1), the previous equation has +8 solutions that read +λ = ± 1 +√ +2 +� +−1± +� +4∆−8Ih +1±4 +� +Ih2 −1 +(13) +The dynamically relevant eigenvalues are those related with +the slow dynamics of the system (i.e., those with smallest +|Re[λ]|). Oscillatory tails appear if the dominant eigenval- +ues are complex-conjugate. When these eigenvalues are all + +4 +(b) +(a) +Figure 3. (a) Modification of the spatial eigenvalues associated with +At +h as a function of S for ∆ = 5. The gray plane corresponds SNl +h, +and the blue one to SM. In this plane, the most relevant eigenvalues +are surrounded by a gray oval. (b) Same than in (a) but for Ab +h. +real numbers, the tails are monotonic. The transition between +these two configurations depends on the parameters of the sys- +tem (for this case ∆ and S). The modification of the spatial +eigenvalues associated with At +h, and Ab +h as a function of S are +depicted in Figs. 3(a) and (b), respectively, for ∆ = 5. +Let’s take a look to the eigenvalues associated with Ab,t +h +around SM (see blue planes in Fig. 3). For Ab +h, the dominant +eigenvalues around SM [see oval gray shape in Fig. 3(b)] have +a large imaginary part and a very small real part, what means +that the oscillatory tails will have small wavelength and weak +decay. This leads to well defined oscillatory tails around Ab +h. +For At +h [see Figs. 3(a)], although the dominant eigenvalues are +also complex conjugate, the imaginary and real parts are re- +spectively smaller and larger than in Figs. 3(b), which leads to +oscillatory tails with a very strong decay and very large wave- +length: effectively a quasi monotonic tail. Therefore, front +H1 +H1 +Figure 4. Collapsed homoclinic snaking for ∆ = 5. Solid thick (thin) +lines correspond to stable (unstable) states. Some examples of dark +LSs are shown on the right panels (i)-(v). +Bi label the solution +branches in-between SNl,r +i , and SM is the uniform Maxwell point +of the system. +locking will be favored around Ab +h, in contrast with At +h. +An example of front locking is shown in Fig. 2(e). This +temporal evolution has been computed through a direct nu- +merical simulation of Eq. (1) starting from a super-Gaussian +profile subtracted to At +h close to SM [shown in blue in +Fig. 2(e)]. Initially, two fronts with opposite polarity form and +approach one-another as time passes. Eventually, they lock at +a fixed separation D, yielding a dark LS with two central dips. +The time evolution of that separation can be described by the +equation19 +∂tD = ρeRe[λ]Dcos(Im[λ]D)+η, +(14) +where ρ depends on the parameters and η ∼ S−SM. Although +the front interaction appearing here is generic, and well de- +scribed by Eq. (14), its derivation from Eq. (1) is not realiz- +able, and we introduce it here for explaning the front locking +mechanism. The fixed points De of this system (∂tDe = 0) +correspond to the locking of fronts, and hence, to the forma- +tion of LSs of width De. Thus, for the same value of η, LSs +of different widths may coexist. As we will see in the com- +ing section, LSs formed through front locking organize in a +particular bifurcation structure which depends directly on the +interaction law (14). +B. +Bifurcation structure: Collapsed homoclinic snaking +To fully understand the formation of these states we per- +form a bifurcation analysis based on the path-continuation + +- +Im +- +1 +- +-2 +- +- +1 +1 +14Q-1- +S1 +2.5Mi +3 +3MI +/ +.5 +4 +SIm2℃4-V +5Q +1-2 +1 +1 +2 +1 +- +- +0 +-2 +Re(入)SN +2.4 +2S +M +2.8 +3 +.6 +S3.2 +3.4I0 +SN: +-2 +Re(入)5 +(a) +(b) +(c) +(d) +(b) +(c) +(d) +MI +h +h +H2 +, +H1 +, +C +Figure 5. (a) (∆,S)-parameter space for β4 = 1. This diagram shows the main bifurcations of the system: the MI, saddle-node bifurcations of +the homogeneous state SNl,r +h , and saddle-node bifurcations associated with the collapsed homoclinic snaking SNl,r +i . Panels (b)-(d) show the +collapsed snaking for ∆ = 3,5 and 8 respectively, corresponding to the vertical dashed lines plotted in (a). Solid thick and thin lines represent +stable and unstable LS solutions. The labels Hl,r +i +mark the position of the Hopf bifurcations leading to breathing behavior. +techniques described in Sec. II A. The output of these com- +putations leads the bifurcation diagram shown in Fig. 4 where +the energy of A, i.e., the L2-norm +||A||2 ≡ L−1 +� L/2 +−L/2 |A(x)|2dx +is plotted as a function of S for ∆ = 5. This diagram is known +as collapsed homoclinic snaking and is generic of systems +where LSs emerge through uniform-front locking. Indeed, the +damped oscillatory shape of the bifurcation curve around SM +is a direct consequence of the front interaction and locking de- +scribed by Eq. (14) (see Ref.19 for a general description). The +collapsed snaking curve emanates from SNl +h. Close to this bi- +furcation, LSs have small amplitude and are unstable. These +small amplitude states can be computed through multi-scale +perturbation theory as done in the standard LL equation6,19. +Following the diagram downwards, the dark LSs undergo a +sequence of saddle-node bifurcations appearing in pairs SNr,l +i , +where the sub-index i represents the number of dips appearing +in each state. In these bifurcations, LSs gain and loss stabil- +ity, and at each SNl +i an extra dip is nucleated in the struc- +ture at x = 0. In this way, the width of the LSs increases +while decreasing ||A||2. In what follows, we mark the solu- +tion branches connecting SNl,r +i +as Bi (see Fig. 4). +C. +Persistence in the (∆,S)-parameter space +Figure 5 shows the modification of the SNl,r +i +bifurcations +(see blue lines) in the (∆,S)-parameter space, together to SNl,r +h +and MI. With decreasing ∆, the pairs SNl,r +i +approach each +other and eventually they meet in a sequence of cusp bifur- +cations Ci (here we only show C1). Those with larger i, cor- +responding to SNs down in the collapsed snaking shown in +Fig. 4, are the one disappearing first, when decreasing ∆. This +phase diagram is common in systems sharing similar HSS +linear stability, such as in passive Kerr cavities with second- +order chromatic dispersion6 and in dispersive cavity enhanced +second-harmonic generation26. +The modification of the collapsed snaking diagram along +the (∆,S)-parameter space is illustrated in Fig. 5(b)-(d) for +∆ = 3, 5 and 8, respectively [see vertical dashed lines in +Fig. 5(a)]. For ∆ = 3, all the Bi branches are stable, and +most of them correspond to narrow states. Increasing ∆, cusp +Ci with larger i appear, and wider states emerge. Moreover, +Bi become wider due to the separation of their corresponding +SNl,r +i . +The linear stability analysis along these diagrams shows +that, eventually, B1 undergoes a pair of Hopf bifurcaitons +Hl,r +1 , where the single-dip LS becomes unstable in favor of +localized oscillations (i.e., breathers). This is the situation de- +picted in Fig. 5(c) for ∆ = 5. We will explore the spatiotem- +poral dynamics of these states in Sec. IV. +Further increasing ∆, B2 also destabilizes through the Hopf +bifurcations Hl,r +2 [see Fig. 5(d) for ∆ = 8], leading to the ap- +pearance of new breather states. A formal comparison with +the standard LL equation in the normal chromatic dispersion +regime (see Ref.6) shows that the destabilization of the Bi +branches occurs for larger values of ∆. +IV. +SPATIO-TEMPORAL DYNAMICS +Figure 6 shows a portion of the bifurcaiton diagram de- +picted in Fig. 5(d) for ∆ = 8, where the maxima and minima +of the oscillatory variation of the breathers’ norm emerging +from Hl,r +1 are plotted using orange circles, whereas those ema- +nating from Hl,r +2 are shown using red circles. In what follows, +we refer to this set of breather branches as Feigenbaum-like +diagrams (F)27: F1 (orange dots) is connected to B1, while F2 +forms along B2 (red dots). These diagrams, and therefore the +breathers, bifurcate supercritically from every Hl,r +i . +Let us first focus on F1. An example of the breather belong- +ing to this set is depicted in Fig. 6(i) for S = 3.52. This state +emerges with a small amplitude form Hl +1 and with a single os- + +6 +b +(ii) +(i) +Figure 6. Top part of the collapsed snaking diagram for ∆ = 8 where we show all the local maxima and minima of the breathers. We use orange +dots for the breathers emerging from B1 at Hl,r +1 (i.e., F1), and red dots for those bifurcating from Hl,r +2 along B2 (i.e., F2). PD corresponds to a +period-doubling bifurcation, and b marks the emergence of beating behavior. Panels (i) and (ii) show two examples of breathers corresponding +to F1 and F2, respectively. +cillatory period. The breather undergoes a sequence of period- +doubling (PD) bifurcations yielding more complex dynamics, +that we do not show here. The PD cascade is observable from +the F1 diagram, although we only mark the first one at each +side. Near ∆ ≈ 3.6, the system evolves to temporal chaos, and +this PD sequence is inverted. Eventually the chaotic state dies, +possibly in a boundary crisis of the attractor (BC)27, and the +system ends up in the closest basin of attraction: the HSS At +h. +This situation share similarities with the one reported in Kerr +cavities when only second-order dispersion is considered6. +The F2 diagram in Fig. 6 shows more complexity than F1, +and furthermore, does not encounter any crisis or instability +destroying the oscillatory states. +An example of a single- +period breather bifurcating from Hl +2 is depicted in Fig. 6(ii) +for S = 3.82. Here, F2 shows the transition between differ- +ent dynamical regimes including beating phenomena (b) be- +tween different frequencies and PD cascades27. Figure 7(a) +shows a close-up view of F2 where these transitions are illus- +trated in more detail. The vertical dashed lines in Fig. 7(a) +correspond to the time series and frequency spectra shown be- +low [see Figs. 7(i)-(viii)], which represent the evolution of the +breather intensity norm ||A||2 and its Fourier transform ||A||2 +f , +respectively. From the modification of the spectra, one can +clearly understand the transition between the different tempo- +ral dynamical regimes. Such modification is depicted in the +color-map shown in Fig. 7(b) all along F2. +At (i) the breather is characterized by a single frequency and +their harmonics. At b, a new frequency arises [see Fig. 7(b)], +and the breather undergoes a beating phenomenon between +the former and latter frequency. In Fig. 7 (ii), the new fre- +quency is marked using a red arrow and its difference with +the former one with δ. Because of this beating, the tempo- +ral series (see top panel) shows modulation in the amplitude, +characterized by the beat period 2π/δ. At (iii), the δ is larger, +resulting in a smaller modulation on the temporal series. In- +creasing S further, the beating becomes more chaotic leading +to the dynamical behavior shown in Fig. 7 (iii). Eventually +this states dies out, leading to a single frequency breather as- +sociated with the time series shown in Fig. 7 (iv). Increasing +more S, a sequence of PD bifurcation occur leading to the +breather dynamics depicted in Fig. 7 (v)-(vii). +V. +IMPLICATIONS OF FOURTH-ORDER DISPERSION +ON THE BIFURCATION STRUCTURE AND STABILITY +Previously, we have performed bifurcation analysis for a +fixed value of d4, namely d4 = 1. Here we explore the effects +of modifying d4 on the stability of LSs, and on the their bifur- +cation structure. We will see that large values of d4 may also +lead to the emergence of bright LSs, similarly to the scenario +described in Ref.11 for TOD. +Figure 8 shows the modification of the collapse snaking di- +agram for ∆ = 5 with increasing d4: d4 = 0 in Fig. 8(a), d4 = 2 + +1060 +t +55190 +t +18510 +I/A//2 +5; S = 3.5204V +10 +1IA2 +58; S = 3.819750 +60 +70 +80 +90100 +110 +120 +130 +140180 +60 +70 +80 +90100 +110 +120 +130 +147 +b +Figure 7. (a) Close-up view of the Fiegenbaum-like diagram F2 between Hl,r +2 (see Fig. 6). (b) Modification of the frequency spectrum of the +oscillatory states along the diagram shown in (a). To plot this spectrum we have filtered out the frequencies below the red dashed line depicted +in panels (i)-(vi). These panels show the time series and frequency spectra of the different oscillatory states [see vertical dashed lines in (a) +and (b)]. +in Fig. 8(b), and d4 = 10 in Fig. 8(c). Regarding the top pan- +els, we can see how by increasing d4, the branches Bi be- +come wider, leading to a wider existence region for the differ- +ent dark LSs. Furthermore, d4 has an stabilizing effect on the +breather states emerging from B1 and B2. This phenomenon +can be easily observed comparing Fig. 8(a) and Fig. 8(b). For +d4 = 2, B2 has stabilized completely while B1 partially. Both +stabilizations occur due to the movement of the Hopf bifurca- +tions along the branches. +Performing a two parameter continuation of the different +saddle-node bifurcations SNl,r +i +in Fig. 8(a)-(c)[top], we are +able to compute their modification in parameters S and d4. +The results are shown in the (d4,S)-phase diagram shown in +Fig. 8(d). The vertical dashed lines correspond to the dia- +grams plotted in Fig. 8(a)-(c)[top]. Decreasing, d4, the pair +SNl,r +i +(for i fixed) comes closer, reducing the extension of Bi +until reaching d4 = 0. For d4 < 0.5 the shrinking process in- +tensifies. In contrast, the pairs SNl,r +i +separate softly with in- + +0 +3.7 +3 +(i) +8.484 +8.461 +8.438 +120 +140 +160 +180 +七 +-2 +-3 +-4.8 +3.9 +4 +(ii) +8.594 +8.539 +8.484 +200 +120 +140 +160 +180 +2 +2 +34.1 +4.2 +(iii) +8.629 +8.539 +8.449 +00 +120 +140 +160 +180 +2 +七 +-2 +3 +-44.3 +4.4 +4 +(iv) +8.66 +8.543 +8.426 +00 +120 +140 +160 +180 +2( +七 +5 +-2 +-2.5 +-3.5.5 +00H? +8.4 +3.7 +3 +(b) +0.8 +0.6 +3 +0.4 +0.2.8 +3.9 +4- +S +4.1 +4.24.3 +4.4 +4..5(i) +8.8 +PD· +(a) +1 +1 +At +1 +1 +8.7 +- +- +- +- +2 +- +8.6 +- +1 +1 +1 +- +8.5 +- +-(iiiii)(iv) +B1(vi)(vii)(viiiH, +2 +B.0.0 +3 +(vi) +8.715 +8.659 +8.603 +200 +120 +140 +160 +180 +2 +七 +.2 +-3 +-4 +1 +0 +0.5 +30.0 +3 +(vii) +8.73 +8.673 +8.616 +00 +120 +140 +160 +180 +2 +七 +-2 +3 +1 +0 +0.5 +30.0 +3 +viii +8.744 +8.691 +8.638 +00 +120 +140 +160 +180 +2 +七 +0 +0.5 +3000.0 +3 +V +8.698 +8.639 +8.58 +120 +140 +160 +180 +-2 +0 +0.5 +38 +d +(i) +(ii) +(iii) +(iv) +(iv) +(i) +(ii) +(iii) +* +* +* +* +H +d +d +d +d +Figure 8. Modification of the collapsed snaking for ∆ = 5 when varying d4: in (a) d4 = 0, in (b) d4 = 2, and d4 = 10 in (c). Examples of bright +LSs of different widths, corresponding to the diagram shown in (c), are depicted in panels (i)-(iv). Panel (d) shows the (∆,d4)-parameter phase +diagram for dark LSs. Panel (e) shows the phase diagram associated with bright LSs. The vertical dashed lines correspond to the diagrams +shown in panels (a)-(c). +creasing d4, and the separation seems to saturate for large val- +ues of d4. The different blue tonalities between the SNl,r +i +lines +correspond to regions where dark LSs with different dips ex- +ist. A similar tendency has also been observed for bright LSs +in the presence of TOD13, where in contrast, the saturation +occurs when SNl,r +i +approach each other with increasing d3. +Figures 8(a)-(c) also show the modification, with increasing +d4, of the bottom part of the collapsed snaking around SM. For +d4 = 0, this curve follows a straight line on top of SM, which +for ||A||2 ≈ 0.9 turns to the right and eventually connects with +MI at Ab +h +6. Increasing d4 [see Fig. 8(b) for d4 = 2], the dia- +gram reaches lower values of ||A||2, and undergoes a pair of +saddle-node bifurcations snl,r +1 +before reaching MI. Between +these folds, a new branch of stable birght LS solutions (B∗ +1) +appears. Increasing β4 further [see Fig. 8(c) for d4 = 10], the +extension of B∗ +1 increases. Two example of bright LSs on this +branch are shown in Figs. 8(i),(ii). Following up this diagram, +new stable branches appear in-between snl,r +2 and snl,r +3 . We la- +bel these branches B∗ +2 and B∗ +3, respectively. With increasing +||A||2, the bright LSs broaden as depicted in Figs. 8(iii),(iv). +The region of existence of these bright states is illustrated in +the (S,d4)-phase diagram shown in Fig. 8(e). The saddle- +node bifurcations snl,r +i +converge rapidly to SM for i > 1. These +states emerge due to the modification of the spatial eigenval- +ues, which now, allow the front locking around Ab +h and At +h. +Finally, to complete our study, we analyse the effect of d4 +on the spatiotemporal dynamics of the system. To do so we +fix (∆,S) = (8,3.97), and scan the variation of the extrema +of the dark LSs norm with d4, as illustrated in Fig. 9. For +d4 = 0, dark LSs are static, and the diagram just shows a sin- +gle branch. Increasing d4 a bit further, the LSs encounter a +Hopf bifurcation H, where they destabilize in favor of a single- +period oscillation, which in a very narrow d4-interval leads to +more complex temporal dynamics characterized by the time +series shown in Figs. 9(i),(ii). Eventually, around d4 ≈ 8.3, the +complex dynamics disappear, yielding a single period breather +again [see temporal series in Fig. 9(iii)]. +VI. +DISCUSSIONS AND CONCLUSIONS +In this paper we have studied the formation, bifurcation +structure and dynamical instabilities of dissipative Kerr LSs + +9 +Figure 9. Modification of the temporal dynamics associated with a single peak dark LSs when varying d4. The plot shows the variation of the +local maxima and minima of ||A||2 while passing from different dynamical regimes (red dots) and the unstable branch associated with the LS +(dashed blue). The time series shown in panels (i)-(iii) correspond to the sections depicted through vertical dashed lines in the panel above. +Here, (∆,S) = (8,3.97). +in the presence of FOD. Here we focus on the case where +d2 = 1 and d4 > 0, a configuration which has not been studied +in previous works. The linear stability analysis of the HSSs in +this configuration shows the presence of bistability between +two coexisting HSS states, namely At +h and Ab +h (see Sec. II). +We show that uniform wave fronts connecting the previous +HSS states can lock, yielding the formation of LSs. For that, a +necessary condition is the presence of oscillatory tails around +either At +h or Ab +h, which is determined by the spatial eigenvalues +of the system. For d4 = 1, our findings show that this locking +is possible only around Ab +h, leading to the formation of dark +LSs (see Sec. III A). +In bifurcation terms, these states undergo collapse homo- +clinic snaking whose extension and stability change with +∆ (see Secs. III B, III C and IV). Collapsed homoclinic +snaking is generic for systems exhibiting uniform-front lock- +ing, and has been reported in a variety of pattern forming sys- +tems including nonlinear quadratic optical resonators8–10,26, +mode-locked vertical external-cavity surface-emitting lasers +(VCSEL)28,29, semiconductor micro-resonators with strong +time-delayed feedback30, and reaction-diffusion systems31–35, +to only cite a few. +For d4 = 1, the scenario presented here is morphologically +very similar to the one found when studying Kerr cavities with +only normal SOD (i.e., d2 = 1)6,36, despite of the extension +of the LS existence regions and the onset of temporal insta- +bilities. This result shows that dark states are robust against +high-order dispersive effects. +This situation changes when increasing d4 (Sec. V). Re- +garding dark LSs, their region of existence initially increases, +although their extension soon saturates for values of d4 ≈ 3. +The most interesting phenomenon is that increasing d4, bright +LS solutions emerge, due to the stabilizing effect of this high- +order dispersion term, and their existence region broadens. +(see Fig. 8 in Sec. V). We find that d4 is also able to tune the +emergence and type of oscillatory states appearing in the sys- +tem, becoming a very relevant parameter to control the tem- +poral dynamics. +The capacity of d4 for stabilizing LSs was discussed in +Ref.12 for a different regime of operation, where dark local- +ized patterns, and their associated homoclinic snaking, were +stabilized. The interaction of LSs, and formation of soliton +molecules, in the presence of FOD effects have been also ana- +lyzed in Kerr cavities37,38. In this context, FOD increases con- +siderably the number of allowed locking distances between +LSs, and therefore the variety of molecules. The stabilization +of bright Kerr LSs in the context of uniform bistable regimes +with collapsed snaking has been also investigated in the pres- +ence of TOD11 and stimulated Raman scattering39. +In summary, dark LSs formation is robust in the presence of +FOD, which furthermore is able to stabilize, bright states. The +advances in dispersion engineering make possible the man- +agement of different dispersion terms and therefore the access +to dispersion regimes previously unattainable3. This capac- +ity makes our work relevant not only from a theoretical point +of view, but also from a experimental perspective, opening the +possibility to observe these type of states and dynamics in real +Kerr cavities. +ACKNOWLEDGEMENTS +PPR acknowledges support from the European Union’s +Horizon 2020 research and innovation programme under the + +10 +Marie Sklodowska-Curie grant agreement no. +101023717. +FL acknowledges support from the European Research Coun- +cil (grant agreement 757800), HORIZON EUROPE Euro- +pean Research Council (57800), and Fonds de la Recherche +Scientifique-FNRS. +REFERENCES +1N. Akhmediev and A. Ankiewicz, eds., Dissipative Solitons. Lecture Notes +in Physics, Berlin Heidelberg: Springer-Verlag, 2005. +2T. J. Kippenberg, A. L. Gaeta, M. Lipson, and M. L. Gorodetsky, “Dis- +sipative Kerr solitons in optical microresonators,” Science, vol. 361, +p. eaan8083, Aug. 2018. +Publisher: American Association for the Ad- +vancement of Science. +3A. F. J. Runge, D. D. Hudson, K. K. K. Tam, C. M. de Sterke, and +A. Blanco-Redondo, “The pure-quartic soliton laser,” Nature Photonics, +vol. 14, pp. 492–497, Aug. 2020. Number: 8 Publisher: Nature Publishing +Group. +4O. Thual and S. Fauve, “Localized structures generated by subcritical in- +stabilities,” Journal de Physique, vol. 49, no. 11, pp. 1829–1833, 1988. +5P. Coullet, “Localized patterns and fronts in nonequilibrium systems,” In- +ternational Journal of Bifurcation and Chaos, vol. 12, pp. 2445–2457, Nov. +2002. +6P. Parra-Rivas, E. Knobloch, D. Gomila, and L. Gelens, “Dark solitons in +the Lugiato-Lefever equation with normal dispersion,” Physical Review A, +vol. 93, p. 063839, June 2016. +7P. Parra-Rivas, L. Gelens, T. Hansson, S. Wabnitz, and F. Leo, “Frequency +comb generation through the locking of domain walls in doubly resonant +dispersive optical parametric oscillators,” Optics Letters, vol. 44, pp. 2004– +2007, Apr. 2019. +8P. Parra-Rivas, L. Gelens, and F. Leo, “Localized structures in dispersive +and doubly resonant optical parametric oscillators,” Physical Review E, +vol. 100, p. 032219, Sept. 2019. +9C. M. Arabí, P. Parra-Rivas, P. Parra-Rivas, T. Hansson, L. Gelens, S. Wab- +nitz, S. Wabnitz, S. Wabnitz, and F. Leo, “Localized structures formed +through domain wall locking in cavity-enhanced second-harmonic gener- +ation,” Optics Letters, vol. 45, pp. 5856–5859, Oct. 2020. +10P. Parra-Rivas, C. Mas Arabí, and F. Leo, “Dissipative localized states and +breathers in phase-mismatched singly resonant optical parametric oscilla- +tors: Bifurcation structure and stability,” Physical Review Research, vol. 4, +p. 013044, Jan. 2022. Publisher: American Physical Society. +11P. Parra-Rivas, D. Gomila, and L. Gelens, “Coexistence of stable dark- and +bright-soliton Kerr combs in normal-dispersion resonators,” Physical Re- +view A, vol. 95, p. 053863, May 2017. +12M. Tlidi and L. Gelens, “High-order dispersion stabilizes dark dissipative +solitons in all-fiber cavities,” Optics Letters, vol. 35, pp. 306–308, Feb. +2010. +13P. Parra-Rivas, D. Gomila, F. Leo, S. Coen, and L. Gelens, “Third-order +chromatic dispersion stabilizes Kerr frequency combs,” Optics Letters, +vol. 39, pp. 2971–2974, May 2014. +14C. Bao, H. Taheri, L. Zhang, A. Matsko, Y. Yan, P. Liao, L. Maleki, and +A. E. Willner, “High-order dispersion in Kerr comb oscillators,” JOSA B, +vol. 34, pp. 715–725, Apr. 2017. Publisher: Optica Publishing Group. +15H. Taheri and A. B. Matsko, “Quartic dissipative solitons in optical Kerr +cavities,” Optics Letters, vol. 44, pp. 3086–3089, June 2019. Publisher: +Optica Publishing Group. +16P. Parra-Rivas, S. Hetzel, Y. V. Kartashov, P. F. d. Córdoba, J. A. Conejero, +A. Aceves, and C. Milián, “Quartic Kerr cavity combs: bright and dark +solitons,” Optics Letters, vol. 47, pp. 2438–2441, May 2022. Publisher: +Optica Publishing Group. +17J. Knobloch and T. Wagenknecht, “Homoclinic snaking near a heteroclinic +cycle in reversible systems,” Physica D: Nonlinear Phenomena, vol. 206, +pp. 82–93, June 2005. +18M. Haelterman, S. Trillo, and S. Wabnitz, “Dissipative modulation instabil- +ity in a nonlinear dispersive ring cavity,” Optics Communications, vol. 91, +pp. 401–407, Aug. 1992. +19P. Parra-Rivas, E. Knobloch, L. Gelens, and D. Gomila, “Origin, bifurcation +structure and stability of localized states in Kerr dispersive optical cavities,” +IMA Journal of Applied Mathematics, vol. 86, pp. 856–895, Oct. 2021. +20E. Doedel, H. B. Keller, and J. P. Kernevez, “Numerical analysis and con- +trol of bifurcation problems (ii): bifurcation in infinite dimensions,” In- +ternational Journal of Bifurcation and Chaos, vol. 01, pp. 745–772, Dec. +1991. +21E. Doedel, H. B. Keller, and J. P. Kernevez, “Numerical analysis and control +of bifurcation problems (i): bifurcation in finite dimensions,” International +Journal of Bifurcation and Chaos, vol. 01, pp. 493–520, Sept. 1991. +22E. J. Doedel, A. R. Champneys, T. F. Fairgrieve, Y. A. Kuznetsov, B. Sand- +stede, and X. Wang, “AUTO-07p: Software for continuation and bifurca- +tion problems in ordinary differential equations,” Department of Computer +Science, Concordia University, Montreal, 2007. +23S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and +Chaos. Texts in Applied Mathematics, New York: Springer-Verlag, 2 ed., +2003. +24A. J. Scroggie, W. J. Firth, G. S. McDonald, M. Tlidi, R. Lefever, and L. A. +Lugiato, “Pattern formation in a passive Kerr cavity,” Chaos, Solitons & +Fractals, vol. 4, pp. 1323–1354, Aug. 1994. +25P. Coullet, C. Elphick, and D. Repaux, “Nature of spatial chaos,” Physical +Review Letters, vol. 58, pp. 431–434, Feb. 1987. +26P. Parra-Rivas, C. M. Arabí, and F. Leo, “Dark quadratic localized states and +collapsed snaking in doubly resonant dispersive cavity-enhanced second- +harmonic generation,” Physical Review A, vol. 104, p. 063502, Dec. 2021. +Publisher: American Physical Society. +27E. Ott, Chaos in Dynamical Systems. Cambridge: Cambridge University +Press, 2 ed., 2002. +28C. Schelte, A. Pimenov, A. G. Vladimirov, J. Javaloyes, and S. V. Gure- +vich, “Tunable Kerr frequency combs and temporal localized states in time- +delayed Gires-Tournois interferometers,” Optics Letters, vol. 44, pp. 4925– +4928, Oct. 2019. Publisher: Optica Publishing Group. +29T. G. Seidel, J. Javaloyes, and S. V. Gurevich, “A normal form for frequency +comb and localized states in time-delayed Kerr-Gires-Tournois interferom- +eters,” Mar. 2022. arXiv: 2203.02411. +30E. R. Koch, T. G. Seidel, J. Javaloyes, and S. V. Gurevich, “Tempo- +ral localized states and square-waves in semiconductor micro-resonators +with strong time-delayed feedback,” Jan. 2023. arXiv:2301.09400 [nlin, +physics:physics]. +31Y. R. Zelnik, P. Gandhi, E. Knobloch, and E. Meron, “Implications of trista- +bility in pattern-forming ecosystems,” Chaos: An Interdisciplinary Journal +of Nonlinear Science, vol. 28, p. 033609, Mar. 2018. +32P. Parra-Rivas and C. Fernandez-Oto, “Formation of localized states in dry- +land vegetation: Bifurcation structure and stability,” Physical Review E, +vol. 101, p. 052214, May 2020. +33F. Al Saadi and A. Champneys, “Unified framework for localized pat- +terns in reaction–diffusion systems; the Gray–Scott and Gierer–Meinhardt +cases,” Philosophical Transactions of the Royal Society A: Mathematical, +Physical and Engineering Sciences, vol. 379, p. 20200277, Dec. 2021. Pub- +lisher: Royal Society. +34F. Al Saadi, A. Worthy, H. Alrihieli, and M. Nelson, “Localised spatial +structures in the thomas model,” Mathematics and Computers in Simula- +tion, vol. 194, pp. 141–158, 2022. +35F. A. Saadi and P. Parra-Rivas, “Transitions between dissipative localized +structures in the simplified Gilad-Meron model for dryland plant ecology,” +Nov. 2022. arXiv:2211.12263 [nlin]. +36P. Parra-Rivas, D. Gomila, E. Knobloch, S. Coen, and L. Gelens, “Origin +and stability of dark pulse Kerr combs in normal dispersion resonators,” +Optics Letters, vol. 41, pp. 2402–2405, June 2016. +37P. Parra-Rivas, D. Gomila, P. Colet, and L. Gelens, “Interaction of solitons +and the formation of bound states in the generalized Lugiato-Lefever equa- +tion,” The European Physical Journal D, vol. 71, p. 198, July 2017. +38A. G. Vladimirov, M. Tlidi, and M. Taki, “Dissipative soliton interaction in +Kerr resonators with high-order dispersion,” Physical Review A, vol. 103, +p. 063505, June 2021. Publisher: American Physical Society. +39P. Parra-Rivas, S. Coulibaly, M. G. Clerc, and M. Tlidi, “Influence of stim- +ulated Raman scattering on Kerr domain walls and localized structures,” +Physical Review A, vol. 103, p. 013507, Jan. 2021. + diff --git a/x9FKT4oBgHgl3EQfLS3X/content/tmp_files/load_file.txt b/x9FKT4oBgHgl3EQfLS3X/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d7a24c699cbc95f3ffde5f29446e80af8eba4dd5 --- /dev/null +++ b/x9FKT4oBgHgl3EQfLS3X/content/tmp_files/load_file.txt @@ -0,0 +1,809 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf,len=808 +page_content='Emergence of collapsed snaking related dark and bright Kerr dissipative solitons with quartic-quadratic dispersion Edem Kossi Akakpo,1 Marc Haelterman,1 Francois Leo,1 and Pedro Parra-Rivas2, 1 1)OPERA-photonics, Université libre de Bruxelles, 50 Avenue F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Roosevelt, CP 194/5, B-1050 Bruxelles, Belgium 2)Dipartimento di Ingegneria dell’Informazione, Elettronica e Telecomunicazioni, Sapienza Universitá di Roma, via Eudossiana 18, 00184 Rome (*Electronic mail: pedro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='parra-rivas@uinroma1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='it) We theoretically investigate the dynamics, bifurcation structure and stability of dark localized states emerging in Kerr cavities in the presence of second- and fourth-order dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' These states form through the locking of uniform wave fronts, or domain walls, connecting two coexisting stable uniform states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' They undergo a generic bifurcation structure known as collapsed homoclinic snaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' We characterize the robustness of these states by computing their stability and bifurcation structure as a function of the main control parameter of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Furthermore, we show that by increasing the dispersion of fourth order, bright localized states can be also stabilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' INTRODUCTION The generation of dissipative solitons, also referred to as localized states (LSs), in externally driven dispersive Kerr op- tical resonators hinges in a double balance condition between four different factors which compensate pairwise: Kerr non- linearity counteracts chromatic dispersion (similarly to sin- gle pass conservative systems), while energy dissipation is compensated through external driving1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' The dynamics of the cavity can be basically controlled by tuning the external energy source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' However, in most of the applications, it is extremely important the careful management of dispersion, which has become a fundamental engineering problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' In mi- croresonators, for example, the dispersion can be engineered through the resonator geometry2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Recently, a new accurate control of dispersion, based on an intracavity pulse shaper, was proposed by Runge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' in soliton lasers, opening new avenues regarding LSs formation and control3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Generally, the formation of dissipative solitons in Kerr res- onators is based on the concept of bistability and front lock- ing: when two different stable states coexist for the same range of parameters, front waves may form, interact and lock, leading to the formation of a plethora of LSs of dif- ferent extensions4,5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' This mechanism has been useful to ex- plain the formation of LSs based on second-order dispersion (SOD)6–10, but also to understand the implications that higher- order dispersion effect may have on the LS dynamics and stability11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' One of the first studies in this topic, showed that fourth-order dispersion (FOD) was able to stabilize dark lo- calized pattern in a regime where otherwise such states were absent12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Later, different studies have tackled the effect of third-order dispersion (TOD) on different bistable configura- tions, showing that, in general, new type of bright solitons can be formed11,13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' In the last few years, dissipative solitons have been also studied in Kerr cavities driven at the pure FOD point14,15, and their bifurcation structure and stability analyzed in the anoma- lous and normal regimes16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' In the first case, single and multi- peak LSs persist and are stable over a wider parameter region than those in the pure SOD case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Moreover, in the second scenario, pure FOD is able to stabilize bright LSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Thus, in (a) (b) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' (a) Sketch of a externally driven Kerr resonator of length L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' θ represents the transmission coefficient of the coupler between the Ein and the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' β2 and β4 are the second- and fourth-order dispersion coefficients (b) Coexistence of bright and dark LSs for large β4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' general, higher-dispersion effects seem to have a positive im- pact on soliton formation and stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' In this work, we analyze a externally driven Kerr resonator [see sketch in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 1(a)] in a operation regime not tackled pre- viously, were both normal SOD and FOD effects contribute to the cavity dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' We show that in this regime, the lock- ing of fronts connecting two uniform bistable states leads to the formation of collapsed snaking related dark LSs17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' This configuration shows morphological similarities with the nor- mal SOD scenario6, despite the different dynamical instabili- ties thresholds and the LSs existence domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Increasing the FOD strength, we find that different types of bright LSs are stabilized and coexist with dark ones [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 1(b)], in a fash- ion similar to the one described in the context of TOD11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' In what follows, we unveil the bifurcation structure organization associated with such coexistence, showing the main dynami- cal regimes of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' II we introduce the model and give some preliminary results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' We present the time-independent problem (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' II A), show the homogenous steady state solution (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' II B), and compute its linear stabil- ity (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' II C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Section III focuses on the bifurcation structure and stability of dark LSs and in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' IV we study the spatio- temporal dynamics of some of the LSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' The effect that the variation of FOD may have on the dynamics and stability of the dark LSs and the stabilization of bright LSs is analyzed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Finally, we discuss the results and draw some conclu- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='11746v1 [nlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='PS] 27 Jan 2023 2 sions in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' THE LUGIATO-LEFEVER MODEL WITH QUARTIC DISPERSION In the mean-field approximation, passive Kerr cavities can be described by the Lugiato-Lefever (LL) equation18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Consid- ering chromatic dispersion up to fourth-order, and neglecting the contribution of TOD, the normalized LL equation reads ∂tA = −(1+i∆)A−id2∂ 2 x A+id4∂ 4 x A+i|A|2A+S, (1) where A is the complex field amplitude, t represents the time coordinate, and x the fast time in fiber cavities or angular vari- able in microresonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' The losses are normalized to 1, ∆ is the phase detuning from the closest cavity resonance and S in the driving field amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' With this normalization, d2 = sign(β2) = ±1 and d4 ≡ β4α/(6L|β2|2), where α repre- sent the losses, L is the cavity length, and β2 and β4 represent the SOD and FOD coefficients, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' In this work we focus on the regime defined by d2 = 1 and d4 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' The time-independent problem: Spatial dynamics LSs and any other type of time-independent states (∂tA = 0) satisfy the complex ordinary differential equation id4∂ 4 x A = id2∂ 2 x A+(1+i∆)A−i|A|2A−S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' (2) Considering A = U + iV, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' (2) can be recast into the 8D spatial dynamical system dY dx = F(Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='d2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='d4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='∆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='S),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' (3) where the new variables read Y = (Y1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='Y2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='Y8) = (U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='∂xU,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='∂xV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='∂ 2 x U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='∂ 2 x V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='∂ 3 x U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='∂ 3 x V),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' and the vector field F is defined as � � � � � � � � � � � Fm = Ym+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' m = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='6 F7 = 1 d4 [∆Y1 −(Y 2 1 +Y 2 2 )Y1 +Y2 +d2Y5] F8 = − 1 d4 [Y1 +∆Y2 −(Y 2 1 +Y 2 2 )Y2 +d2Y6 +S] By using the dynamical system (3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' we can apply well known results of dynamical systems and bifurcation theory to study our problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Furthermore, this approach allows us to apply a correspondence between the time-independent so- lutions of the system [i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=', solutions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' (2)] and different solutions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' In this context, a uniform front corre- sponds to a heteroclinic orbit, a LS is a homoclinic orbit, a spatially periodic pattern a limit cycle, and the homogeneous state of the system a fixed point19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Equation 3 will be used for computing LSs through nu- merical path-continuation algorithms based on a predictor- corrector method20,21 using the free software package AUTO- 07p22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' With this procedure, we are able to compute not only t m b t m b C Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' (a) Homogeneous steady state in the bistable regime for ∆ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' (b) Marginal instability curve corresponding to (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Panel (c) shows the (∆,S)-parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' The vertical line corresponds to the situation shown in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' The horizontal line to the nonlinear cavity resonance shown in (d) for S = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' (e) Shows the formation of a dark LS for ∆ = 5 and S = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' The blue curve in (e) represents the initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' the stable but also the unstable state solutions, unveiling their connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' The stability is determined after continuation by solving the eigenvalue problem L ψ = σψ, (4) where L is the linear operator associated with the right hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' (1) evaluated a given steady state, and ψ is the eigenmode corresponding to the eigenvalue σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' The time- independent state is stable if Re[σ] < 0, and unstable oth- erwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' If by varying a control parameter of the system, let’s say p, this transition occurs at the value p = pc (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=', Re[σ(pc)] = 0), we say that a local bifurcation takes place at pc23,24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 3 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Homogeneous steady state The simplest time-independent solution is the uniform or homogeneous steady state (HSS) solution Ah (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=', ∂xA = 0), which satisfies the algebraic equation (1+i∆)Ah −i|Ah|2Ah −S = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' (5) This equation can be rewritten in the form S2 = I3 h −2∆I2 h +(1+∆2)Ih, (6) with Ih ≡ |Ah|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' For a fixed value of ∆, this expression defines a nonlinear dependence between the intracavity intensity Ih and the pump S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' An example of such dependence is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 2(a) for ∆ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' For this value of ∆, the system shows multistability, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=', for the same value of S three HSSs, namely Ab h, Am h , and At h, coexist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' These states are connected at two folds, or turning points, located at the positions Il,r h ≡ 1 3 � 2∆± � ∆2 −3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' (7) As a function of ∆, these folds points define the two solid lines plotted in the (∆,S)-parameter diagram shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 2(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Decreasing ∆, eventually, these two folds meet and disappear in a cusp bifurcation Ch occurring exactly at ∆ = √ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' The vertical dashed line shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 2(c) corresponds to the dia- gram plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' For a fixed value of S, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' (6) is solved by the expression ∆ = Ih ± � (S/Ih)2 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' (8) An example of these solution branches is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 2(d) for S = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='8, and represents the nonlinear resonance of the cav- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Linear stability analysis of the uniform state The linear stability of HSS against perturbations ∼ eσtψk can be determined analitically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' In this case, eigenmodes read ψk = eikx + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=', while the eigenvalues depend on the wavenumber k through the dispersion relation σ(k) = −1± � −K(k)2 −2(∆−2Ih)K(k)−C, (9) with K(k) = −d2k2 −d4k4, (10a) and C = (∆−Ih)2 −2Ih(∆−Ih) (10b) For non-uniform perturbations (k ̸= 0), the transition sta- ble/unstable takes place at the Turing or modulational in- stability (MI), which satisfies simultaneously the conditions ∂kσ(k) = 0 and σ(k) = 0 for a critical wavenumber k = kc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' From this condition, MI occurs at Ih = 1, and the growing perturbation at this point has the wavenumber kc = ± � � � �−d2 ± � d2 2 −4d4(2−∆) 2d4 , (11) with d2 = 1 for our regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' The condition σ(k) = 0 leads also to the marginal instabil- ity curve Ih = 1 3 �� K2 +2K∆+∆2 −3±2(∆+K) � , (12) which separates stable from unstable regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Figure 2(b) shows the marginal instability curve corre- sponding to the the HSS shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' The minimum of this curve corresponds to the MI, and the area inside it to the unstable HSSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' In correspondence, stable (unstable) HSSs are marked using solid (dashed) lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' At k = 0, two homogeneous instabilities [i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=', saddle-node (SN) bifurca- tions] occur at the fold positions [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' (7)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' In what follows we label these points SNl,r h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' BIFURCATION ANALYSIS FOR DARK LOCALIZED STATES A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Bistability and plane-front locking In the bistability region shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 2, fronts connecting At h and Ab h forwards and backwards can form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' These fronts drift at a constant speed which depends on the parameters of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' The speed increases (decreases) as the sys- tem parameters approach (separate) from the Maxwell point of the system, where it cancels out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Around this zero-speed point, fronts can lock if oscillatory tails exist in the front pro- files, leading to the formation of LSs of different widths5,25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Asymptotically, these tails can be described through the ex- pression A(x) − Ah ∼ eλx, where λ is the spatial eigenvalue of the system evaluated at Ah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' The spatial eigenvalues can be computed through the Jacobian associated with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' (3), or equivalently, by solving the equation σ(−iλ) = 0, namely d2 4λ 8 −2d2d4λ 6 +(4Ihd4 −2∆d4 +d2 2)λ 4 −(4Ihd2 +2∆d2)λ 2 +(∆2 −4Ih∆+3I2 h +1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' In our regime (d2 = 1 and d4 = 1), the previous equation has 8 solutions that read λ = ± 1 √ 2 � −1± � 4∆−8Ih +1±4 � Ih2 −1 (13) The dynamically relevant eigenvalues are those related with the slow dynamics of the system (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=', those with smallest |Re[λ]|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Oscillatory tails appear if the dominant eigenval- ues are complex-conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' When these eigenvalues are all 4 (b) (a) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' (a) Modification of the spatial eigenvalues associated with At h as a function of S for ∆ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' The gray plane corresponds SNl h, and the blue one to SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' In this plane, the most relevant eigenvalues are surrounded by a gray oval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' (b) Same than in (a) but for Ab h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' real numbers, the tails are monotonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' The transition between these two configurations depends on the parameters of the sys- tem (for this case ∆ and S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' The modification of the spatial eigenvalues associated with At h, and Ab h as a function of S are depicted in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 3(a) and (b), respectively, for ∆ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Let’s take a look to the eigenvalues associated with Ab,t h around SM (see blue planes in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' For Ab h, the dominant eigenvalues around SM [see oval gray shape in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 3(b)] have a large imaginary part and a very small real part, what means that the oscillatory tails will have small wavelength and weak decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' This leads to well defined oscillatory tails around Ab h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' For At h [see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 3(a)], although the dominant eigenvalues are also complex conjugate, the imaginary and real parts are re- spectively smaller and larger than in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 3(b), which leads to oscillatory tails with a very strong decay and very large wave- length: effectively a quasi monotonic tail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Therefore, front H1 H1 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Collapsed homoclinic snaking for ∆ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Solid thick (thin) lines correspond to stable (unstable) states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Some examples of dark LSs are shown on the right panels (i)-(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Bi label the solution branches in-between SNl,r i , and SM is the uniform Maxwell point of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' locking will be favored around Ab h, in contrast with At h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' An example of front locking is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 2(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' This temporal evolution has been computed through a direct nu- merical simulation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' (1) starting from a super-Gaussian profile subtracted to At h close to SM [shown in blue in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 2(e)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Initially, two fronts with opposite polarity form and approach one-another as time passes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Eventually, they lock at a fixed separation D, yielding a dark LS with two central dips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' The time evolution of that separation can be described by the equation19 ∂tD = ρeRe[λ]Dcos(Im[λ]D)+η, (14) where ρ depends on the parameters and η ∼ S−SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Although the front interaction appearing here is generic, and well de- scribed by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' (14), its derivation from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' (1) is not realiz- able, and we introduce it here for explaning the front locking mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' The fixed points De of this system (∂tDe = 0) correspond to the locking of fronts, and hence, to the forma- tion of LSs of width De.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Thus, for the same value of η, LSs of different widths may coexist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' As we will see in the com- ing section, LSs formed through front locking organize in a particular bifurcation structure which depends directly on the interaction law (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Bifurcation structure: Collapsed homoclinic snaking To fully understand the formation of these states we per- form a bifurcation analysis based on the path-continuation Im 1 2 1 1 14Q-1- S1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='5Mi 3 3MI / .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='5 4 SIm2℃4-V 5Q 1-2 1 1 2 1 0 2 Re(入)SN 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='4 2S M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='8 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='6 S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='4I0 SN: 2 Re(入)5 (a) (b) (c) (d) (b) (c) (d) MI h h H2 , H1 , C Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' (a) (∆,S)-parameter space for β4 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' This diagram shows the main bifurcations of the system: the MI, saddle-node bifurcations of the homogeneous state SNl,r h , and saddle-node bifurcations associated with the collapsed homoclinic snaking SNl,r i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Panels (b)-(d) show the collapsed snaking for ∆ = 3,5 and 8 respectively, corresponding to the vertical dashed lines plotted in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Solid thick and thin lines represent stable and unstable LS solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' The labels Hl,r i mark the position of the Hopf bifurcations leading to breathing behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' techniques described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' II A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' The output of these com- putations leads the bifurcation diagram shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 4 where the energy of A, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=', the L2-norm ||A||2 ≡ L−1 � L/2 −L/2 |A(x)|2dx is plotted as a function of S for ∆ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' This diagram is known as collapsed homoclinic snaking and is generic of systems where LSs emerge through uniform-front locking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Indeed, the damped oscillatory shape of the bifurcation curve around SM is a direct consequence of the front interaction and locking de- scribed by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' (14) (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='19 for a general description).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' The collapsed snaking curve emanates from SNl h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Close to this bi- furcation, LSs have small amplitude and are unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' These small amplitude states can be computed through multi-scale perturbation theory as done in the standard LL equation6,19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Following the diagram downwards, the dark LSs undergo a sequence of saddle-node bifurcations appearing in pairs SNr,l i , where the sub-index i represents the number of dips appearing in each state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' In these bifurcations, LSs gain and loss stabil- ity, and at each SNl i an extra dip is nucleated in the struc- ture at x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' In this way, the width of the LSs increases while decreasing ||A||2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' In what follows, we mark the solu- tion branches connecting SNl,r i as Bi (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Persistence in the (∆,S)-parameter space Figure 5 shows the modification of the SNl,r i bifurcations (see blue lines) in the (∆,S)-parameter space, together to SNl,r h and MI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' With decreasing ∆, the pairs SNl,r i approach each other and eventually they meet in a sequence of cusp bifur- cations Ci (here we only show C1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Those with larger i, cor- responding to SNs down in the collapsed snaking shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 4, are the one disappearing first, when decreasing ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' This phase diagram is common in systems sharing similar HSS linear stability, such as in passive Kerr cavities with second- order chromatic dispersion6 and in dispersive cavity enhanced second-harmonic generation26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' The modification of the collapsed snaking diagram along the (∆,S)-parameter space is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 5(b)-(d) for ∆ = 3, 5 and 8, respectively [see vertical dashed lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 5(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' For ∆ = 3, all the Bi branches are stable, and most of them correspond to narrow states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Increasing ∆, cusp Ci with larger i appear, and wider states emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Moreover, Bi become wider due to the separation of their corresponding SNl,r i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' The linear stability analysis along these diagrams shows that, eventually, B1 undergoes a pair of Hopf bifurcaitons Hl,r 1 , where the single-dip LS becomes unstable in favor of localized oscillations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=', breathers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' This is the situation de- picted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 5(c) for ∆ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' We will explore the spatiotem- poral dynamics of these states in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Further increasing ∆, B2 also destabilizes through the Hopf bifurcations Hl,r 2 [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 5(d) for ∆ = 8], leading to the ap- pearance of new breather states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' A formal comparison with the standard LL equation in the normal chromatic dispersion regime (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='6) shows that the destabilization of the Bi branches occurs for larger values of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' SPATIO-TEMPORAL DYNAMICS Figure 6 shows a portion of the bifurcaiton diagram de- picted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 5(d) for ∆ = 8, where the maxima and minima of the oscillatory variation of the breathers’ norm emerging from Hl,r 1 are plotted using orange circles, whereas those ema- nating from Hl,r 2 are shown using red circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' In what follows, we refer to this set of breather branches as Feigenbaum-like diagrams (F)27: F1 (orange dots) is connected to B1, while F2 forms along B2 (red dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' These diagrams, and therefore the breathers, bifurcate supercritically from every Hl,r i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Let us first focus on F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' An example of the breather belong- ing to this set is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 6(i) for S = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' This state emerges with a small amplitude form Hl 1 and with a single os- 6 b (ii) (i) Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Top part of the collapsed snaking diagram for ∆ = 8 where we show all the local maxima and minima of the breathers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' We use orange dots for the breathers emerging from B1 at Hl,r 1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=', F1), and red dots for those bifurcating from Hl,r 2 along B2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=', F2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' PD corresponds to a period-doubling bifurcation, and b marks the emergence of beating behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Panels (i) and (ii) show two examples of breathers corresponding to F1 and F2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' cillatory period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' The breather undergoes a sequence of period- doubling (PD) bifurcations yielding more complex dynamics, that we do not show here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' The PD cascade is observable from the F1 diagram, although we only mark the first one at each side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Near ∆ ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='6, the system evolves to temporal chaos, and this PD sequence is inverted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Eventually the chaotic state dies, possibly in a boundary crisis of the attractor (BC)27, and the system ends up in the closest basin of attraction: the HSS At h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' This situation share similarities with the one reported in Kerr cavities when only second-order dispersion is considered6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' The F2 diagram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 6 shows more complexity than F1, and furthermore, does not encounter any crisis or instability destroying the oscillatory states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' An example of a single- period breather bifurcating from Hl 2 is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 6(ii) for S = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Here, F2 shows the transition between differ- ent dynamical regimes including beating phenomena (b) be- tween different frequencies and PD cascades27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Figure 7(a) shows a close-up view of F2 where these transitions are illus- trated in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' The vertical dashed lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 7(a) correspond to the time series and frequency spectra shown be- low [see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 7(i)-(viii)], which represent the evolution of the breather intensity norm ||A||2 and its Fourier transform ||A||2 f , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' From the modification of the spectra, one can clearly understand the transition between the different tempo- ral dynamical regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Such modification is depicted in the color-map shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 7(b) all along F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' At (i) the breather is characterized by a single frequency and their harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' At b, a new frequency arises [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 7(b)], and the breather undergoes a beating phenomenon between the former and latter frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 7 (ii), the new fre- quency is marked using a red arrow and its difference with the former one with δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Because of this beating, the tempo- ral series (see top panel) shows modulation in the amplitude, characterized by the beat period 2π/δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' At (iii), the δ is larger, resulting in a smaller modulation on the temporal series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' In- creasing S further, the beating becomes more chaotic leading to the dynamical behavior shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 7 (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Eventually this states dies out, leading to a single frequency breather as- sociated with the time series shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 7 (iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Increasing more S, a sequence of PD bifurcation occur leading to the breather dynamics depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 7 (v)-(vii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' IMPLICATIONS OF FOURTH-ORDER DISPERSION ON THE BIFURCATION STRUCTURE AND STABILITY Previously, we have performed bifurcation analysis for a fixed value of d4, namely d4 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Here we explore the effects of modifying d4 on the stability of LSs, and on the their bifur- cation structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' We will see that large values of d4 may also lead to the emergence of bright LSs, similarly to the scenario described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='11 for TOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Figure 8 shows the modification of the collapse snaking di- agram for ∆ = 5 with increasing d4: d4 = 0 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 8(a), d4 = 2 1060 t 55190 t 18510 I/A//2 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' S = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='5204V 10 1IA2 58;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' S = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='819750 60 70 80 90100 110 120 130 140180 60 70 80 90100 110 120 130 147 b Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' (a) Close-up view of the Fiegenbaum-like diagram F2 between Hl,r 2 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' (b) Modification of the frequency spectrum of the oscillatory states along the diagram shown in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' To plot this spectrum we have filtered out the frequencies below the red dashed line depicted in panels (i)-(vi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' These panels show the time series and frequency spectra of the different oscillatory states [see vertical dashed lines in (a) and (b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 8(b), and d4 = 10 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 8(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Regarding the top pan- els, we can see how by increasing d4, the branches Bi be- come wider, leading to a wider existence region for the differ- ent dark LSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Furthermore, d4 has an stabilizing effect on the breather states emerging from B1 and B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' This phenomenon can be easily observed comparing Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 8(a) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 8(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' For d4 = 2, B2 has stabilized completely while B1 partially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Both stabilizations occur due to the movement of the Hopf bifurca- tions along the branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Performing a two parameter continuation of the different saddle-node bifurcations SNl,r i in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 8(a)-(c)[top], we are able to compute their modification in parameters S and d4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' The results are shown in the (d4,S)-phase diagram shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 8(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' The vertical dashed lines correspond to the dia- grams plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 8(a)-(c)[top].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Decreasing, d4, the pair SNl,r i (for i fixed) comes closer, reducing the extension of Bi until reaching d4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' For d4 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='5 the shrinking process in- tensifies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' In contrast, the pairs SNl,r i separate softly with in- 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='7 3 (i) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='484 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='461 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='438 120 140 160 180 七 2 3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='9 4 (ii) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='594 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='539 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='484 200 120 140 160 180 2 2 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='2 (iii) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='629 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='539 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='449 00 120 140 160 180 2 七 2 3 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='4 4 (iv) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='66 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='543 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='426 00 120 140 160 180 2( 七 5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='5 00H?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='7 3 (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='6 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='9 4- S 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='.5(i) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='8 PD· (a) 1 1 At 1 1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='7 2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='6 1 1 1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='5 (iiiii)(iv) B1(vi)(vii)(viiiH, 2 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='0 3 (vi) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='715 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='659 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='603 200 120 140 160 180 2 七 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='2 3 4 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='5 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='0 3 (vii) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='73 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='673 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='616 00 120 140 160 180 2 七 2 3 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='5 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='0 3 viii 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='744 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='691 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='638 00 120 140 160 180 2 七 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='5 3000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='0 3 V 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='698 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='639 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='58 120 140 160 180 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='5 38 d (i) (ii) (iii) (iv) (iv) (i) (ii) (iii) H d d d d Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Modification of the collapsed snaking for ∆ = 5 when varying d4: in (a) d4 = 0, in (b) d4 = 2, and d4 = 10 in (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Examples of bright LSs of different widths, corresponding to the diagram shown in (c), are depicted in panels (i)-(iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Panel (d) shows the (∆,d4)-parameter phase diagram for dark LSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Panel (e) shows the phase diagram associated with bright LSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' The vertical dashed lines correspond to the diagrams shown in panels (a)-(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' creasing d4, and the separation seems to saturate for large val- ues of d4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' The different blue tonalities between the SNl,r i lines correspond to regions where dark LSs with different dips ex- ist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' A similar tendency has also been observed for bright LSs in the presence of TOD13, where in contrast, the saturation occurs when SNl,r i approach each other with increasing d3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Figures 8(a)-(c) also show the modification, with increasing d4, of the bottom part of the collapsed snaking around SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' For d4 = 0, this curve follows a straight line on top of SM, which for ||A||2 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='9 turns to the right and eventually connects with MI at Ab h 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Increasing d4 [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 8(b) for d4 = 2], the dia- gram reaches lower values of ||A||2, and undergoes a pair of saddle-node bifurcations snl,r 1 before reaching MI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Between these folds, a new branch of stable birght LS solutions (B∗ 1) appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Increasing β4 further [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 8(c) for d4 = 10], the extension of B∗ 1 increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Two example of bright LSs on this branch are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 8(i),(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Following up this diagram, new stable branches appear in-between snl,r 2 and snl,r 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' We la- bel these branches B∗ 2 and B∗ 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' With increasing ||A||2, the bright LSs broaden as depicted in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 8(iii),(iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' The region of existence of these bright states is illustrated in the (S,d4)-phase diagram shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 8(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' The saddle- node bifurcations snl,r i converge rapidly to SM for i > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' These states emerge due to the modification of the spatial eigenval- ues, which now, allow the front locking around Ab h and At h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Finally, to complete our study, we analyse the effect of d4 on the spatiotemporal dynamics of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' To do so we fix (∆,S) = (8,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='97), and scan the variation of the extrema of the dark LSs norm with d4, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' For d4 = 0, dark LSs are static, and the diagram just shows a sin- gle branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Increasing d4 a bit further, the LSs encounter a Hopf bifurcation H, where they destabilize in favor of a single- period oscillation, which in a very narrow d4-interval leads to more complex temporal dynamics characterized by the time series shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 9(i),(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Eventually, around d4 ≈ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='3, the complex dynamics disappear, yielding a single period breather again [see temporal series in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 9(iii)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' DISCUSSIONS AND CONCLUSIONS In this paper we have studied the formation, bifurcation structure and dynamical instabilities of dissipative Kerr LSs 9 Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Modification of the temporal dynamics associated with a single peak dark LSs when varying d4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' The plot shows the variation of the local maxima and minima of ||A||2 while passing from different dynamical regimes (red dots) and the unstable branch associated with the LS (dashed blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' The time series shown in panels (i)-(iii) correspond to the sections depicted through vertical dashed lines in the panel above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Here, (∆,S) = (8,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='97).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' in the presence of FOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Here we focus on the case where d2 = 1 and d4 > 0, a configuration which has not been studied in previous works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' The linear stability analysis of the HSSs in this configuration shows the presence of bistability between two coexisting HSS states, namely At h and Ab h (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' We show that uniform wave fronts connecting the previous HSS states can lock, yielding the formation of LSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' For that, a necessary condition is the presence of oscillatory tails around either At h or Ab h, which is determined by the spatial eigenvalues of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' For d4 = 1, our findings show that this locking is possible only around Ab h, leading to the formation of dark LSs (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' III A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' In bifurcation terms, these states undergo collapse homo- clinic snaking whose extension and stability change with ∆ (see Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' III B, III C and IV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Collapsed homoclinic snaking is generic for systems exhibiting uniform-front lock- ing, and has been reported in a variety of pattern forming sys- tems including nonlinear quadratic optical resonators8–10,26, mode-locked vertical external-cavity surface-emitting lasers (VCSEL)28,29, semiconductor micro-resonators with strong time-delayed feedback30, and reaction-diffusion systems31–35, to only cite a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' For d4 = 1, the scenario presented here is morphologically very similar to the one found when studying Kerr cavities with only normal SOD (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=', d2 = 1)6,36, despite of the extension of the LS existence regions and the onset of temporal insta- bilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' This result shows that dark states are robust against high-order dispersive effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' This situation changes when increasing d4 (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Re- garding dark LSs, their region of existence initially increases, although their extension soon saturates for values of d4 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' The most interesting phenomenon is that increasing d4, bright LS solutions emerge, due to the stabilizing effect of this high- order dispersion term, and their existence region broadens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 8 in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' We find that d4 is also able to tune the emergence and type of oscillatory states appearing in the sys- tem, becoming a very relevant parameter to control the tem- poral dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' The capacity of d4 for stabilizing LSs was discussed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='12 for a different regime of operation, where dark local- ized patterns, and their associated homoclinic snaking, were stabilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' The interaction of LSs, and formation of soliton molecules, in the presence of FOD effects have been also ana- lyzed in Kerr cavities37,38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' In this context, FOD increases con- siderably the number of allowed locking distances between LSs, and therefore the variety of molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' The stabilization of bright Kerr LSs in the context of uniform bistable regimes with collapsed snaking has been also investigated in the pres- ence of TOD11 and stimulated Raman scattering39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' In summary, dark LSs formation is robust in the presence of FOD, which furthermore is able to stabilize, bright states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' The advances in dispersion engineering make possible the man- agement of different dispersion terms and therefore the access to dispersion regimes previously unattainable3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' This capac- ity makes our work relevant not only from a theoretical point of view, but also from a experimental perspective, opening the possibility to observe these type of states and dynamics in real Kerr cavities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' ACKNOWLEDGEMENTS PPR acknowledges support from the European Union’s Horizon 2020 research and innovation programme under the 10 Marie Sklodowska-Curie grant agreement no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 101023717.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' FL acknowledges support from the European Research Coun- cil (grant agreement 757800), HORIZON EUROPE Euro- pean Research Council (57800), and Fonds de la Recherche Scientifique-FNRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' REFERENCES 1N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Akhmediev and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Ankiewicz, eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=', Dissipative Solitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Lecture Notes in Physics, Berlin Heidelberg: Springer-Verlag, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 2T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Kippenberg, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Gaeta, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Lipson, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Gorodetsky, “Dis- sipative Kerr solitons in optical microresonators,” Science, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 361, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' eaan8083, Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Publisher: American Association for the Ad- vancement of Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 3A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Runge, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Hudson, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Tam, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' de Sterke, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Blanco-Redondo, “The pure-quartic soliton laser,” Nature Photonics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 14, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 492–497, Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Number: 8 Publisher: Nature Publishing Group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 4O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Thual and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Fauve, “Localized structures generated by subcritical in- stabilities,” Journal de Physique, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 49, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 11, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 1829–1833, 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 5P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Coullet, “Localized patterns and fronts in nonequilibrium systems,” In- ternational Journal of Bifurcation and Chaos, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 12, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 2445–2457, Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 6P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Parra-Rivas, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Knobloch, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Gomila, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Gelens, “Dark solitons in the Lugiato-Lefever equation with normal dispersion,” Physical Review A, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 93, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 063839, June 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 7P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Parra-Rivas, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Gelens, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Hansson, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Wabnitz, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Leo, “Frequency comb generation through the locking of domain walls in doubly resonant dispersive optical parametric oscillators,” Optics Letters, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 44, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 2004– 2007, Apr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 8P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Parra-Rivas, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Gelens, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Leo, “Localized structures in dispersive and doubly resonant optical parametric oscillators,” Physical Review E, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 100, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 032219, Sept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 9C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Arabí, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Parra-Rivas, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Parra-Rivas, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Hansson, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Gelens, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Wab- nitz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Wabnitz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Wabnitz, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Leo, “Localized structures formed through domain wall locking in cavity-enhanced second-harmonic gener- ation,” Optics Letters, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 45, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 5856–5859, Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 10P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Parra-Rivas, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Mas Arabí, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Leo, “Dissipative localized states and breathers in phase-mismatched singly resonant optical parametric oscilla- tors: Bifurcation structure and stability,” Physical Review Research, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 4, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 013044, Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Publisher: American Physical Society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 11P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Parra-Rivas, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Gomila, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Gelens, “Coexistence of stable dark- and bright-soliton Kerr combs in normal-dispersion resonators,” Physical Re- view A, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 95, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 053863, May 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 12M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Tlidi and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Gelens, “High-order dispersion stabilizes dark dissipative solitons in all-fiber cavities,” Optics Letters, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 35, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 306–308, Feb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 13P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Parra-Rivas, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Gomila, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Leo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Coen, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Gelens, “Third-order chromatic dispersion stabilizes Kerr frequency combs,” Optics Letters, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 39, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 2971–2974, May 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 14C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Bao, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Taheri, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Zhang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Matsko, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Yan, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Liao, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Maleki, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Willner, “High-order dispersion in Kerr comb oscillators,” JOSA B, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 34, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 715–725, Apr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Publisher: Optica Publishing Group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 15H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Taheri and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Matsko, “Quartic dissipative solitons in optical Kerr cavities,” Optics Letters, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 44, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 3086–3089, June 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Publisher: Optica Publishing Group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 16P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Parra-Rivas, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Hetzel, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Kartashov, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Córdoba, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Conejero, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Aceves, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Milián, “Quartic Kerr cavity combs: bright and dark solitons,” Optics Letters, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 47, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 2438–2441, May 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Publisher: Optica Publishing Group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 17J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Knobloch and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Wagenknecht, “Homoclinic snaking near a heteroclinic cycle in reversible systems,” Physica D: Nonlinear Phenomena, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 206, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 82–93, June 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 18M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Haelterman, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Trillo, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Wabnitz, “Dissipative modulation instabil- ity in a nonlinear dispersive ring cavity,” Optics Communications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 91, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 401–407, Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 19P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Parra-Rivas, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Knobloch, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Gelens, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Gomila, “Origin, bifurcation structure and stability of localized states in Kerr dispersive optical cavities,” IMA Journal of Applied Mathematics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 86, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 856–895, Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 20E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Doedel, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Keller, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Kernevez, “Numerical analysis and con- trol of bifurcation problems (ii): bifurcation in infinite dimensions,” In- ternational Journal of Bifurcation and Chaos, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 01, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 745–772, Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 21E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Doedel, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Keller, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Kernevez, “Numerical analysis and control of bifurcation problems (i): bifurcation in finite dimensions,” International Journal of Bifurcation and Chaos, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 01, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 493–520, Sept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 22E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Doedel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Champneys, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Fairgrieve, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Kuznetsov, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Sand- stede, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Wang, “AUTO-07p: Software for continuation and bifurca- tion problems in ordinary differential equations,” Department of Computer Science, Concordia University, Montreal, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 23S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Texts in Applied Mathematics, New York: Springer-Verlag, 2 ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=', 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 24A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Scroggie, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Firth, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' McDonald, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Tlidi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Lefever, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Lugiato, “Pattern formation in a passive Kerr cavity,” Chaos, Solitons & Fractals, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 1323–1354, Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 25P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Coullet, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Elphick, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Repaux, “Nature of spatial chaos,” Physical Review Letters, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 58, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 431–434, Feb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 26P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Parra-Rivas, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Arabí, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Leo, “Dark quadratic localized states and collapsed snaking in doubly resonant dispersive cavity-enhanced second- harmonic generation,” Physical Review A, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 104, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 063502, Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Publisher: American Physical Society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 27E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Ott, Chaos in Dynamical Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Cambridge: Cambridge University Press, 2 ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=', 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 28C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Schelte, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Pimenov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Vladimirov, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Javaloyes, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Gure- vich, “Tunable Kerr frequency combs and temporal localized states in time- delayed Gires-Tournois interferometers,” Optics Letters, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 44, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 4925– 4928, Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Publisher: Optica Publishing Group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 29T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Seidel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Javaloyes, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Gurevich, “A normal form for frequency comb and localized states in time-delayed Kerr-Gires-Tournois interferom- eters,” Mar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' arXiv: 2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='02411.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 30E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Koch, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Seidel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Javaloyes, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Gurevich, “Tempo- ral localized states and square-waves in semiconductor micro-resonators with strong time-delayed feedback,” Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='09400 [nlin, physics:physics].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 31Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Zelnik, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Gandhi, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Knobloch, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Meron, “Implications of trista- bility in pattern-forming ecosystems,” Chaos: An Interdisciplinary Journal of Nonlinear Science, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 28, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 033609, Mar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 32P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Parra-Rivas and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Fernandez-Oto, “Formation of localized states in dry- land vegetation: Bifurcation structure and stability,” Physical Review E, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 101, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 052214, May 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 33F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Al Saadi and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Champneys, “Unified framework for localized pat- terns in reaction–diffusion systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' the Gray–Scott and Gierer–Meinhardt cases,” Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 379, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 20200277, Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Pub- lisher: Royal Society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 34F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Al Saadi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Worthy, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Alrihieli, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Nelson, “Localised spatial structures in the thomas model,” Mathematics and Computers in Simula- tion, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 194, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 141–158, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 35F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Saadi and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Parra-Rivas, “Transitions between dissipative localized structures in the simplified Gilad-Meron model for dryland plant ecology,” Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content='12263 [nlin].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 36P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Parra-Rivas, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Gomila, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Knobloch, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Coen, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Gelens, “Origin and stability of dark pulse Kerr combs in normal dispersion resonators,” Optics Letters, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 41, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 2402–2405, June 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 37P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Parra-Rivas, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Gomila, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Colet, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Gelens, “Interaction of solitons and the formation of bound states in the generalized Lugiato-Lefever equa- tion,” The European Physical Journal D, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 71, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 198, July 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 38A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Vladimirov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Tlidi, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Taki, “Dissipative soliton interaction in Kerr resonators with high-order dispersion,” Physical Review A, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 103, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 063505, June 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Publisher: American Physical Society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 39P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Parra-Rivas, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Coulibaly, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Clerc, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' Tlidi, “Influence of stim- ulated Raman scattering on Kerr domain walls and localized structures,” Physical Review A, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 103, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 013507, Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FKT4oBgHgl3EQfLS3X/content/2301.11746v1.pdf'} diff --git a/ztFQT4oBgHgl3EQfCjXs/vector_store/index.pkl b/ztFQT4oBgHgl3EQfCjXs/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..a86e03339bee73410e55ea40fa12c0ae161f2c09 --- /dev/null +++ b/ztFQT4oBgHgl3EQfCjXs/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:fcfe080aeea7e345fc812293786fe45793e7f94d16baed6e80070597fbce9edc +size 211995